Computational Algebraic and Analytic Geometry

572

AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA

Mika Seppälä Emil Volcheck Editors

American Mathematical Society

Computational Algebraic and Analytic Geometry

AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA

Mika Seppälä Emil Volcheck Editors

572

Computational Algebraic and Analytic Geometry

AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA

Mika Seppälä Emil Volcheck Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classiﬁcation. Primary 14HXX, 30FXX, and 68WXX.

Library of Congress Cataloging-in-Publication Data Computational algebraic and analytic geometry : AMS special sessions on computational algebraic and analytic geometry for low-dimensional varieties, January 8, 2007, New Orleans, LA, January 6, 2009, Washington, DC, [and] January 6, 2011, New Orleans, LA / Mika Seppälä, Emil Volcheck, editors. p. cm. — (Contemporary mathematics ; v. 572) Includes bibliographical references. ISBN 978-0-8218-6869-0 (alk. paper) 1. Curves, Algebraic–Data processing–Congresses. 2. Riemann surfaces–Congresses. I. Seppälä, Mika. II. Volcheck, Emil, 1966–

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Contents

Preface vii Large Hyperbolic Polygons and Hyperelliptic Riemann Surfaces Anthony Arnold and Klaus-Dieter Semmler 1 On Isolated Strata of Pentagonal Riemann Surfaces in the Branch Locus of Moduli Spaces Gabriel Bartolini, Antonio F. Costa, and Milagros Izquierdo 19 Finite Group Actions of Large Order on Compact Bordered Surfaces E. Bujalance, F. J. Cirre, and M. D. E. Conder 25 Surfaces of Low Degree Containing a Canonical Curve Izzet Coskun 57 Ideals of Curves Given by Points E. Fortuna, P. Gianni, and B. Trager 71 Non-genera of Curves with Automorphisms in Characteristic p Darren Glass 89 Numerical Schottky Uniformizations of Certain Cyclic L-gonal Curves Ruben´ A. Hidalgo and Mika Seppal¨ a¨ 97 Generalized Lantern Relations and Planar Line Arrangements Eriko Hironaka 113 Effective p-adic Cohomology for Cyclic Cubic Threefolds Kiran S. Kedlaya 127 Generating Sets of Affine Groups of Low Genus K. Magaard, S. Shpectorov, and G. Wang 173 Classification of Algebraic ODEs with Respect to Rational Solvability L. X. Chauˆ Ngo,ˆ J. Rafael Sendra, and Franz Winkler 193 Circle Packings on Conformal and Affine Tori Christopher T. Sass, Kenneth Stephenson, and G. Brock Williams 211 Effective Radical Parametrization of Trigonal Curves Josef Schicho and David Sevilla 221

Preface

Distinct communities of mathematicians have grown around analytical and algebraic approaches to geometry. Even though both approaches are deeply connected through results such as Chow’s Theorem and GAGA, mathematicians in- frequently collaborate across these communities. Computational methods make these connections explicit and increase our understanding of geometry in ways not possible when each approach is pursued as its own form of pure mathematics. In this way, computational methods help bring together these different communities of mathematicians. Uniformization of Riemann surfaces is a prime example of a topic where computational methods are bringing important new insights. In the late 19th and early 20th centuries, mathematicians such as Burnside, Koebe, Myrberg, Rankin, and Whittaker labored to make uniformization explicit, developing numerical techniques even when no computers were available. Their work is considered to be a crowning achievement of geometry in that era. The Abel-Jacobi and Torelli Theo- rems represent another prominent example of a theory that relates analytical and algebraic representations of geometric objects, in this case, relating complex lattices to curves and their Jacobians. During the last twenty years, practical numerical and symbolic computations have become commonplace and possible for anybody. This has given new life to some of the old ideas, and has led to new approaches to some of the classical problems. Here are some examples of such work. Uniformization has advanced in both theory and practice through the development of effective computational methods for special cases of the problem. Ideally one would like to find explicit symbolic methods to pass, for example, from an algebraic plane curve, given by a polynomial, to a uniformization of the curve in question. In the case of genus one, the symbolic approach is part of the classical analysis of elliptic curves. For curves of higher genus, symbolic methods have suc- ceeded in special cases only. Numeric methods have yielded more general results, but a solution to the general case still looms far in the future. These numeric methods lead one to study, for example, algebraic curves given by approximations of the actual polynomials defining the curve, which is a major topic in numerical algebraic geometry. An example of a completely new theory whose development was supported by computational methods is a discrete version of the Riemann mapping theorem offered by circle packings. Research on explicit methods connecting Riemann surfaces and their corresponding Fuchsian groups has also benefited from computational methods. This series is also inspired by work of Curtis McMullen, who, in

vii

viii PREFACE his AMS Colloquium Lectures in 2000, connected dynamics on a Riemann surface to rational points on the corresponding algebraic curve. This volume is a collection of research papers on computational methods in algebraic and analytic geometry. It has its roots in the series of AMS Special Sessions on Computational Algebraic and Analytic Geometry that have taken place at the Joint AMS-MAA National Meetings every odd year since 1999, and in the large European research projects that coordinated the work in this area during 1991–1996. Usually AMS Special Sessions and other similar meetings are characterized by the methods used in the papers presented. This volume and the preceding AMS Special Sessions form an exception to this rule: papers published here and those presented earlier in the Special Sessions entitled Computational Algebraic and Analytic Geometry on Low-Dimensional Varieties have, as their unifying factor, the same object of study. Compact Riemann surfaces are algebraic curves. They are also characterized by their Jacobian variety and their Fuchsian group. Hence the same object can be studied by a variety of methods: analytic, algebraic, and geometric. It is this extraordinary variety of methods that makes this area challenging, interesting, and very fertile. The editors are grateful for the contributions of the authors, and the referees who helped to create this volume. The editors thank all the sponsoring institutions that helped to advance research in this ﬁeld. Most importantly, the editors thank the American Mathematical Society and its expert publishing oﬃcers.

Mika Sepp¨al¨a and Emil Volcheck

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11364

Large hyperbolic polygons and hyperelliptic Riemann surfaces

Anthony Arnold and Klaus–Dieter Semmler

Abstract. The following pages will present large hyperbolic polygons as a useful tool to study hyperelliptic Riemann surfaces. A large hyperbolic polygon is a set of vertices in the hyperbolic plane, where one can draw lines, one through each vertex, such that for any given such line all others lie entirely in the same half-space with respect to the given one. Identifying a vertex with the half-turn around it, the vertices of a polygon generate a group of Möbius transformations Γ. If the polygon is large, the corresponding group, and in particular its subgroup Γ2 of words of even length in the generators, act discontinuously on the hyperbolic plane and its quotient is a Riemann surface with an involution. By a closing mechanism that consists of adding two additional points to the set of vertices of the polygon (and thus adding generators to the Fuchsian group), we will obtain a closed hyperelliptic Riemann surface. Now all hyperelliptic Riemann surfaces of genus greater than one carry large hyperbolic polygons, and we get the full Teichmüller space of such surfaces this way. The picture of large hyperbolic polygons will give a useful parametrization of Teichmüller space for hyperelliptic Riemann surfaces of any genus greater than one. We give algorithms to produce random large hyperbolic polygons such that any open set of Teichmüller space will come up with non-zero probability. We also give algorithms that output an explicit fundamental domain for the surface and explicit generators (matrices) where the input is a large hyperbolic polygon. Furthermore, we propose reduction algorithms implementing modular transformations on a large hyperbolic polygon. The goal of these reduction algorithms is to obtain a standard presentation in form of a fundamental domain in Teichmüller space for the Teichmüller modular group.

1. Hyperbolic Polygons We assume the reader is familiar with the upper half-plane model of hyperbolic geometry as it can be found in many books (e.g., Beardon[1], Buser[2], etc.), to the extent that points in this theory are the elements of the set

U = {r + si ∈ C | r, s ∈ R,s>0}, lines are half circles with centers on the real axes or vertical rays from real points, carrying the shortest curves between any pair of points thereon, and SL(2, R)acts

Key words and phrases. Hyperbolic geometry, Riemann surfaces. The ﬁrst author has been supported by FNS-Grant No. 511760 (Fonds national suisse).

c 2012 American Mathematical Society 1

2 A. ARNOLD AND K.-D. SEMMLER by orientation preserving isometries via the Möbius formula ab a(r + si)+b m = : r + si → m(r + si)= cd c(r + si)+d The elements of SL(2, R) fall into the three classes, elliptic, parabolic, and loxodromic, characterized by their trace, governing their geometric behavior, fixpoints etc. We use the term loxodromic here, although the reader might note that in the case of SL(2, R) the term hyperbolic is often used for loxodromics. Many authors work in PSL(2, R) instead of SL(2, R), but keeping track of signs will prove essential to our approach. Furthermore much geometric information is hidden in the sign of traces of products of matrices and the restriction to the use of absolute values of traces is the main obstacle to proper geometric interpretation of formulas. This will become clear in the sequel. A (hyperbolic) polygon is a finite ordered set of points in the hyperbolic plane. We do not intend to speak of a polygon as a domain but rather as a polyline, so, for instance, it may be self-crossing. We will identify each point with the half-turn around it. More explicitly we identify the point r+si in the upper half-plane model with the (point-) matrix 1 −rr2 + s2 r + si p = ∈ SL(2, R), s −1 r whichthenactsasaMöbius transformation having the given point as only fixpoint and as such being of order two. A matrix obtained by this identification will be called point matrix. Yetthesquareofsuchamatrixis− id and p−1 = −p, its matrix inverse, is not a point matrix. Fenchel uses point-matrices with the opposite sign in [4], but this choice of sign is of minor importance. Then a m ∈ SL(2, R) acts by conjugation m(r + si)=m · p · m−1 Lemma 1.1. The point matrices are exactly the elements of SL(2, R) with zero trace and a negative (2,1)-entry. They are mapped to each other by conjugation with elements in SL(2, R). Proof. Such a matrix m ∈ SL(2, R) with zero trace is of the form −ac −ba − 2 1+a2 1 with non-negative a, b,andbc a = 1. Hence b>0andc = b . Now put s = b a and r = b and we get the desired form. For the second part we only have to check if the (2,1)-entry remains negative by conjugation, an easy calculation. Now a polygon can be regarded as a set of generators of a subgroup of SL(2, R), and a basic question would be to decide when such a group is Fuchsian and what is the corresponding Riemann surface. More precisely:

Definition 1.2. Given a polygon P = p1p2 ...pn we deﬁne 2 2 Γ:=Γ(P ):=pi | pi ∈ P , and Γ := Γ (P ):=pipj | pi,pj ∈ P . Now we want to discuss when Γ2 is purely hyperbolic. This leads to the deﬁnition of large polygons.

LARGE HYPERBOLIC POLYGONS AND HYPERELLIPTIC RIEMANN SURFACES 3

Definition 1.3. A polygon P = p1p2 ...pn is large iﬀ there exist (hyperbolic) lines l1,l2,...,ln such that li passes through pi and for any li all other lj ; j = i lie in the same half-plane w.r.t. the given line. We add the following technical condition: the intersection of the “non-empty” half-planes has inﬁnite volume, or equivalently, at least one line can be slightly moved, still satisfying the condition that all the other lines lie in the same half-plane as above. This is to avoid discussions of limiting cases.

Figure 1. A large quadrilateral

Given a large polygon P = p1p2 ...pn we may re-order the points such that the endpoints of the corresponding lines at inﬁnity are cyclicly ordered consecutively (say in the unit circle model) on the boundary circle of the hyperbolic plane. Such a polygon will be called an ordered large polygon. The name comes from the observation, that, when you take two points quite close, then their lines are quite close to each other as well. Then you have to go far out to squeeze another circle in between. This is a nice visualization for the collar lemma. On the other hand − tr(p1p2p3) equals the hyperbolic sine of the base times the hyperbolic sine of the height of the triangle p1p2p3, so this number is also a measure of largeness. This is documented in more generality by the following. 1.1. The Main Theorem. Here is the main theorem: Theorem 1.4 (A.Arnold).

• Given a large ordered polygon P = p1p2 ...pn.Then

(1.1) tr(pkpk−1 ...p1) < −1 for all k =3..n.

• If we are given a polygon consisting of n points P = p1p2 ...pn satisfying condition ( 1.1), then this polygon is large and the given order yields the order as explained above. Remark 1.5.

• Here the points are matrices and hk := pkpk−1 ...p1 is a matrix product in SL(2, R) and tr means the half-trace ab 1 tr := (a + d). cd 2

4 A. ARNOLD AND K.-D. SEMMLER

• For two distinct points p, q in the upper half plane we have

(1.2) − cosh disthyp(p, q)=tr(p · q) < −1 • Observe that the signs of the traces are important for the positioning of the following and previous points with respect to the axes of the hyperbolic transformations hk.Thatispk+1 and pk lie on diﬀerent sides of the axis of hk, because

(1.3) tr(pk · hk) · tr(pk+1 · hk)=− tr(hk−1) · tr(hk+1) < −1.

Proof. Let P = p1p2 ...pn be a large polygon. We prove the first statement by induction on k. Move the lines slightly such that two consecutive lines “touch” at infinity opening up wide the boundary region between the first point of the first line, say q, and the last point of the k-th line, call it qk, while the last point of the i- th line, qi, equals the first point of the i+1-st one. Hence as Möbius transformation hk maps q to qk. Conjugating this to 0 and ∞ respectively in the upper half- plane model hk takes the form −a 1 h = c k −c 0 with a>2. All real parts of the pi’s and qi’s for i =1..k − 1andpk become positive. The line through pk+1 can only lie on the negative real-part half plane but any point pk+1 with negative real-part admits such a line. Furthermore if hk had a negative real number as fixpoint we could move the point q, residing now at 0, continuously to the negative value, then all the other qi’s will decrease as well, staying positive. But once q hits the fixpoint, the image of q, qk, should become this fixpoint as well, this is impossible. So the fixpoints of hk are positive real numbers as well and the axis of hk intersects two lines, in fact the first l1 and the last lk. This implies that in the matrix above, c must be positive. From this we get for c, sk+1, −rk+1 > 0andpk+1,rk+1 + sk+1 i tr(h )=tr(p · h ) k+1 k+1 k 1 −r r2 + s2 −a 1 =tr k+1 k+1 k+1 · c s −1 rk+1 −c 0 k+1 (1.4) 1 − 2 2 − 1 = rk+1a c(rk+1 + sk+1) 2sk+1 c 1 1 < − csk+1 − < −1. 2 csk+1 For the second part we assume being given a polygon satisfying (1.1). We prove by induction that for any k =3..n

(1) The points pj; j =1..k are on the same side of the axis of hk := pkpk−1 ...p1. (2) The axes of the products hj := pj pj−1 ...p1 do not intersect, and lie entirely on the same side of the axis of hk := pkpk−1 ...p1 as the points. (3) Taking any infinite point q of the other, “empty” side of the axis of hk, reflecting consecutively about the points, we get lines lj connecting qj−1 −1 with qj := pj qj−1pj , having the defining properties of a large polygon. Let us look more closely to the case k = 3 in the following lemma, setting h = p2p1 and p = p3:

LARGE HYPERBOLIC POLYGONS AND HYPERELLIPTIC RIEMANN SURFACES 5

Lemma 1.6. Let h be loxodromic with negative trace, and p a point, such that tr(p · h) < −1.Thenp and h are on the same side of p · h, in particular the axis of h and p · h do not intersect. Proof of the Lemma. By conjugation we may assume that the axis of h is the imaginary axis in the upper half plane model and p has non-positive real part. In other words −a 0 1 −rr2 + s2 h = − 1 and p = − 0 a s 1 r with a, s > 0andr ≤ 0. Then r(a2 − 1) tr(p · h)= < −1 2as which implies, that a>1, i.e., 0 is repelling fixpoint. Then observe that the matrix with determinant one and trace less than minus one bc m = de has the euclidian circle of radius ρ and center σ as invariant set, where b − e tr(m)2 − 1 σ = and ρ = . 2d d Applied to 1 − r2+s2 · ra a m = p h = − r s a a we see that 1 a + 1 1 1 σ = r a = r (1 + )and 2 a 2 a2 1 ra r 2 2 − 2 − 2 1 s ( − ) − 4 1 r (a a ) 4s ρ = s sa = 2 a 2 a Thus σ<0 <ρ, σ2 >ρ2,and|r + si− σ|2 =(r − σ)2 + s2 >ρ2. We conclude, that p and the axis of h lie “outside” of the invariant euclidean circle of p · h.Let’s remark that σ + ρ is the repelling, and σ − ρ is the attracting fixpoint of p · h.This finishes the proof of the lemma. We continue the proof of the theorem. For the case k = 3 we still need to construct the lines, taking any infinite point q within the invariant circle, the “empty” side of p · h.Byh it is moved to another negative real value q

6 A. ARNOLD AND K.-D. SEMMLER

1.2. Large polygons yield Riemann surfaces. This is why large polygons are so useful: Theorem 1.7. The group generated by a large polygon acts properly discontinuously. In fact, given the lines li through the points pi, each deﬁnes two half-planes Hj− and Hj+ and, say the latter, contains the other lines. Then FD(P ):= Hj+ j is a fundamental domain of the group Γ as deﬁned before. Proof. The argument is of Schottky type: Given any reduced word in the generators ··· w = pjk pjk−1 pj1 ,

(no two consecutive indices are equal), then we prove that w maps FD into Hjk− by induction on the word length. It is certainly true for k =1.Andwhenaword − ⊂ of length k 1mapsFDintoHjk−1−,thenpjk will map Hjk−1− Hjk+ into Hjk−. In order to show that the images of FD tile the whole hyperbolic plane, one has to make sure, that these images do not accumulate. For this we modify slightly the given lines in order to remove any common point at infinity (no two lines are asymptotic), still passing through the given points and being large (not separating one from another and not intersecting). All lines then have positive distance to each other, say greater than ε>0. We thus replace FD from above by FD built by the modified lines, trading some sectors with FD. Assume a point p is not covered by the translates of FD but a ∈ FD is. Consider the segment from a to p and divide it in segments of length between ε/2andε. There is a first point not covered by the translates of FD,callitp again, and the segment crosses a line, translate of our original lines with a translate of our original points. But the half-turn around this one, conjugate of our generating half-turns, will cover all points of distance ε of this line. Hence p will be covered. A contradiction. 1.3. How to get large polygons. Now it is time to give our algorithms to produce randomly large hyperbolic polygons Algorithm 1.8 (A. Arnold). We work in the unit circle. (1) Choose 2n points on the unit circle (at infinity) (with uniform probability) (2) Order these points and join consecutive points by a line (circular arc), two by two. (3) Choose a point pi on each line. The last step is choosing a point on a hyperbolic line of infinite length. There is no obvious probability measure for this choice, and in particular there is no homogenous probability distribution in the Teichmüller space of large polygons. By this we mean, that there is no random function selecting each point on such a line with equal probability. But in practice we can use the euclidean probability measure on these circular arcs, making sure that no points at infinity are chosen. This will prefer points close together and large polygons with far away points are not likely chosen. In order to observe what the algorithms in later sections do to such a polygon, one should modify the random function to favor points close to the (infinite) boundary of the plane. Otherwise some examples (for instances with very long side length) will rarely be outputted.

LARGE HYPERBOLIC POLYGONS AND HYPERELLIPTIC RIEMANN SURFACES 7

A more complicated algorithm is the following: Algorithm 1.9 (K.-D. Semmler). We work in the upper half-plane, or rather in the rectangle R := {r + is ||r|≤MAX,0

(1) Choose two distinct points p1,p2 ∈ R. (2) For kfrom3tondo: (3) Put hk−1 = pk−1 ...p1;(hk−1 ∈ SL(2, R)andtr(hk−1) < −1) and choose pk ∈ R. If tr(pk · hk−1) < −1 then take the next k. Else put 2 Ak−1 := hk−1 − tr(hk−1) · 1 and AA := tr(hk−1) − 1, then • If tr(pk · hk−1)=0then choose a random number x<−1 endif (This is an exceptional case with probability 0). • If tr(pk · hk−1) > 0 then (wrong side!, reﬂect over hk−1) put x := − tr(pk · hk−1) endif. Now put 2 AA + x x − α tr(pk · hk−1) α := 2 ,β:= ; AA +tr(pk · hk−1) AA and (new) pk := α · pk + β · A

• While −1 < tr(pk · hk−1) < 0 do (new point is too close to the line, increase the distance) put x =2tr(p · h − ) and put again k k 1 2 AA + x x − α tr(pk · hk−1) α := 2 ,β:= ; AA +tr(pk · hk−1) AA and (new) pk := α · pk + β · A endwhile end for The ﬁrst algorithm prefers polygons close to symmetric ones, while the second prefers small dividing curves on the quotient. 1.4. How to get more of them. Observe the following surprising corollary to the main theorem: Proposition 1.10. Given n points satisfying condition ( 1.1) then for any k indices with 1 ≤ i1 < ···

tr(p1p2p3) < −1andtr(p1p2p3p4) < −1 imply tr(p2p3p4) < −1 is not obvious. With large polygons in mind the proof is just the observation:

8 A. ARNOLD AND K.-D. SEMMLER

Proof. If you take a large polygon and you drop some points, the sub-polygon will be large.

Some more action on the set of large polygons: Proposition 1.11. Any cyclic permutation of the points of an ordered large polygon yields an ordered large polygon. Jumping with one point over a neighboring point of an ordered large polygon yields an ordered large polygon, i.e., we pass from P = p1 ...pj−1pj pj+1 ...pn to P = p1 ...pj−1 pj pj+1 pj+2 ...pn − where pj = pj+1 (changing the index) and pj+1 = pj+1pj pj+1 (jumped point) or to P = p1 ...pj−1 pj pj+1 pj+2 ...pn − where pj = pj pj+1pj (jumped point) and pj+1 = pj (changing the index). Proof. The first is obvious. The second needs the following observation: Let lj, joining the infinite points qj−1 with qj , contain the point pj and lj+1, joining the infinite points qj+1 with qj+2, contain the point pj+1. Then we put lj+1 the line connecting −pj+1qj−1pj+1 and −pj+1qj pj+1,inotherwordswetaketheim- − age of the line lj by the half-turn pj+1, which will pass through pj+1 = pj+1pj pj+1 − (see fig. 2). And we put lj the line connecting qj−1 with pj+1qj−1pj+1, containing pj = pj+1. These two new lines will, together with the old lines of the rest of the old polygon, realize P as a large polygon. For P we use the inverse construction.

pj+1 lj

p j lj+1 p = −p · p · p • j+1 j+1 j j+1 lj lj+1

Figure 2. Replacing lines when jumping over a neighboring point

The classical Dehn twists can be applied to large polygons:

Proposition 1.12. Given a large polygon P = p1 ...pj−1pjpj+1 ...pn and an index j,puth := hj = pj pj−1 ...p1. Then the polygon P obtained from P by replacing the sub-polygon Pj = p1 ...pj by a conjugate by h is large. m −m m −m m −m P = h · p1 · h ...h · pj−1 · h h · pj · h pj+1 ...pn for some m ∈ Z

We may also replace the remaining sub-polygon P \ Pj = pj+1 ...pk by a conjugate by h, because it diﬀers from P by a global conjugation.

LARGE HYPERBOLIC POLYGONS AND HYPERELLIPTIC RIEMANN SURFACES 9

Proof. The transformed sub-polygon is of course large. But the words later in condition (1.1) for k>jdo not see the conjugation m −m m −m tr(pk ...pj+1 · h · pj ...p1 · h )=tr(pk ...pj+1 · h · h · h )

=tr(pk ...pj+1 · pj ...p1) < −1 for all k = j +1..n.

2. Closing large polygons Up to now the large polygons provided Fuchsian groups with non-compact fundamental domain (see theorem 1.7), hence the quotient U/Γ2 was a non-compact Riemann surface with an involution, in some sense, a generalized hyperelliptic Rie- mann surface. The quotient U/Γ will be a disc with n order two cone points.

Proposition 2.1. Let p1,...,pn be a large hyperbolic n-gon. U 2 n−1 Then /Γ is a (generalized) hyperelliptic surface of signature ([ 2 ], 1 or 2). The surface contains one half-cylinder if n is odd and two if n is even. The hyperelliptic involution keeps p1,...,pn ﬁxed, which descend to Weierstrass points.

if n is odd

if n is even

Figure 3. Quotient of hyperbolic plane by a large polygon with Weierstrass points

In order to have genuine compact hyperelliptic Riemann surfaces we have to work a bit more. From now on we consider only large polygons with an even number vertices, i.e., given an integer g>2 we are given a large 2g-gon P = p1p2 ...p2g.By (1.1) h2g = p1p2 ...p2g is a loxodromic element. The following elementary lemma is included only because we want to keep track of the signs. Lemma 2.2. Let h ∈ SL(2, R) be a loxodromic element with negative trace. Then to any point p on its axis, there exists a unique point q on its axis such that pq = h. And conversely, for any two points p, q, the element h = pq is loxodromic and has trace less than −1. Proof. We may assume s, a > 0and −a 0 1 0 s2 h = 1 and p = − . 0 −a s 10 s Then the pointmatrix corresponding to a i is the only solution. The second part is a reformulation of equation (1.2).

10 A. ARNOLD AND K.-D. SEMMLER

Definition 2.3. Let P = p1p2 ...p2g be an ordered large polygon. Closing the polygon consists of adding two points p2g+1,p2g+2 obtaining (matrices in SL(2, R) satisfying)

(2.1) p2g+2p2g+1 ...p1 =id

A polygon P = p1p2 ...p2gp2g+1p2g+2 is called closed if, by dropping the last two points, we get an ordered large polygon P = p1p2 ...p2g. Of course now we formulate the theorem

Theorem 2.4. Any large polygon can be closed, i.e., any group Γ2 of a large polygon can be extended to the uniformizing group of a hyperelliptic Riemann surface.

Proof. This follows from lemma 2.2: Let P = p1 ...p2g be a large polygon. Pick any point pg+1 on the axis of h2g = p2g ...p1 and then, according to lemma 2.2 the other point pg+2 such that h2g = pg+1pg+2 implying equation (2.1).

Proposition 2.5. Let P = p1 ...p2gp2g+1p2g+2 be a closed polygon, then, dropping any two points, you get a large polygon.

Proof. First we drop two consecutive points from the P = p1p2 ...,p2g+2 starting with p2g,p2g+1. The only part in condition (1.1) for the polygon P = p1p2 ...p2g−1p2g+2 to check is

tr(p2g+2p2g−1 ...p1) < −1orequivalentlytr(p2g−1 ...p1p2g+2) < −1 but equation (2.1) implies

p2g+1p2gp2g−1 ...p1p2g+2 = id hence p2g−1 ...p1p2g+2 = p2g+1p2g

The lemma 2.2 proves that P = p2g+2p1p2 ...p2g−1 is large and p2g,p2g+1 lie on the axis of the product p2g−1 ...p1p2g+2. Inductively we can walk around the polygon to get, that all sub-polygons obtained from P = p1 ...p2g+2 by dropping two consecutive points are large. Now we show that we can drop the last, p2g+2 and some other point pa diﬀerent from p2g+1 and p2g+2. The polygon made of all the points except pa,p2g+1,p2g+2 is large because it is a sub-polygon of P = p1p2 ...p2g, assumed to be large to begin with. It remains to show that

tr(p2g+1 ...pa ...p1) < −1 where the hat indicates that this point is to be removed, as usual. Now it suﬃces to observe that p2g+2p2g+1 ...p1 = id implies that −1 −1 p2g+2 (p2g+1 ...pa+1papa+1 ...p2g+1) p2g+1 ...pa+1pa ...pa+1 ...p1 =id and so −1 −1 p2g+1 ...pa ...p1 =(p2g+1 ...pa+1papa+1 ...p2g+1) p2g+2 −1 −1 Again lemma 2.2 and lemma 1.1 conclude, because (p2g+1 ...pa+1papa+1 ...p2g+1) is a point.

LARGE HYPERBOLIC POLYGONS AND HYPERELLIPTIC RIEMANN SURFACES 11

3. Groups from Polygons We will now consider the groups (see definition 1.2) obtained from closed polygons. In particular we look for fixpoint free and thus Fuchsian groups. Warning: The polygon P generating a group Γ(P ) is not a fundamental polygon of the action of Γ(P )onU even if this group happens to act properly discontinuously. Yet it is often easy to construct such a fundamental domain out of the generating polygon. Polygons will be related if the corresponding groups are the same or conjugate, and we will try to have “nice” polygons for some groups. Obviously acting by the 2 symmetric group Sn on a polygon will not change Γ or Γ . And, for example, Γ(P )=Γ(P )if ··· · · −1 (3.1) P = p1p2 ...pi pn where pi = pj pi pj , i.e., we don’t change the groups if we jump with the point pi over the point pj, replacing pi by the symmetric point pi (see fig. 4). These permutations and jumps yield actions on the space of n-gons. Other actions will appear later. We will try to relate geometric properties of polygons to those of their groups. In particular we will construct Fuchsian groups by constructing “nice” polygons. Let us now propose a domain for a closed polygon P = p1 ...p2g+2 which will turn out to be the fundamental domain for the group Γ2(P ) acting on the upper half plane.

p1 p2

− · · p3 p4 p3 = p4 p3 p4

Figure 4. A simple jump

Algorithm 3.1. Choose two consecutive points, say p2g+1,p2g+2,ofthegiven −1 closed polygon. Put q0 = p2g+2,andqk = pkqk−1pk , i.e., −1 −1 qk = pk ...p1q0p1 ...pk for k =1..2g

12 A. ARNOLD AND K.-D. SEMMLER

Observe now that p2g+1 is now the midpoint of the line passing through the ﬁnite points q0 and q2g, which is the axis of h2g = p2g ...p1, because −1 −1 −1 −1 −1 p2g+1q2gp2g+1 = p2g+1p2g ...p1q0p1 ...p2g p2g+1 = p2g+2q0p2g+2 = p2g+2 and so all points pk are on one side of this line. We now turn this domain around p2g+1 and get the points on the other side. −1 q2g+k = p2g+1qkp2g+1 for k =1..2g hence the polygon FD(P ):=q0 ...q4g−1.

−1 The original points pk; k =1...2g and the points p2g+1pkp2g+1; k =1...2g become the midpoints of the sides qk−1qk of FD(P ):=q0 ...q4g−1.

Figure 5. Large hexagon and fundamental domain for the genus two Bolza surface

Proposition 3.2. Given a closed polygon P = p1 ...p2g+2.Thepolygon FD(P ):=q0 ...q4g−1 constructed above is a fundamental domain for the group 2 Γ (P ). Opposite sides qk−1qk and q2g+k−1q2g+k are identiﬁed by the side-pairers si := p2g+1pi. There is only one vertex cycle, that is, all qi are identiﬁed to one point by Γ2(P ). The quotient S(P ):=U/Γ2(P ) is a hyperelliptic Riemann surface of genus g and the hyperelliptic involution is represented by p2g+1.

LARGE HYPERBOLIC POLYGONS AND HYPERELLIPTIC RIEMANN SURFACES 13

Proof. Poincar´e’s theorem as in, e.g., [8] shows, that the group generated by 2 the si has FD(P ) as (closure of) a fundamental domain, and we have Γ (P )= si = p2g+1pi. The only vertex cycle is exactly 2 2 (p2g+1 ...p1) =(−p2g+2) =id To get the genus, we can use the Euler characteristic with one polygon, one vertex and 2g sides: χ(S)=1− 2g +1=2− 2g

By applying a cyclic permutation to a closed polygon prior to the algorithm above, you get many (2g + 2) essentially different fundamental domains for the same group Γ2(P ). This is very helpful to test numeric or algorithmic programs, like calculating eigenvalues or systoles (see, e.g., section 4). 3.1. Standard generators. Constructing a closed polygon amounts to creating explicit generators for the uniformizing group of a hyperelliptic Riemann surface. 2 Γ (P )=si = p2g+2pi | i =1...2g − 1 We want to show, that these generators are not too far off from standard generators g 2 Γ (P )= αj ,βj ; j =1..g | [αj ,βj]=±1 j=1 This shows the role of some words on the surface. Here is the construction 2 α1 := p1p2,β1 := p3p2, [α1,β1]=p1p2 p3p2 p2p1 p2p3 = −(p1p2p3) 2 As observed, (p1p2p3) represents a dividing geodesic cutting off a (hyperelliptic invariant) torus from the surface. Now

α2 = p3p2p1p4,β2 = p5p4 and

[α2,β2]=p3p2p1p4 p5p4 p4p1p2p3 p4p5

= −p3p2p1p4 p5p1p2p3 p4p5 implying 2 [α1,β1][α2,β2]=−(p1p2p3) (−p3p2p1p4 p5p1p2p3 p4p5) 2 = −(p1p2p3p4p5) 2 Now (p1p2p3p4p5) represents a dividing geodesic cutting oﬀ a (hyperelliptic invariant) genus two piece from the surface. This gives a pretty good idea how this will continue. For j =1..g − 1:

αj = p2j−1 ...p1p2j ,βj = p2j+1p2j which implies [αj ,βj ]=−p2j−1 ...p1p2j p2j+1p1 ...p2j+1 and g−1 2 [αj ,βj ]=−(p1 ...p2g−1) j=1

14 A. ARNOLD AND K.-D. SEMMLER by induction. Now for the last one we simply put 2 αg := p2gp2g+1,βg := p2g+2p2g+1, [αg,βg]=−(p2gp2g+1p2g+2) and g 2 2 [αj ,βj ]=(p1 ...p2g−1) (p2gp2g+1p2g+2) =id j=1 because p1 ...p2g−1 p2gp2g+1p2g+2 = id. We remark that words with an odd number of consecutive vertices represent square roots of dividing geodesics, and we know the topology of the parts.

4. Reduction Algorithms on Polygons On polygons with certain properties (large, closed) we can act creating different polygons, i.e., different sets of generators of the same group. Giving as input such a polygon we get a different polygon as output, for instance, by jumping a point over its neighbor, and renumbering, or by a cyclic permutation, etc. (see 1.11). A strategy consists of giving geometric criteria to say whether we act on the input polygon or not. Why do we prefer the new polygon over the old? An action on polygons of a certain type together with a strategy yields an algorithm that stops with output a final set of objects optimal for our strategy, or deciding that the input was not admissible. Typical strategies are trying to find accidental elliptics or for the resulting group deciding that the quotient S = U/Γ2(P ) for the input polygon P will not be a Riemann surface, or for the the resulting Riemann surface S = U/Γ2(P ) • decreasing the distances of points, finding short closed geodesics on the surface, • finding short dividing geodesics. All these strategies also have the eventual goal of computing systoles, Bers’ constants, invariant pants decompositions or (length-) spectral information. 4.1. Triangles. Keen, Gilman, Maskit, and other (e.g., [5, 16]) have given criteria to decide when two loxodromic elements generate a Fuchsian group. For the case of loxodromics, the result reads: Theorem 4.1 (Gilman, Keen, Maskit, et al.). • Two loxodromic g, h ∈ SL(2, R) with intersecting axes generate a purely loxodromic Fuchsian group Γ iff the commutator [g, h] is loxodromic. • In this case (tr(g)2 − 1) · (tr(h)2 − 1) > 1. • The Riemann surface S = U/Γ is a one-holed torus. • (Collar Lemma) If g sits in a fixpoint free Fuchsian group Γ representing a simple closed geodesic of length lg = 2 arccosh | tr(g)| in the Riemann surface S = U/Γ, then there is an embedded hyperbolic cylinder in S of length at least (see [16]) 1 l = 2 arcsinh lg sinh 2 • There is an algorithm (see [5]), replacing the generators g, h via Nielsen transformations by g,h, generating the same group, such that 1 < | tr(g)|≤|tr(h)|≤min(|tr(gh)|, | tr(gh−1)|) ≤|tr(g) · tr(h)|

LARGE HYPERBOLIC POLYGONS AND HYPERELLIPTIC RIEMANN SURFACES 15

and these numbers are uniquely determined by the group. They represent a fundamental domain for the action of the Teichmüller modular group on Teichmüller space of one-holed (non-punctured) torus. The Gilman algorithm consists of acting by two-step Nielsen transformations with the strategy to decrease the absolute values of the traces. The final set is the given fundamental domain on Teichmüller space (in trace parameters). Similar results and slight generalizations have been achieved by Buser-Semmler [3], Rosenberger [11] (for elliptic generators), and others, e.g., [22], [15], [13]. In our setting we can replace the two loxodromic g, h ∈ SL(2, R) with intersecting axes by a triangle P = p1p2p3, such that g = p1p2 and h = p3p1 and the result above reads Theorem 4.2. The group Γ2(P ) of a triangle P is purely loxodromic iff P is large. There is an algorithm replacing the triangle by another, jumping one of the points over the other and renaming the vertices, in order to get a weakly acute triangle (all inner angles ≤ π/2), generating the same group Γ2(P ).

Proof. First, we see [g, h]=−(p1p2p3) and being large implies that p3p2p1 and hence its inverse squared is loxodromic. The following algorithm “Reduce- Torus” will stop, when

(4.1) 1 < − tr(p1p2), − tr(p3p2) ≤−tr(p1p3) ≤ tr(p1p2) · tr(p3p2), and one can check that this condition is equivalent to the triangle being weakly acute. Again renaming will yield the result of the cited theorem 4.1 above. We want to generalize the preceding results to higher polygons. One step will be the following algorithm ”ReduceTorus”. This algorithm will do slightly more, it will replace the triangle P = p1p2p3 by a triangle P = p1p2p3, preserving the product p3p2p1 = p3p2p1 in oder to be useful within a context of large or closed polygons.

Algorithm 4.3 (ReduceTorus). Given a triangle P = p1p2p3 do replace as follows: While not acute do

• If tr(p1p3) · tr(p2p3) < − tr(p1p2) then put −1 (p1,p2,p3)=(p1,p3,p3 p2p3),

• else if tr(p1p2) · tr(p1p3) < − tr(p2p3) then put −1 (p1,p2,p3)=(p1p2p1 ,p1,p3),

• else if tr(p1p2) · tr(p2p3) < − tr(p1p3) then put −1 (p1,p2,p3)=(p1,p2p3p2 ,p2), • else acute = true end do If − tr(p1p2) > − tr(p2p3), − tr(p1p3) then put −1 −1 (p1,p2,p3)=(p1p2p1 ,p1p3p1 ,p1) else if − tr(p2p3) > − tr(p1p2), − tr(p1p3) then put −1 −1 (p1,p2,p3)=(p3,p3 p1p3,p3 p2p3).

16 A. ARNOLD AND K.-D. SEMMLER

This stops because in each step − tr(p1p3) − tr(p2p3) − tr(p1p2) > 3 decreases and produces a weakly acute triangle, i.e., satisfying equation (4.1).

We can apply this routine to any sub-triangle p1p2p3, p4p5p6, etc., of a large or closed polygon P = p1 ...pn. But this does not ﬁnd short dividing geodesics of the form h3 = p3p2p1. For this we use the following algorithm “FindSmallTorus”:

Algorithm 4.4 (FindSmallTorus). Given a large ordered polygon P = p1 ...pn − do ﬁnd the smallest number among tr(pi1 pi2 pi3 ), with i1

Algorithm 4.5 (Unwind). Given a large or a closed polygon P = p1 ...pn and a sub-polygon of consecutive points Q = p1 ...pk,awordw in those ﬁrst k generators and one of the remaining points p = pj , with j>k. To unwind Q with respect to p consists of the following process:

If w is not specified, we unwind for w = pk ...p2 and w = pk−1 ...p1. The effect of this algorithm is best described, when we look for a small (one-holed) torus (triangle) in our polygon: First look for the smallest torus with “FindSmallTorus.” 2 So h3 is a candidate of a small geodesic cutting off a one-holed torus off our 2 surface S = U/Γ and the triangle p1p2p3 is fairly small. Now we reduce this triangle to a (weakly) acute triangle with “ReduceTorus.” Then we “Unwind” the corresponding triangle Q = p1p2p3 with respect to p4 or pn (or any other remaining point for that matter) and h3 = h. So it may be, that we find a m −m m −m m smaller torus, e.g., h p2h h p3h p4 for some integer m in the polygon P = m −m m −m m −m h p1h h p2h h p3h p4 ...pn. Observe that largeness is still guaranteed m −m m −m m −m because h p1h h p2h h p3h = p1p2p3. Now we combine these steps:

Algorithm 4.6. Given a closed polygon P = p1 ...p2g+2

• “FindSmallTorus” and put it in ﬁrst place T = p1 ...p3. • “ReduceTorus” in the polygon three by three consecutive points.

LARGE HYPERBOLIC POLYGONS AND HYPERELLIPTIC RIEMANN SURFACES 17

• “Unwind” the first torus with respect to the other points. • Retry “FindSmallTorus” to see if we get yet a smaller torus. – If we find a smaller torus then restart. – Else “Unwind” quadrilateral T = p1 ...p4 w.r.t p5. – If changed restart else “Unwind” higher sub-polygons, etc. • stop if no improvement has been achieved. Other steps or sub-algorithms can be inserted. Combining these steps we get a final set. We don’t know if this will suffice to get a unique representation of such a (closed) hyperelliptic Riemann surface. Yet for the case genus two, given by a closed hexagon, the second author proposed a fundamental domain for the action of the modular group on Teichmüller space in [14] characterizing the fact, that a given dividing geodesic is minimal. For this he gave a list of sixteen competing curves, which appear all in the algorithm above as competing tori (sub-triangles). In this case the above algorithm yields a final 2 set of closed hexagons P = p1 ...p6 unique up to isometry of the surface U/Γ for interior polygons in this final set. Theorem 4.7. A fundamental domain for the action of the modular group on the Teichmüller space of genus two closed surfaces is given by the set of closed hexagons P = p1 ...p6, where the two tori p1 ...p3 and p4 ...p6 satisfy

− tr(p3p2p1) ≤ a ﬁnite set of competing tori as in the above algorithms and

min(− tr(p2p1), − tr(p3p2)) ≤ min(− tr(p5p4), − tr(p6p2p5)). Proof. Check that the sixteen curves in [14] are among the triangles tested against T = p1 ...p3. The details of the translation of the methods used there to large polygons here is rather technical.

References [1] Beardon, A. F. The Geometry of Discrete Groups. Springer, New York Heidelberg Berlin, 1982. MR1393195 (97d:22011) [2] Buser, P. Geometry and Spectra of Compact Riemann Surfaces.Birkhäuser, Boston, 1992. MR1183224 (93g:58149) [3] Buser, P. and Semmler, K.-D. The geometry and spectrum of the one holed torus. Comment. Math. Helvetici 63 (1988), 259-274. MR948781 (89k:58286) [4] Fenchel, W. Elementary Geometry in Hyperbolic Space. deGruyter, 1989. MR1004006 (91a:51009) [5] Gilman, J. Two-generator discrete subgroups of PSL(2,R).Mem.Amer.Math.Soc.117 (1995),no. 561 MR1290281 (97a:20082) [6] Gilman, J. and B. Maskit, An algorithm for 2-generator Fuchsian groups. Michigan Math. J. 38 (1991),no. 1, pp. 13-32 MR1091506 (92f:30062) [7] Helling, H. Uber¨ den Raum der kompakten Riemannschen Flächen vom Geschlecht 2.J.Reine Angew. Math. 268/269 (1974), pp. 286-293 Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday MR0361167 (50:13613) [8] Maskit, B. Kleinian groups. Grundlehren der Mathematischen Wissenschaften, vol. 287. Springer-Verlag, Berlin, (1988). MR90a:30132 [9] Lehner, J. Analytic Number Theory. Springer Lecture Notes 899 (1981), 315-324. MR654537 (83h:10056) [10] Milnor, J. On the 3-dimensional Brieskorn manifolds M(p,q,r). Ann. Math. Studies 84 (1975), 175-225 . MR0418127 (54:6169)

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[11] Maclachlan, C., and Rosenberger, G.: Small volume isospectral, non-isometric, hyperbolic 2-orbifolds.Arch.Math.,62 (1994) , 33-37 . MR1249582 (96g:11054) [12] Horowitz, R. D. Characters of free groups represented in the two-dimensional special linear group. Comm. Pure Appl. Math. 25 (1972), 635-649. MR0314993 (47:3542) [13] Griffiths, D. . Doctoral Thesis. [14] Semmler, K.-D. An explicit Fundamental Domain for the Teichmüller space of Riemann surfaces of genus 2. Doctoral Thesis , EPFL (No 766),(1988). [15] Gauglhofer, Th., and Semmler, K.-D. Trace coordinates of Teichmüller space of Riemann surfaces of signature (0,4). Conform. Geom. Dyn. 9 (2005), 46-75. MR2133805 [16] Keen L. Collars on Riemann surfaces. In Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., (1973), pp. 263–268. Ann. of Math. Studies, No. 79. Princeton Univ. Press, Princeton, N.J., 1974. MR0379833 (52:738) [17] Schmutz Schaller P. Collars on Riemann surfaces. [18] A. Aigon-Dupuis, P. Buser and K.-D. Semmler, Hyperbolic Geometry In Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology Edited by Jens Bolte and Frank Steiner Lond. Math. Soc. Lec. Not. Series 397 (2012) [19] A. Arnold, Largehyperbolicn-gons, projet de master EPFL, hiver 2006-2007. [20] A. F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, 1983. MR698777 (85d:22026) [21] A. Binotto, Essai sur les groupesadeuxg´ ` enérateurs, projet de diplôme EPFL, Hiver 1994- 1995. [22] T. Gauglhofer, Trace coordinates of Teichmüller spaces of Riemann surfaces,Thèse No 3521, EPFL, 2006. [23] B. Maskit, On Klein’s combination theorem, Trans. Amer. Math. Soc. 120, pp. 499-509, 1965. MR0192047 (33:274)

Ecole Polytechnique Fed´ erale´ de Lausanne, FSB–Mathematics Institute of Geom- etry and Applications–MATHGEOM, Station 8, CH–1015 Lausanne E-mail address: [email protected] Ecole Polytechnique Fed´ erale´ de Lausanne, FSB—Mathematics Institute of Ge- ometry and Applications–MATHGEOM, Station 8, CH–1015 Lausanne E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11372

On isolated strata of pentagonal Riemann surfaces in the branch locus of moduli spaces

Gabriel Bartolini, Antonio F. Costa, and Milagros Izquierdo

Abstract. The moduli space Mg of compact Riemann surfaces of genus g has orbifold structure, and the set of singular points of such an orbifold is the branch locus Bg.Forg ≡ 3 mod 4 there exists isolated strata corresponding to families of pentagonal Riemann surfaces.

1. Introduction In this article we study the topology of moduli spaces of (compact) Riemann surfaces of genus g ≥ 2. The moduli space Mg of compact Riemann surfaces of genus g being the quotient of the Teichm¨uller space by the discontinuous action of the mapping class group, has the structure of a complex orbifold, whose set of singular points is called the branch locus Bg. The branch locus Bg, g ≥ 3 consists of the Riemann surfaces with symmetry, i.e. Riemann surfaces with non- trivial automorphism group. Our goal is to study the topology of Bg through its connectedness. The connectedness of moduli spaces of hyperelliptic, p−gonal and real Riemann surfaces has been widely studied, for instance by [BSS], [CI1], [G], [Se], [BCIP]. It is known that B2 is not connected, since R. Kulkarni (see [K]and[BCI]) 2 5 showed that the curve w = z − 1 is isolated in B2, i. e. this single surface is an isolated component of B2,furthermoreB2 has exactly two connected components (see [BI]and[Bo]). It is also known that the branch loci B3, B4 and B7 are connected and B5, B6, B8 are connected with the exception of isolated points (see [BCIP]and[CI2]). In this article we prove that Bg is disconnected for g ≡ 3mod4, g ≥ 18; more concretely we ﬁnd equisymmetric isolated strata induced by order 5 automorphisms of Riemann surfaces of genera g ≡ 3mod4. In[CI5]itisprovedthatBg is disconnected for g ≥ 65.

2. Riemann surfaces and Fuchsian groups From now on we shall consider compact surfaces with genus g ≥ 2andFuchsian groups with fundamental region of ﬁnite hyperbolic area.

2010 Mathematics Subject Classiﬁcation. Primary 14H15, 30F10, 30F60. Partially supported by MTM2011-23092. Partially supported by the Swedish Research Council(VR).

c 2012 American Mathematical Society 19

20 GABRIEL BARTOLINI, ANTONIO F. COSTA, AND MILAGROS IZQUIERDO

Let X be a Riemann surface of genus g ≥ 2 and assume that Aut(X) = {1}. The surface X may be uniformized by a surface Fuchsian group Γg (torsion free Fuchsian group isomorphic to π1(X)) and X = D/Γg has ﬁnite hyperbolic area. Hence X/Aut(X) is an orbifold and there is a Fuchsian group Γ ≤ PSL(2, R), such that Γg ≤ Γ and:

D→X = D/Γg → X/Aut(X)=D/Γ If the Fuchsian group Γ is isomorphic to an abstract group with canonical presentation | m1 ··· mk (2.1) a1,b1,...,ag,bg,x1 ...xk x1 = = xk = xi [ai,bi]=1 . we say that Γ has signature

(2.2) s(Γ) = (g; m1,...,mk). The generators in presentation (2.1) will be called canonical generators. Let X be a Riemann surface uniformized by a surface Fuchsian group Γg, i.e. a group with signature (g; −). A ﬁnite group G is a group of automorphisms of X, i.e. there is an action a of G on X, if and only if there is a Fuchsian group Δ and an epimorphism θa :Δ→ G such that ker θa =Γg. The epimorphism θa is the monodromy of the covering fa : X → X/G = D/Δ. The relationship between the signatures of a Fuchsian group and subgroups is given in the following theorem of Singerman: Theorem 2.1. ([Si1])LetΓ be a Fuchsian group with signature ( 2.2)and canonical presentation ( 2.1). Then Γ contains a subgroup Γ of index N with signature s(Γ )=(h; m11,m12, ..., m1s1 , ..., mr1, ..., mrsr ). if and only if there exists a transitive permutation representation θ :Γ→ ΣN satisfying the following conditions: 1. The permutation θ(xi) has precisely si cycles of lengths less than mi,the lengths of these cycles being m /m , ..., m /m . i i1 i isi 2. The Riemann-Hurwitz formula μ(Γ)/μ(Γ) = N. where μ(Γ),μ(Γ) are the hyperbolic areas of the surfaces D/Γ, D/Γ. For G, an abstract group isomorphic to all the Fuchsian groups of signature s =(h; m1, ..., mr), the Teichm¨uller space of Fuchsian groups of signature s is:

{ρ : G→PSL(2, R):s(ρ(G)) = s}/ conjugation in PSL(2, R)=Ts.

The Teichm¨uller space Ts is a simply-connected complex manifold of dimension 3g − 3+r. The modular group, M(Γ), of Γ, acts on T (Γ) as [ρ] → [ρ ◦ α]where α ∈ M(Γ). The moduli space of Γ is the quotient space M(Γ) = T (Γ)/M (Γ), then M(Γ) is a complex orbifold and its singular locus is B(Γ), that is called the branch locus of M(Γ). If Γg is a surface Fuchsian group, we denote Mg = Tg/Mg and the branch locus by Bg. The branch locus Bg consists of surfaces with non-trivial symmetries for g>2. If X/Aut(X)=D/Γandgenus(X)=g, there is a natural inclusion i : Ts ⊂ Tg : G→ R ⊂G | → R ρ : PSL(2, ), π1(X) , ρ = ρ π1(X): π1(X) PSL(2, ).

ON ISOLATED STRATA OF PENTAGONAL RIEMANN SURFACES 21

If we have π1(X) G, then there is a topological action of a ﬁnite group G = G/π1(X) on surfaces of genus g given by the inclusion a : π1(X) →G.This G,a inclusion a : π1(X) →Gproduces ia(Ts) ⊂ Tg. The image of ia(Ts)isM , G,a where M is the set of Riemann surfaces with automorphism group containing a subgroup acting in a topologically equivalent way to the action of G on X given by G,a the inclusion a,see[H]. The subset MG,a ⊂ M is formed by the surfaces whose automorphism group acts in the same topological way as a. The branch locus, B , g →M B MG,a of the covering Tg g can be described as the union g = G= {1} ,where {MG,a} is the equisymmetric stratiﬁcation of the branch locus [B]:

Theorem 2.2. ([B])LetMg be the moduli space of Riemann surfaces of genus g, G a finite subgroup of the corresponding modular group Mg.Then: MG,a M (1) g is a closed, irreducible algebraic subvariety of g. MG,a (2) g , if it is non-empty, is a smooth, connected, locally closed algebraic M MG,a subvariety of g, Zariski dense in g . MG,a There are finitely many strata g . An isolated stratum MG,a in the above stratification is a stratum that satisfies G,a H,b M ∩ M = ∅, for every group H and action b on surfaces of genus g.Thus G,a M = MG,a Since each non-trivial group G contains subgroups of prime order, we have the following remark: Remark . 2.3 ([Co]) Cp,a Bg = M p prime C ,a where M p is the set of Riemann surfaces of genus g with an automorphism group containing Cp, the cyclic group of order p, acting on surfaces of genus g in the topological way given by a.

3. Disconnectedness by pentagonal Riemann surfaces By the Castelnuovo-Severi inequality [A], the p-gonal or elliptic-p-gonal mor- 2 phism of a Riemann surface Xg of genus g is unique if g ≥ 2hp +(p − 1) +1,where h, h =0, 1, is the genus of the quotient surface. Let Xg, g ≥ 10h + 17, be an elliptic-pentagonal or pentagonal surface, such ∈ MC5,a ∈ \ that Xg g for some action a. Consider an automorphism b Aut(X) α , by the Castelnuovo-Severi inequality, b induces an automorphism ˆb of order p on the Riemann surface Xg/a = Yh,ofgenush, according to the following diagram: b Xg = D/Γg → Xg = D/Γg fa ↓↓fa ˆb Xg/α = Yh(P1,...,Pr) → Xg/α = Yh(P1,...,Pr) where Γg is a surface Fuchsian group and fa : Xg = D/Γg → Xg/α is the morphism induced by the group of automorphisms α with action a. S = {P1,...,Pr} r is the branch set in Yh of the morphism fa with monodromy θa :Δ(h;5, ...,5) → C5 ti deﬁned by θa(xi)=α ,whereti ∈{1, 2, 3, 4} for 1 ≤ i ≤ r.Letnj denote the

22 GABRIEL BARTOLINI, ANTONIO F. COSTA, AND MILAGROS IZQUIERDO number of times that the exponents j occurs among t1,...,tr,for1≤ j ≤ 4. Then n1 + n2 + n3 + n4 = r and 1n1 +2n2 +3n3 +4n4 ≡ 0mod5. Now, ˆb induces a permutation on S that either takes singular points with monodromy αj to points with monodromy α5−j, takes points with monodromy αj to points with monodromy α2j , or it acts on each subset formed by points in S with same monodromy αtj . Therefore the following conditions force ˆb to be the identity on Yh:

1. |n1 − n4| + |n2 − n3|≥3+h, | − |≥ ≤ ≤ (3.1) 2. n1 nj 3+h, for some nj such that 2 j 4and 3. nj mod p ≥ 3+h.

Theorem 3.1. Assume g ≥ 18 is even, then there exist isolated strata formed by pentagonal surfaces.

r Proof. We will construct monodromies θ :Δ(0;5, ...,5) → C5,wherer = g 2 + 2 by the Riemann-Hurwitz formula, such that the conditions (3.1) above are ti satisfied. Assume θ(xi)=α , i =1,...,r.Letnj = |{ti = j; i =1,...,r}|,then 2 3 4 we will define the epimorphism θ by the generating vector (n1α, n2α ,n3α ,n4α ), j j where njα means that α is the monodromy of nj different singular points Pi.

g mod 5 r mod 5 n1 n2 n3 n4 g ≡ 0mod5 r ≡ 2mod5 (r − 13) 5 1 7 g ≡ 1mod5 r ≡ 0mod5 (r − 7)511 g ≡ 2mod5 r ≡ 3mod5 (r − 9)135 g ≡ 3mod5 r ≡ 1mod5 (r − 7)151 g ≡ 4mod5 r ≡ 4mod5 (r − 9)513

We see that the given epimorphisms satisfy the conditions (3.1) except for g = 20, r = 12. However, in this case, g = 20, r = 12, let the epimorphism θ : 12 2 3 4 Δ(0; 5, ...,5) → C5 be deﬁned by the generating vector (α, 7α ,α , 3α ). θ clearly satisﬁes the conditions 3.1 above.

Remark 3.2. The complex dimension of the isolated strata given in the proof of theorem 3.1 is 0 × 3 − 3+r = g/2+2− 3=g/2 − 1.

Remark 3.3. There are several isolated strata of dimension g/2 − 1inBg for even genera g ≥ 22. For instance consider g ≡ 3 mod 5. The monodromy θ deﬁned by the generating vector ((r − 9)α, 5α2, 3α3, 5α4) induces an isolated stratum diﬀerent from the one given in the proof of Theorem 3.1 since the actions determined by θ and θ are not topologically equivalent, see [H].

Theorem 3.4. Assume g ≥ 29, g ≡ 1mod4, g =37 , then there exist isolated strata formed by elliptic-pentagonal surfaces.

Proof. Similarly to the proof of Theorem 3.1, using the conditions above r → g−1 (3.1), we will construct epimorphisms θ :Δ(1;5, ...,5) C5,wherer = 2 by ti the Riemann-Hurwitz formula. Assume θ(xi)=α , i =1,...,r.Letnj = |{ti = j; i =1,...,r}|, the epimorphism θ will be deﬁned by the generating vector (n1α, 2 3 4 j j n2α ,n3α ,n4α ), where nj α means that α appears as the monodromy of nj

ON ISOLATED STRATA OF PENTAGONAL RIEMANN SURFACES 23 diﬀerent singular points Pi.

g mod 5 r mod 5 n1 n2 n3 n4 g ≡ 0mod5 r ≡ 2mod5 (r − 13) 5 1 7 g ≡ 1mod5 r ≡ 0mod5 (r − 7)511 g ≡ 2mod5 r ≡ 3mod5 (r − 19) 11 3 5 g ≡ 3mod5 r ≡ 1mod5 (r − 7)151 g ≡ 4mod5 r ≡ 4mod5 (r − 11) 1 5 5 We see that the given epimorphisms satisfy the conditions set except for g = 37, r = 18. Remark 3.5. The complex dimension of the isolated strata given in the proof of theorem 3.4 is 1 × 3 − 3+r =(g − 1)/2.

Remark 3.6. Let g ≡ 3 mod 4. Then there is no isolated stratum in Bg of dimension (g − 1)/2. Such a stratum will consist of elliptic-pentagonal surfaces r given by epimorphisms θ :Δ(1;5, ...,5) → C5,wherer =(g − 1)/2 ≡ 1mod2. Now such epimorphisms cannot satisfy the third condition in (3.1). Acknowledgment. The authors wish to express their thanks to the referees for several helpful corrections.

References [A] Accola, R. D. M. (1984) On cyclic trigonal Riemann surfaces. I. Trans. Amer. Math. Soc. 283 no. 2, 423–449. MR737877 (85j:14052) [BCIP] Bartolini, G., Costa, A.F., Izquierdo, M., Porto, A.M., (2010) On the connectedness of the branch locus of the moduli space of Riemann surfaces, RACSAM Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. 104 no.1 81-86. MR2666443 (2011e:32013) [BI] Bartolini, G., Izquierdo, M. (2010) On the connectedness of branch loci of moduli spaces of Riemann surfaces of low genus. Proc. Amer. Math. Soc., (2011) doi:10.1090/S0002-9939- 2011-10881-5. [Bo] Bolza, O. (1888) On binary sextics with linear transformations between themselves, Amer. J. Math. 10, 47–70. [B] Broughton, S. A. (1990) The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups. Topology Appl. 37 101–113. MR1080344 (92d:57013) [BCI] Bujalance, E.; Costa, A. F.; Izquierdo, M. (1998) A note on isolated points in the branch locus of the moduli space of compact Riemann surfaces. Ann. Acad. Sci. Fenn. Math. 23 no. 1, 25–32. MR1601914 (99a:30038) [BSS] Buser, P., Seppälä, M., Silhol, R.(1995) Triangulations and moduli spaces of Riemann surfaces with group actions. Manuscripta Math. 88 209-224. MR1354107 (96k:32040) [CI1] Costa, A. F., Izquierdo, M. (2002) On the connectedness of the locus of real Riemann surfaces. Ann. Acad. Sci. Fenn. Math. 27 341-356. MR1922193 (2003e:14018) [CI2] Costa, A. F., Izquierdo, M. (2010) On the connectedness of the branch locus of the moduli space of Riemann surfaces of genus 4. Glasg. Math. J. 52 (2010), no. 2, 401-408. MR2610983 (2011k:32013) [CI3] Costa, A. F., Izquierdo, M. (2009) On the existence of connected components of dimension one in the branch loci of moduli spaces of Riemann surfaces. To appear in Mathematica Scandinavica. [CI4] Costa, A. F., Izquierdo, M. (2010) Equisymmetric strata of the singular locus of the moduli space of Riemann surfaces of genus 4. LMS Lect. Note Series 368 130-148. MR2665007 (2011g:30095) [CI5] Costa, A. F., Izquierdo, M. (2011) On the connectivity of branch loci of moduli spaces. Preprint 2011. [Co] Cornalba, M. (1987) On the locus of curves with automorphisms. Annali di Matematica Pura e Applicata (4) 149, 135-151. MR932781 (89b:14038)

24 GABRIEL BARTOLINI, ANTONIO F. COSTA, AND MILAGROS IZQUIERDO

[G] González-D´ıez, G. (1995). On prime Galois covering of the Riemann sphere. Ann. Mat. Pure Appl. 168 1-15 MR1378235 (97c:14033) [H] Harvey, W. (1971) On branch loci in Teichmüller space. Trans. Amer. Math. Soc. 153 387-399. MR0297994 (45:7046) [K] Kulkarni, R. S. (1991) Isolated points in the branch locus of the moduli space of compact Riemann surfaces. Ann. Acad. Sci. Fen. Ser. A I Math. 16 71-81. MR1127697 (93a:30048) [Se] Seppälä, M. (1990) Real algebraic curves in the moduli space of complex curves. Comp. Math., 74 259-283. MR1055696 (91j:14020) [Si1] Singerman, D. (1970) Subgroups of Fuchsian groups and finite permutation groups Bull. London Math. Soc. 2 319-323. MR0281805 (43:7519) [Si2] Singerman, D. (1972) Finitely maximal Fuchsian groups. J. London Math. Soc. 6 29-38. MR0322165 (48:529)

Matematiska institutionen, Linkopings¨ Universitet, 581 83 Linkoping,¨ Sweden E-mail address: [email protected] Dept. Matematicas´ Fundamentales, Facultad de Ciencias, UNED, 28040 Madrid, Spain E-mail address: [email protected] Matematiska institutionen, Linkopings¨ Universitet, 581 83 Linkoping,¨ Sweden E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11357

Finite group actions of large order on compact bordered surfaces

E. Bujalance, F. J. Cirre, and M. D. E. Conder

Abstract. In this paper we determine, up to topological equivalence, all the ﬁnite group actions of order at least 6(g − 1) on compact bordered surfaces of algebraic genus g for 2 ≤ g ≤ 101. The topological types of the surfaces where these actions occur are also given.

1. Introduction A bordered surface may be seen as the quotient space of an unbordered orientable surface under the action of a symmetry. This was realised by Klein, who started the study of groups of automorphisms of bordered surfaces at the end of the 19th century. An interesting problem in this topic is to determine the largest automorphism groups acting on such surfaces of a given algebraic genus g ≥ 2. The algebraic genus of a bordered surface is the genus of the orientable surface which double-covers it. The problem is inspired in the classical one for Riemann surfaces, after Schwarz showed in 1879 [22] the finiteness of the automorphism group of a Riemann surface of genus g ≥ 2. The largest order of such groups is 84(g − 1) and those attaining this bound are known as Hurwitz groups, [14]. There exist infinitely many values of g for which this bound is attained, and also infinitely many for which this does not occur. It is an open problem to determine the precise values of g for which it is attained. Just as an unbordered orientable surface may be endowed with a conformal structure, a bordered surface may be endowed with a dianalytic structure, see [1]. A surface with a dianalytic structure is called a Klein surface. Bordered Klein surfaces of algebraic genus g ≥ 2 admit at most 12(g − 1) automorphisms, as May showed in [17]. Groups of automorphisms attaining this bound are called M∗- groups. A table with some M∗-groups of order up to 240 was given in [19]. As with Hurwitz groups, it is an open problem to determine the (infinitely many) values of g for which this bound is attained. Also related to the results of this paper we mention two more problems. The first one asks for the order ν(g) of the largest group of automorphisms of a bordered surface of algebraic genus g, for given g ≥ 2. The second one seeks the minimum

2010 Mathematics Subject Classification. Primary 30F50, 20H10; Secondary 57M60. Key words and phrases. Riemann surfaces, finite group actions. The first and second authors were partially supported by MTM2011-23092. The third author was partially supported by N.Z. Marsden Fund (grant no. UOA1012).

c 2012 American Mathematical Society 25

26 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER algebraic genus of a bordered surface on which a given finite group G acts as a group of automorphisms. Unlike Riemann surfaces, which are topologically classified by a single parameter (the genus), a bordered Klein surface requires three parameters to classify it topologically — namely, its algebraic genus g, the number k of its boundary components, and its orientability. Taking ε = + if the surface is orientable and ε = − otherwise, the triple (g, k, ε)iscalledthetopological type of X. For a fixed value of g there are [(3g +4)/2]−1 different possibilities for the topological type of bordered surfaces, where [t] stands for the integer part of t. This is a classical result of Wei- chold. Hence a proper classification of bordered Klein surfaces with large group of automorphisms has to be undertaken not only in terms of the algebraic genus, but also in terms of the number of boundary components and the orientability. Our classification takes this into account and, accordingly, in this paper we classify the topological types of bordered Klein surfaces of algebraic genus g for 2 ≤ g ≤ 101 with at least 6(g − 1) automorphisms. Furthermore, the algebraic structure of such groups of automorphisms is also determined in some cases. This problem was considered by May [20] for groups of the largest possible order (M∗-groups), obtaining the list of topological types of bordered Klein surfaces with maximal symmetry of genus g ≤ 40. The method employed in [20]isbased on previous work by Greenleaf and May [13] where full covers of surfaces with primitive maximal symmetry are studied. Our approach here is combinatorial, based on the theory of non-euclidean crystallographic groups. This allows us to develop a procedure for determining in a reasonable time all M∗-groups up to order 1200, and classifying the topological types on which they act. We extend these results to groups of order 8(g − 1), 20 − − 3 (g 1) and 6(g 1), which are, respectively, the second, third and fourth largest possible orders of automorphism groups acting on bordered Klein surfaces of algebraic genus g ≥ 2. There are many results concerning the classification of groups of automorphisms of bordered surfaces, see the references in [9] or the more recent survey [6]. In most of these, interest is focused on classifying non-isomorphic abstract groups. Here we are interested in classifying inequivalent topological group actions. This is a finer classification, as the same abstract group may act on the same surface in different topological ways. We will use the theory of non-euclidean crystallographic (NEC) groups to deal with this topological classification. Compact connected Klein surfaces may also be viewed as real algebraic curves, due to the well-known functorial equivalence between such surfaces and algebraic function fields in one variable over R, see [1]. Thus our results can be stated in terms of real algebraic curves and their groups of birational transformations defined over R. Namely, we classify all possible large actions of birational transformations on real algebraic curves with real points of genus g,for2≤ g ≤ 101. The paper is organised as follows. In Section 2 we introduce the basic results on NEC groups and equivalent actions of finite groups which will be used subsequently. In Section 3 we compute the seven signatures with which a group of order at least 6(g − 1) may act on a bordered surface of algebraic genus g ≥ 2. Then the computational procedure developed to determine the topological types of bordered surfaces of algebraic genus g with at least 6(g − 1) automorphisms is explained in Section 4. Sections 5, 6, 7 and 8 contain the lists of such topological types up to

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 27 genus g ≤ 101, and also presentations of the corresponding groups. It is not difficult to identify some common patterns in the presentations and find infinite families of groups to which they belong. This is done in Section 9.

2. Preliminaries Let S be a bordered surface (which may or may not be orientable), and let Hom(S) be the group of its homeomorphisms (including those that reverse orientation, if S is orientable). Following Broughton [3], we define an (effective) action of a finite group G on S to be an injective homomorphism ε : G → Hom(S). Let Aut G be the group of automorphisms of G. Two actions ε1,ε2 : G → Hom(S)of the same abstract group G are topologically equivalent if there exist β ∈ Aut G and h ∈ Hom(S) such that

−1 ε2(g)=hε1(β(g))h for all g ∈ G. A natural tool to study finite groups acting on bordered surfaces is the theory of non-euclidean crystallographic groups (or NEC groups, for short), as introduced by Macbeath in [16]. Every such finite group action may be obtained by means of a pair of NEC groups Γ and Λ, with Γ a normal subgroup of finite index in Λ. For general background on the theory of NEC groups, we refer the reader to Chapters 0and1in[9], but in this section we briefly introduce the main ideas and results to be used in what follows. Recall that the signature σ(Λ) of an NEC group Λ is a collection of non-negative integers and symbols of the form ± { } (2.1) σ(Λ) = (γ; ;[m1,...,mr]; (n11,...,n1s1 ),...,(nk1,...,nksk ) ) which collects algebraic and topological features of Λ. If k>0 (which is the only case we will have to consider here) then the corresponding NEC group is called a bordered NEC group. A bordered NEC group with no non-trivial orientation-preserving elements of finite order is called a bordered surface group. The name comes from the fact that if Γ is such a group then the quotient U/Γisacompactbordered surface of topological genus γ with k boundary components, and is orientable if + occurs, and non-orientable otherwise. The algebraic genus g of such a surface is defined as 2γ + k − 1 if the surface is orientable, g = γ + k − 1otherwise. Conversely, it follows from the Uniformization Theorem that any bordered surface S of algebraic genus g ≥ 2 is the orbit space U/Γ of the hyperbolic plane U under the action of some bordered surface NEC group Γ. An important fact in the study of finite group actions is that, given a surface S so represented, a finite group G acts on S if and only if there exist an NEC group Λ and an epimorphism from Λ onto G with Γ as its kernel. All finite group actions on bordered surfaces arise in this way. An epimorphism whose kernel is a bordered surface NEC group is called a bordered smooth epimorphism. Consequently, we may study finite group actions by means of bordered smooth epimorphisms. To that end, we will use the following translation of the relation of (topological) equivalent group actions defined above into an (algebraic) equivalence relation on bordered smooth epimorphisms. For a proof, see [5, Section 2].

28 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Proposition 2.1. Let ε1,ε2 : G → Hom(S) be two finite group actions on a bordered surface S,andletθi :Λi → G be the corresponding bordered smooth epimorphism, for i =1, 2. Then the group actions ε1 and ε2 are topologically conjugate if and only if there exist an isomorphism α :Λ1 → Λ2 and an automorphism β : G → G such that βθ1 = θ2α. Since any bordered smooth epimorphism θ :Λ→ G with kernel Γ is equivalent to the natural epimorphism Λ → Λ/Γ, it is easier to say that two actions θ1 :Λ1 → G and θ2 :Λ2 → G are equivalent if there exists an isomorphism α :Λ1 → Λ2 such α that (ker θ1) =kerθ2. Accordingly, finding inequivalent actions associated with a given NEC group Λ is the same as finding normal subgroups (of finite index) in Λ up to conjugacy within Aut Λ (the automorphism group of Λ), and hence within Out Λ = Aut Λ/Inn Λ (the outer automorphism group of Λ). In this paper we will only have to deal with presentations of NEC groups whose signature is of the form

(0;+;[m1,...,mr]; {(n1,...,ns)}).

A presentation of such groups is the following. It has generators x1,...,xr (elliptic mi ≤ ≤ elements), c0,...,cs (reﬂections), and deﬁning relations: xi =1for1 i r, 2 ni ≤ ≤ − ··· ··· ci =(cici+1) =1for0 i s 1, and cs x1 xr = x1 xr c0.

3. Signatures associated with groups of order ≥ 6(g − 1) Let G be a finite group acting on a compact bordered surface S of algebraic genus g ≥ 2 and assume |G|≥6(g − 1). Let us write G =Λ/ΓwhereΓisasurface NEC group of algebraic genus g uniformizing S. In this section we enumerate the possible signatures of the NEC group Λ. The first condition comes from the following lemma, see [10]. Lemma 3.1. An NEC group Λ contains a normal bordered surface subgroup with finite index if and only if the signature of Λ has an empty period cycle or a period cycle with two consecutive link periods equal to 2. The second condition comes from the Riemann-Hurwitz formula. Recall that the area of a fundamental region for an NEC group Λ with signature (2.1) is 2πμ(Λ), where r 1 1 k si 1 μ(Λ) = αγ + k − 2+ 1 − + 1 − , m 2 n i=1 i i=1 j=1 ij with α = 2 if the sign is +, and α = 1 otherwise. If Λ is a subgroup of finite index of Λ, then Λ is also an NEC group, and its area is given by the so called Riemann-Hurwitz formula: μ(Λ)=[Λ:Λ] · μ(Λ). In our case, since the surface NEC group Γ satisfies μ(Γ) = g − 1wehave ⎛ ⎞ r 1 1 k si 1 g − 1=|G| ⎝αγ + k − 2+ 1 − + 1 − ⎠ . m 2 n i=1 i i=1 j=1 ij r − k si − Let us write R = i=1 (1 1/mi)andS = i=1 j=1 (1 1/nij) /2. Observe that if R>0thenr/2 ≤ R

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 29 k ≤ k | |≥ if S>0then i=1 si/4 S< i=1 si/2. Since we are assuming that G 6(g − 1) > 0, we require 13 2 <αγ+ k + R + S ≤ . 6 In particular, 1 ≤ αγ + k<3 because k ≥ 1. If αγ + k =2then0

This forces s = 4 and, moreover, (n3,n4)=(2, 3), (2, 4), (2, 5), (2, 6) or (3, 3), so we obtain the following possible signatures for Λ :

σ1 = (0; +; [−]; {(2, 2, 2, 3)}); σ2 = (0; +; [−]; {(2, 2, 2, 4)});

σ3 = (0; +; [−]; {(2, 2, 2, 5)}); σ4 = (0; +; [−]; {(2, 2, 2, 6)});

σ5 = (0; +; [−]; {(2, 2, 3, 3)}). If r =1then 1 1 1

σ6 = (0; +; [3]; {(2, 2)}). Assume ﬁnally that S =0, so that 1

σ7 = (0; +; [2, 3]; {(−)}). From the above analysis and the Riemann-Hurwitz formula we have the following. Proposition 3.2. Let G be a ﬁnite group acting on a bordered Klein surface of algebraic genus g ≥ 2. Assume |G|≥6(g − 1). Then one of the following occurs : 1) |G| = 12(g−1), in which case G acts with signature (0; +;[−];{(2, 2, 2, 3)}); 2) |G| =8(g −1), in which case G acts with signature (0; +;[−];{(2, 2, 2, 4)}); | | 20 − − { } 3) G = 3 (g 1), in which case G acts with signature (0; +;[ ]; (2, 2, 2, 5) ); 4) |G| =6(g−1), in which case G acts with signature (0;+;[−]; {(2, 2, 2, 6)}), (0;+;[−]; {(2, 2, 3, 3)}), (0;+;[3];{(2, 2)}) or (0;+;[2, 3]; {(−)}). We analyse each case separately in the following sections. Observe that for each of the above signatures σ1,...,σ7, there does exist a bordered smooth epimorphism from an NEC group Γ with signature σi onto a ﬁnite group G. This is a consequence, for instance, of Lemma 3.1.

30 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

4. A computational approach To find the topological types of all compact bordered surfaces of algebraic genus g ≤ 101 with at least 6(g − 1) automorphisms, we fix one of the seven signatures in Proposition 3.2, and then determine all possible groups G of the corresponding order acting with such signature on a bordered surface of genus g. Once this is done, the topological type of the surface on which G acts can be obtained from a presentation of G (see below). To determine all such groups G, we take an NEC group Λ with that signature, and find all normal subgroups of index |G| in Λ, eliminating those which cannot be the kernel of a bordered smooth epimorphism. Such normal subgroups can be found using recent variants of the well-established algorithm (due to Sims et al) for determining subgroups of a given index in a finitely-presented group. One possibility (the one we have used here) is the lowx computer program, as described in [11]. This involves a back-track search through a tree, with nodes at level k corresponding to certain subgroups generated by conjugacy classes of k distinct elements. Another, faster, version is the new LowIndexNormalSubgroups process developed recently by Derek Holt and his student David Firth (at the University of Warwick) and available in the computational algebra system Magma [2]. The latter process can find normal subgroups of index up to 100,000 by systematic enumeration of possibilities for a composition series of the quotient. In practice, the application of either method for finding normal subgroups of small index in an NEC group Λ with relatively few generators and relations takes very little time indeed. One can obtain all normal subgroups Γ of the specified index in Λ, together with a set of representatives of conjugacy classes of elements generating each one (or equivalently a set of additional relators which yield the associated factor group Λ/Γ when adjoined to the presentation of Λ). Also one can obtain a coset table indicating the natural permutation representation of Λ on the cosets of the normal subgroup Γ in each case. From the information provided, it is then a simple matter to find the orders of the images of particular elements of Λ in the quotient group G =Λ/Γ, and other properties of the epimorphism Λ → Λ/Γ. In particular, we find which normal subgroups Γ can be the kernel of a bordered smooth epimorphism θ :Λ→ Λ/Γ, and thereby obtain the list of all finite groups G =Λ/Γ of given order that act with signature σ(Λ) on the corresponding surface. For each such subgroup Γ we then determine the algebraic genus and other parameters of the corresponding action of G. In fact, rather than checking for those normal subgroups that are bordered surface groups, we can ensure in advance that we have a bordered surface group as kernel by forcing one of the generating reflections ci into the kernel. This is equivalent to killing that generator in the presentation of the NEC group, and gives rise to an even simpler presentation. As an illustration, for the signature σ1 = (0; +; [−]; (2, 2, 2, 3)), we note that at least one of the generating reflections {c0,c1,c2,c3} must belong to the kernel Γ. Neither c0 nor c3 can belong to Γ since c0c3 has order 3. So we may assume, after composing if necessary with the automorphism (c0,c1,c2,c3) → (c3,c2,c1,c0), that c1 ∈ Γ; so c2 ∈/ Γ. This leaves a group with just three generators a = θ(c0),c= θ(c2)andd = θ(c3) subject to the defining relations a2 = c2 = d2 =(cd)2 =(ad)3 =1. In particular, it follows that every M∗-group admits a presentation of the form a, c, d | a2,c2,d2, (cd)2, (ad)3,...,

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 31 where the remaining relators (indicated by dots) make the group finite. The same computational approach can be taken for the other signatures. For each normal subgroup Γ of Λ, the topological type of the bordered surface U/ΓonwhichG =Λ/Γ acts is easily determined as follows. The algebraic genus g is given by the Riemann-Hurwitz formula; the number k of boundary components can be found from a presentation of the normal subgroup Γ (obtainable using the Reidemeister-Schreier process), or more directly using the method described in [9, Section 2.3]; and the orientability ε is + if the image θ(Λ+) of the canonical Fuchsian subgroup Λ+ of Λ has index 2 in G,or− if that index is 1. As an illustration, we now consider signature σ6 = (0; +; [3]; {(2, 2)}). The NEC | 3 2 2 2 −1 2 group Λ = x, c0,c1 x = c0 = c1 =(c0c1) =(c1x c0x) =1 has five normal subgroups of index 336 which are the kernel of some bordered smooth epimorphism Λ → G, corresponding to an action of the quotient G on a surface of algebraic genus 57. The quotients via the normal subgroups are groups G1,G2,G3,G4 and G5 which act on surfaces of topological type (57, 6, +), (57, 6, +), (57, 24, +) (57, 42, +) and (57, 42, +) respectively. The groups G1 and G2 are isomorphic, and the same happens with G3 and G4. The actions of G3 and G4 cannot be equivalent, however, since they act on different topological types. So just the actions of G1 and G2 need to be checked for possible equivalence; in fact, they are equivalent under the −1 −1 outer automorphism of Λ that takes (x, c0,c1)to(x ,c0,xc1x ). Hence we get just four inequivalent actions, on surfaces of topological type (57, 6, +), (57, 24, +) (57, 42, +) and (57, 42, +). To enumerate actions up to equivalence, it is still necessary to take care of normal subgroups that are conjugate to at least one other by some automorphism α ∈ Aut Λ of the given NEC group Λ. The computation of Aut Λ is in general a rather difficult problem, but for the NEC groups Λ we are dealing with, this is an easy consequence of results in [5, Section 4]. Proposition 4.1. Every automorphism α ∈ Aut Λ of an NEC group Λ with signature σ1,σ2,...or σ7 is a composite α = ι◦ϕ,whereι is an inner automorphism of Λ, and either ϕ is the identity or

• ϕ :(c0,c1,c2,c3) → (c3,c2,c1,c0) if Λ has signature σ1,σ2,σ3 or σ4; • ϕ :(c0,c1,c2,c3) → (c2,c1,c0,c3) if Λ has signature σ5; −1 −1 −1 −1 −1 −1 • ϕ :(x, c0,c1) → (x ,c1,c0), (x ,xc0x ,x c1x) or (x, x c1x, xc0x ) if Λ has signature σ6; • → −1 ϕ :(x1,x2,c0) (x1,x2 ,x1c0x1) if Λ has signature σ7.

Remark 4.2. Suppose θ1,θ2 :Λ→ G are two smooth epimorphisms that are equivalent via α ∈ Aut Λ and β ∈ Aut G, so that θ2α = βθ1.Ifα = ια for some inner automorphism ι ∈ Aut Λ,thenθ2α = β θ1,whereβ = ι β for some inner automorphism ι ∈ Aut G. Indeed, if ι is conjugation by λ ∈ Λ,thenι is −1 conjugation by θ2(λ ). This means that, in order to classify epimorphisms, we may suppose that the inner automorphism ι in Proposition 4.1 is trivial. To illustrate how we deal with automorphisms of Λ, let us consider signa- { − } | 2 3 2 ture σ7 = (0; +; [2, 3]; ( ) ). An NEC group Λ = x1,x2,c0 x1 = x2 = c0 = [c0,x1x2]=1 has ﬁve normal subgroups of index 336 which are the kernel of some bordered smooth epimorphism Λ → G, corresponding to an action of the quotient G on a surface of algebraic genus 57. The epimorphism has to kill c0 and so a partial presentation of G is x, y | x2 = y3 = ··· =1. The quotients via the normal

32 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER subgroups are groups G1,G2,G3,G4 and G5 which act on surfaces of topological type (57, 42, +), (57, 42, +), (57, 28, +) (57, 28, +) and (57, 24, +) respectively. The groups G1 and G2 are isomorphic, and the same happens with G3 and G4. Both pairs of actions need to be checked for possible equivalence since each pair acts on the same topological type. It turns out that the assignment (x, y) → (x, y−1) yields an isomorphism between G3 and G4, but not between G1 and G2. According to Proposition 4.1, this is the only assignment that needs to be checked for possible equivalence between groups acting with signature σ7. Hencewegetjustoneaction on a surface of topological type (57, 28, +) and two inequivalent actions on surfaces of topological type (57, 42, +). If G is a group acting with signature σ1,σ2,σ3 or σ4, then as noted above, we may assume, after composing with the outer automorphism ϕ in Proposition 4.1, that the corresponding bordered smooth epimorphism θ :Λ→ G kills the generating reﬂection c1. With this assumption, no other outer automorphism needs to be considered for possible equivalence between groups acting with any of these signatures. For groups G acting with signature σ5, which have a partial presentation G = a, c, d | a2 = c2 = d2 =(cd)3 =(da)3 = ···=1 (with the second generating reﬂection b (= c1) being killed), we have to test for equivalence under the automorphism that takes (a, c, d)to(c, a, d). For groups G acting with signature σ6, which have a partial presentation G = x, c | x3 = c2 = ···=1, we check for equivalence under the automorphism taking (x, c)to(x−1,c).

5. Finite group actions of order 12(g − 1) Let S be a bordered surface of algebraic genus g ≥ 2 admitting a group G with 12(g − 1) automorphisms. We know that G admits the following (partial) presentation

a, c, d | a2,c2,d2, (cd)2, (ad)3,..., where the remaining relations make the group finite. The topological type of S can be obtained from these generators as follows. The number of its boundary components equals |G|/(2 ord(ac)), and S is orientable if and only if all the relations have an even number of letters a, c and d,see[12]. Observe that the same group G may have different sets of generators a,c,d with a presentation of the above form; sothesameabstractgroupG may act on surfaces of different topological types, as the next table shows. Moreover, G may also act on the same topological type but in topologically inequivalent ways. When this occurs we write the different sets of extra relations in the same row (using a, c, d instead of a,c,d), and point it out in the fourth column. On the other hand, when two non-isomorphic groups of the same order act on surfaces of the same topological type, these are given in separate rows of the table. In some cases, G can be easily identified as a well known group, such as Cn (cyclic group), Dn (dihedral group), An (alternating group) or Sn (symmetric group), or a direct product of two or more of these.

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 33

Bordered topological types (g, k, ε)withanM∗-group action for 2 ≤ g ≤ 101 (g, k, ε) |G| Extra relations Comments 2 ∼ (2, 3, +) 12 (ac) G = D6 2 ∼ (2, 1, +) 12 (ac) ad G = D6 3 ∼ (3, 4, +) 24 (ac) G = S4 4 ∼ (3, 3, −)24(ac) , acadacadc G = S4 2 2 ∼ (4, 3, +) 36 ((ac) d) G = S3 × S3 4 ∼ (5, 6, +) 48 (ac) G = C2 × S4 6 2 2 ∼ (5, 4, +) 48 (ac) , ((ac) ad) G = C2 × S4 5 2 2 ∼ (6, 6, −)60(ac) , (ac) ad(ac) dacad G = A5 (9, 6, +) 96 ((ac)2dacd)2 (9, 4, +) 96 ((ac)3d)2 (10, 9, +) 108 (ac)6, ((ac)2ad)3 5 ∼ (11, 12, +) 120 (ac) G = C2 × A5 2 3 2 2 ∼ (11, 6, −) 120 ((ac) d) or (ac) ad(ac) dacad Two actions of G = C2 × A5 6 2 4 ∼ (13, 12, +) 144 (ac) , ((ac) d) G = S3 × S4 4 2 2 2 ∼ (13, 6, +) 144 ((ac) d) , (ac) ad(ac) dacdacad G = S3 × S4 (17, 16, +) 192 (ac)6, (ac)2ad(ac)2adacadcacdacad (17, 12, +) 192 (ac)8, ((ac)2ad)3 4 2 ∼ (21, 12, +) 240 ((ac) ad) G = C2 × C2 × A5 (25, 12, +) 288 (ac)3ad(ac)2dacadcacad (25, 6, +) 288 ((ac)4d)2 (26, 25, +) 300 (ac)6, ((ac)2ad)5 (26, 15, +) 300 (cacada)3, (ac)10 (28, 27, +) 324 (ac)6, ((ac)2d)6 (28, 9, +) 324 ((ac)2ad)3, ((ac)6d)2 (29, 24, +) 336 (ac)7, (ac)2ad(ac)2adacadcacdacad (29, 21, −) 336 (ac)8, ((ac)2ad)4, (ac)4d(ac)3ad(ac)3ad Two actions or (ac)8, (ac)2ad(ac)2ad(ac)2dacad (31, 18, −) 360 (ac)10, ((ac)3d)3 (31, 12, +) 360 ((ac)5d)2, (ac)3ad(ac)3dacd(ac)2ad (33, 24, +) 384 (ac)8, ((ac)3d(ac)2d)2 (33, 16, +) 384 ((ac)2ad)4, (ac)4ad(ac)4adcacd (37, 36, +) 432 (ac)6, ((ac)2ad)6 (37, 18, +) 432 ((ac)2ad)3, (ac)12 (41, 12, +) 480 ((ac)5d)2, (ac)3ad(ac)2ad(ac)3dacdacad (43, 36, −) 504 (ac)7, (ac)2ad(ac)2ad(ac)2dacdacdacad (43, 28, −) 504 (ac)9, (ac)2ad(ac)2ad(ac)2dacad (49, 48, +) 576 (ac)6, ((ac)2d)8 (49, 24, +) 576 ((ac)2d)4, (ac)12 (49, 12, +) 576 ((ac)2ad)3, ((ac)8d)2 (50, 49, +) 588 (ac)6, ((ac)2ad)7 (50, 21, +) 588 ((ac)2ad)3, (ac)14 (55, 36, +) 648 (ac)9, ((ac)3d(ac)2ad)2 (55, 27, −) 648 ((ac)3d(ac)2d)2, (ac)12, ((ac)2ad)4acada (57, 42, +) 672 (ac)8, ((ac)3d)4 Two actions or (ac)8, ((ac)2ad)4 (57, 24, +) 672 ((ac)2ad)4, (ac)6dacd(ac)5ad (61, 45, −) 720 (ac)8, ((ac)2d)5 (61, 36, −) 720 (ac)10, ((ac)2ad)4, ((ac)2d)5 (61, 36, +) 720 (ac)10, (ac)3ad(ac)2ad(ac)3dacdacad (61, 12, +) 720 ((ac)5d)2, ((ac)2ad)5

34 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Bordered topological types (g, k, ε)withanM∗-group action for 2 ≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (65, 64, +) 768 (ac)6, ((ac)2ad)8 (65, 24, +) 768 ((ac)2ad)3, (ac)16 (65, 24, +) 768 ((ac)3d(ac)2d)2, (ac)3ad(ac)3dacd(ac)2dacd(ac)2ad (65, 16, +) 768 ((ac)3d(ac)2ad)2, ((ac)6d)2 (73, 36, +) 864 ((ac)5ad)2, (ac)3d(ac)2ad(ac)2dacdadcacdacdacd(ac)2d (73, 18, +) 864 (ac)4dadcacd(ac)3dacad, (ac)9d(ac)2dacd(ac)2d (76, 75, +) 900 (ac)6, ((ac)2d)10 (76, 15, +) 900 ((ac)2ad)3, ((ac)10d)2 (82, 81, +) 972 (ac)6, ((ac)2ad)9 (82, 27, +) 972 ((ac)2ad)3, (ac)18 (82, 27, +) 972 ((ac)6d)2, (ac)4ad(ac)4dacd(ac)3ad, ((ac)2d)6 (85, 72, +) 1008 (ac)7, (ac)2ad(ac)3d(ac)2d(ac)2dacd(ac)2d(ac)2d (85, 63, −) 1008 (ac)8, ((ac)2d)6, (ac)4d(ac)3dacd(ac)2dacd(ac)2ad (85, 56, +) 1008 (ac)9, (ac)4ad(ac)2ad(ac)4dacdacad (85, 36, −) 1008 (ac)2ad(ac)2ad(ac)2dacdacdacad, (ac)6dacd(ac)5ad Two actions or (ac)6dacd(ac)5ad, (ac)4d(ac)2ad(ac)2dadcacd(ac)2d (85, 28, −) 1008 (ac)6d(ac)2ad(ac)2d(ac)2d Two actions or (ac)2ad(ac)2ad(ac)2dacad, ((ac)4d(ac)3d)2 (85, 24, +) 1008 (ac)2ad(ac)2adacadcacdacad, (ac)3d(ac)3d(ac)3dacd(ac)2dacd (92, 78, −) 1092 (ac)7, ((ac)2d)7 (92, 42, −) 1092 (ac)2ad(ac)2ad(ac)2dacad, (ac)13, ((ac)5d)3 (97, 48, +) 1152 ((ac)5ad)2, ((ac)2d)8 (97, 48, +) 1152 (ac)12, (ac)4d(ac)2dacd(ac)3d(ac)2ad (97, 24, +) 1152 ((ac)2d)4 (97, 24, +) 1152 (ac)4dadcacd(ac)3dacad, ((ac)8d)2 (97, 24, +) 1152 (ac)4d(ac)2dacd(ac)3d(ac)2ad, (ac)9ad(ac)2d(ac)2dacad (101, 100, +) 1200 (ac)6, ((ac)2ad)10 (101, 30, +) 1200 ((ac)2ad)3, (ac)20 This list largely extends the one in [20], which gives the 32 topological types in 18 different genera g ≤ 40 with a group action of order 12(g − 1). The list here gives a finer classification, since we distinguish different topological actions in the same topological type. For instance, this occurs for topological types (11, 6, +) and (29, 21, −), with two inequivalent group actions of orders 120 and 336 respectively. Another difference with [20] is that an explicit presentation by generators and relations of each group is given here.

6. Finite group actions of order 8(g − 1)

We know that in this case, groups act with signature σ2 =(0; +; [−];{(2, 2, 2, 4)}) and admit the following partial presentation: a, c, d | a2,c2,d2, (cd)2, (ad)4,....

Bordered topological types (g, k, ε) with a group action of order 8(g − 1) for 2 ≤ g ≤ 101 (g, k, ε) |G| Extra relations Comments 2 ∼ (2, 2, −)8(ac) , adadc G = D4 ∼ (2, 1, +) 8 acad G = D4 2 ∼ (3, 4, +) 16 (ac) G = D4 × C2 2 4 ∼ (3, 2, +) 16 (acd) , (ac) G = D4 × C2 3 2 ∼ (4, 4, −)24(ac) , (adac) dG= S4

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 35

Bordered topological types (g, k, ε) with a group action of order 8(g−1) for 2≤g ≤101 (cont.) (g, k, ε) |G| Extra relations Comments (5, 4, +) 32 (acad)2, (ac)4 (5, 2, +) 32 (acad)2, (ac)3dacd (7, 8, +) 48 (ac)3 (7, 4, −)48(acd)3 or (adac)2d, (ac)6 Two actions (7, 4, +) 48 (acad)2, (ac)6 (7, 2, +) 48 (acad)2, (ac)5dacd (9, 8, +) 64 (ac)4, (acd)4 (9, 4, +) 64 (acad)2, (ac)8 (9, 4, +) 64 acd(ac)2dac (9, 2, +) 64 (acad)2, (ac)7dacd (10, 9, +) 72 (ac)4, (acad)3 (10, 6, −)72(ac)6, (acd)4, (ac)3d(ac)2ad (10, 4, −)72(ac)9, acadacdad (11, 4, +) 80 (ac)10, (acad)2 (11, 2, +) 80 (acad)2, (ac)9dacd (13, 8, +) 96 (ac)6, ((ac)2ad)2 (13, 4, −)96(ac)12, acadacdad or ((ac)2ad)2, (ac)3(acd)3 Two actions (13, 4, +) 96 (acad)2, (ac)12 (13, 2, +) 96 (acad)2, (ac)11dacd (15, 4, +) 112 (acad)2, (ac)14 (15, 2, +) 112 (acad)2, (ac)13dacd (16, 12, −) 120 (ac)5, (acad)2acdad (16, 10, −) 120 (ac)6, (acad)3, (acd)5 (16, 4, −) 120 acadacdad, (ac)15 (17, 16, +) 128 (ac)4, (acad)4 (17, 8, +) 128 (acd)4, (ac)8, ((ac)3ad)2 (17, 8, +) 128 (ac)2ad(acd)2ad, (ac)8 (17, 4, +) 128 (acad)2, (ac)16 (17, 4, +) 128 (ac)2ad(acd)2ad, (ac)6d(ac)2d (17, 2, +) 128 (acad)2, (ac)15dacd (19, 18, +) 144 (ac)4, (acd)6 (19, 12, +) 144 (ac)6, (acd)4 (19, 8, +) 144 ((ac)2ad)2, (ac)9 (19, 4, −) 144 acadacdad, (ac)18 or ((ac)2ad)2, (ac)6(acd)3 Two actions (19, 4, +) 144 (acad)2, (ac)18 (19, 2, +) 144 (acad)2, (ac)17dacd (21, 16, −) 160 (ac)5, (acd)5 (21, 4, +) 160 (acad)2, (ac)20 (21, 2, +) 160 (acad)2, (ac)19dacd (22, 4, −) 168 acadacdad, (ac)21 (23, 4, +) 176 (acad)2, (ac)22 (23, 2, +) 176 (acad)2, (ac)21dacd (25, 16, +) 192 (ac)6, ((ac)2dacd)2 (25, 16, −) 192 (ac)6, ((ac)2d)3 (25, 8, +) 192 ((ac)3d)2, (adac)4 (25, 8, +) 192 ((ac)2ad)2, (ac)12 (25, 8, +) 192 (ac)2ad(acd)2ad, (ac)12 (25, 4, −) 192 acadacdad, (ac)24 or ((ac)2ad)2, (ac)9(acd)3 Two actions (25, 4, +) 192 (acad)2, (ac)24 (25, 4, +) 192 (ac)2ad(acd)2ad, (ac)10d(ac)2d (25, 2, +) 192 (acad)2, (ac)23dacd

36 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Bordered topological types (g, k, ε) with a group action of order 8(g−1) for 2≤g ≤101 (cont.) (g, k, ε) |G| Extra relations Comments (26, 25, +) 200 (ac)4, (acad)5 (26, 10, −) 200 (acd)4, (ac)10, ((ac)4ad)2c (27, 4, +) 208 (acad)2, (ac)26 (27, 2, +) 208 (acad)2, (ac)25dacd (28, 18, −) 216 (ac)6, (acad)2acdad (28, 9, +) 216 (acad)3, ((ac)4d)2 (28, 4, −) 216 acadacdad, (ac)27 (29, 4, +) 224 (acad)2, (ac)28 (29, 2, +) 224 (acad)2, (ac)27dacd (31, 24, +) 240 (ac)5, (acd)6 (31, 20, +) 240 (ac)6, (acad)3 (31, 20, −) 240 (ac)6, (acd)5 or (ac)6, (ac)2adacd(ac)2dacd Two actions (31, 12, −) 240 (ac)3dacad(acd)2 or (acad)2acdad, (acd)6 Two actions (31, 8, +) 240 ((ac)2ad)2, (ac)15 (31, 4, −) 240 acadacdad, (ac)30 or ((ac)2ad)2, (ac)12(acd)3 Two actions (31, 4, +) 240 (acad)2, (ac)30 (31, 2, +) 240 (acad)2, (ac)29dacd (33, 32, +) 256 (ac)4, (acd)8 (33, 16, +) 256 (acd)4, (acd)8 (33, 16, +) 256 (ac)8, (ac)2adacadacd(ac)2d Two actions or (ac)8, ((ac)3ad)2, (acacd)2(acad)2 (33, 16, +) 256 ((ac)3ad)2, (acad)4, (ac)8 (33, 8, +) 256 (ac)2ad(acd)2ad, (ac)16 (33, 8, +) 256 ((ac)3ad)2, (acad)4, (ac)4(acd)4 (33, 4, +) 256 (acad)2, (ac)32 (33, 4, +) 256 (ac)2ad(acd)2ad, (ac)14d(ac)2d (33, 2, +) 256 (acad)2, (ac)31dacd (34, 4, −) 264 acadacdad, (ac)33 (35, 4, +) 272 (acad)2, (ac)34 (35, 2, +) 272 (acad)2, (ac)33dacd (37, 36, +) 288 (ac)4, (acad)6 (37, 12, +) 288 (acd)4, (ac)12, ((ac)5ad)2 (37, 8, +) 288 ((ac)2ad)2, (ac)18 (37, 4, −) 288 acadacdad, (ac)36 or ((ac)2ad)2, (ac)15(acd)3 Two actions (37, 4, +) 288 (acad)2, (ac)36 (37, 2, +) 288 (acad)2, (ac)35dacd (39, 4, +) 304 (acad)2, (ac)38 (39, 2, +) 304 (acad)2, (ac)37dacd (40, 4, −) 312 acadacdad, (ac)39 (41, 32, +) 320 (ac)5, (acad)4 (41, 16, −) 320 (ac)5, (acad)4 Two actions or (acad)4, (ac)3ad(acd)2acdad, (ac)10 (41, 8, +) 320 (ac)2ad(acd)2ad, (ac)20 (41, 4, +) 320 (acad)2, (ac)40 (41, 4, +) 320 (ac)2ad(acd)2ad, (ac)18d(ac)2d (41, 2, +) 320 (acad)2, (ac)39dacd

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 37

Bordered topological types (g, k, ε) with a group action of order 8(g−1) for 2≤g ≤101 (cont.) (g, k, ε) |G| Extra relations Comments (43, 28, −) 336 (ac)6, (acad)3acdad, ((ac)2ad)3 (43, 24, +) 336 (acad)3, (ac)7 (43, 21, −) 336 (ac)8, ((ac)3d)3, (acad)2acdad Two actions or (ac)8, ((ac)2ad)3, (acad)3acdad (43, 8, +) 336 ((ac)2ad)2, (ac)21 (43, 4, −) 336 acadacdad, (ac)42 Two actions or ((ac)2ad)2, (ac)18(acd)3 (43, 4, +) 336 (acad)2, (ac)42 (43, 2, +) 336 (acad)2, (ac)41dacd (45, 4, +) 352 (acad)2, (ac)44 (45, 2, +) 352 (acad)2, (ac)43dacd (46, 4, −) 360 acadacdad, (ac)45 (47, 4, +) 368 (acad)2, (ac)46 (47, 2, +) 368 (acad)2, (ac)45dacd (49, 32, +) 384 (ac)6, (acad)4, (acd)6 (49, 32, +) 384 (ac)6, (ac)2dacad(acd)3ad (49, 16, +) 384 ((ac)3d)2 (49, 16, +) 384 ((ac)3ad)2, (acad)4, (ac)12 (49, 16, +) 384 (acad)4, (ac)3adacd(ac)2dad, (ac)12 (49, 16, −) 384 ((ac)2d)3, (acad)4 Two actions or (acad)4, (ac)3ad(ac)2dacad, (ac)12 (49, 8, +) 384 ((ac)3d)2 (49, 8, +) 384 ((ac)2ad)2, (ac)24 (49, 8, +) 384 (ac)6, (ac)2dacad(acd)3ad (49, 8, +) 384 (acad)4, (ac)3adacdacadcad, (ac)9d(ac)3d (49, 8, +) 384 ((ac)3ad)2, (acad)4, (ac)12 (49, 8, +) 384 (ac)2ad(acd)2ad, (ac)24 (49, 4, −) 384 acadacdad, (ac)48 Two actions or ((ac)2ad)2, (ac)21(acd)3 (49, 4, +) 384 (acad)2, (ac)48 (49, 4, +) 384 (ac)2ad(acd)2ad, (ac)22d(ac)2d (49, 2, +) 384 (acad)2, (ac)47dacd (50, 49, +) 392 (ac)4, (acad)7 (50, 14, −) 392 (acd)4, (ac)5ad(ac)5dadacad (51, 50, +) 400 (ac)4, (acd)10 (51, 20, +) 400 (acd)4, (ac)10 (51, 4, +) 400 (acad)2, (ac)50 (51, 2, +) 400 (acad)2, (ac)49dacd (52, 4, −) 408 acadacdad, (ac)51 (53, 4, +) 416 (acad)2, (ac)52 (53, 2, +) 416 (acad)2, (ac)51dacd (55, 36, +) 432 (ac)6, ((ac)2dacad)2 (55, 36, +) 432 (ac)6, ((ac)2adacad)2 (55, 18, +) 432 ((ac)3ad)2, (acd)6 (55, 18, +) 432 ((ac)4d)2, (acd)6 (55, 8, +) 432 ((ac)2ad)2, (ac)27 (55, 4, +) 432 (acad)2, (ac)54 (55, 4, −) 432 acadacdad, (ac)54 or ((ac)2ad)2, (ac)24(acd)3 Two actions (55, 2, +) 432 (acad)2, (ac)53dacd

38 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Bordered topological types (g, k, ε) with a group action of order 8(g−1) for 2≤g ≤101 (cont.) (g, k, ε) |G| Extra relations Comments (57, 8, +) 448 (ac)2ad(acd)2ad, (ac)28 (57, 4, +) 448 (acad)2, (ac)56 (57, 4, +) 448 (ac)2ad(acd)2ad, (ac)26d(ac)2d (57, 2, +) 448 (acad)2, (ac)55dacd (58, 4, −) 456 acadacdad, (ac)57 (59, 4, +) 464 (acad)2, (ac)58 (59, 2, +) 464 (acad)2, (ac)57dacd (61, 40, +) 480 (ac)6, ((ac)2d)4, (ac)2(adac)2dac(da)2cad (61, 24, +) 480 (acd)6, (ac)10, ((ac)4ad)2 (61, 24, −) 480 (ac)3d(acd)2(ac)2ad Two actions or (acad)2acdad, (ac)10, ((ac)2d)4 (61, 20, +) 480 ((acad)3, ((ac)2d)4 or (ac)4dacad(ac)2d Two actions (61, 16, −) 480 (acad)4, (ac)3ad(acd)3ad, (ac)15 (61, 8, +) 480 ((ac)2ad)2, (ac)30 (61, 4, −) 480 acadacdad, (ac)60 or ((ac)2ad)2, (ac)27(acd)3 Two actions (61, 4, +) 480 (acad)2, (ac)60 (61, 2, +) 480 (acad)2, (ac)59dacd (63, 4, +) 496 (acad)2, (ac)62 (63, 2, +) 496 (acad)2, (ac)61dacd (64, 4, −) 504 acadacdad, (ac)63 (65, 64, +) 512 (ac)4, (acad)8 (65, 32, +) 512 (ac)8, ((ac)3ad)2, (acd)8 Two actions or (ac)8, ((ac)2dacad)2, ((ac)3dacd)2 (65, 32, +) 512 (ac)8, (acad)4, ((ac)3dacd)2 (65, 32, +) 512 (ac)8, ((ac)3ad)2, (ac)2ad(acd)6ad (65, 16, +) 512 ((ac)3ad)2, ((ac)2d)4 Two actions or ((ac)4d)2, ((ac)2dacad)2 (65, 16, +) 512 (acad)4, ((ac)4d)2 (65, 16, +) 512 ((ac)3ad)2, (acad)4, (ac)16 (65, 16, +) 512 (acd)4, (ac)6ad(ac)5dad(ac)3d (65, 16, +) 512 ((ac)2dacad)2, ((ac)3dacd)2, (ac)5adacdacadacd (65, 8, +) 512 (ac)2ad(acd)2ad, (ac)32 (65, 8, +) 512 ((ac)3ad)2, (acad)4, (ac)12(acd)4 (65, 4, +) 512 (acad)2, (ac)64 (65, 4, +) 512 (ac)2ad(acd)2ad, (ac)30d(ac)2d (65, 2, +) 512 (acad)2, (ac)63dacd (67, 8, +) 528 ((ac)2ad)2, (ac)33 (67, 4, −) 528 acadacdad, (ac)66 or ((ac)2ad)2, (ac)30(acd)3 Two actions (67, 4, +) 528 (acad)2, (ac)66 (67, 2, +) 528 (acad)2, (ac)65dacd (69, 4, +) 544 (acad)2, (ac)68 (69, 2, +) 544 (acad)2, (ac)67dacd (70, 4, −) 552 acadacdad, (ac)69 (71, 4, +) 560 (acad)2, (ac)70 (71, 2, +) 560 (acad)2, (ac)69dacd

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 39

Bordered topological types (g, k, ε) with a group action of order 8(g−1) for 2≤g ≤101 (cont.) (g, k, ε) |G| Extra relations Comments (73, 72, +) 576 (ac)4, (acd)12 (73, 36, +) 576 (ac)8, ((ac)3ad)2, (acad)6 (73, 36, +) 576 (ac)8, ((ac)3ad)2, (ac)2d(acad)5c (73, 24, +) 576 (ac)4, (acd)12 (73, 16, −) 576 (acad)4, (ac)3ad(ac)2dacad, (ac)18 (73, 16, +) 576 (acad)4, (ac)3adacda(cad)2, (ac)18 (73, 12, +) 576 ((ac)2dacad)2, ((ac)3dacd)2, ((ac)5ad)2 (73, 12, +) 576 ((ac)2dacad)2, ((ac)3dacd)2, (ac)3ad(ac)3(d(ac)2)2ad (73, 8, +) 576 ((ac)2ad)2, (ac)36 (73, 8, +) 576 (ac)2ad(acd)2ad, (ac)36 (73, 8, +) 576 (acad)4, (ac)3adacda(cad)2, (ac)15d(ac)3d (73, 4, −) 576 acadacdad, (ac)72 or ((ac)2ad)2, (ac)33(acd)3 Two actions (73, 4, +) 576 (acad)2, (ac)72 (73, 4, +) 576 (ac)2ad(acd)2ad, (ac)34d(ac)2d (73, 2, +) 576 (acad)2, (ac)71dacd (75, 4, +) 592 (acad)2, (ac)74 (75, 2, +) 592 (acad)2, (ac)73dacd (76, 4, −) 600 acadacdad, (ac)75 (77, 4, +) 608 (acad)2, (ac)76 (77, 2, +) 608 (acad)2, (ac)75dacd (79, 8, +) 624 ((ac)2ad)2, (ac)39 (79, 4, −) 624 acadacdad, (ac)78 or ((ac)2ad)2, (ac)36(acd)3 Two actions (79, 4, +) 624 (acad)2, (ac)78 (79, 2, +) 624 (acad)2, (ac)77dacd (81, 64, +) 640 (ac)5, ((ac)2dacdacad)2 (81, 32, +) 640 (acad)4, (ac)10, ((ac)4ad)2 (81, 32, +) 640 (ac)10, ((ac)4ad)2, (ac)3d(acad)2(ac)2d (81, 16, −) 640 (acad)4, (ac)3ad(acd)3ad, (ac)20 Two actions or (acad)4, ((ac)4ad)2, (ac)4dac(acd)3(ac)2d (81, 16, −) 640 (acd)5, ((ac)4ad)2 Two actions or ((ac)4ad)2, (ac)2(acd)2(ac)2adacdad (81, 16, +) 640 ((ac)3ad)2, (acad)4, (ac)20 (81, 8, +) 640 (ac)2ad(acd)2ad, (ac)40 (81, 8, +) 640 ((ac)3ad)2, (acad)4, (ac)16(acd)4 (81, 4, +) 640 (acad)2, (ac)80 (81, 4, +) 640 (ac)2ad(acd)2ad, (ac)38d(ac)2d (81, 2, +) 640 (acad)2, (ac)79dacd (82, 81, +) 648 (ac)4, (acad)9 (82, 36, −) 648 ((ac)2d)3, (ac)9 (82, 18, −) 648 (acd)4, (ac)7ad(ac)5dad(ac)3ad (82, 4, −) 648 acadacdad, (ac)81 (83, 4, +) 656 (acad)2, (ac)82 (83, 2, +) 656 (acad)2, (ac)81dacd (85, 56, −) 672 (ac)6, (acad)3acdad, (acd)8 Two actions or (ac)6, (ac)2a(dac)2a(dac)2dad (85, 56, +) 672 (ac)6, ((ac)2ad)3 (85, 48, +) 672 (ac)7, (ac)2(adac)2dacdadacad

40 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Bordered topological types (g, k, ε) with a group action of order 8(g−1) for 2≤g ≤101 (cont.) (g, k, ε) |G| Extra relations Comments (85, 42, −) 672 (ac)8, (acad)3acdad, (acd)6 Four actions or (ac)8, ((ac)2adacad)2, (acd)7 or (ac)8, (acd)6, (ac)4dacd(ac)3dad or (acad)2acdad, (ac)8, ((ac)2d(ac)2dacd)2 (85, 42, +) 672 (ac)8, ((ac)2ad)3 Two actions or (ac)8, ((ac)2adacad)2, (ac)3d(acd)3(ac)2ad (85, 24, +) 672 (ac)5dacad(ac)2d Two actions or (acad)3, ((ac)3d(ac)2d)2 (85, 8, +) 672 ((ac)2ad)2, (ac)42 (85, 4, −) 672 acadacdad, (ac)84 Two actions or ((ac)2ad)2, (ac)39(acd)3 (85, 4, +) 672 (acad)2, (ac)84 (85, 2, +) 672 (acad)2, (ac)83dacd (87, 4, +) 688 (acad)2, (ac)86 (87, 2, +) 688 (acad)2, (ac)85dacd (88, 4, −) 696 acadacdad, (ac)87 (89, 8, +) 704 (ac)2ad(acd)2ad, (ac)44 (89, 4, +) 704 (acad)2, (ac)88 (89, 4, +) 704 (ac)2ad(acd)2ad, (ac)42d(ac)2d (89, 2, +) 704 (acad)2, (ac)87dacd (91, 72, +) 720 (ac)5, (acad)5 (91, 60, −) 720 (ac)6, (ac)3d((ac)2ad)2 (91, 45, −) 720 (acd)5, (ac)8, (acad)4acdad (91, 36, −) 720 (acad)3acdad, ((ac)2ad)3, (ac)10, ((ac)2d)4 (91, 24, +) 720 (acd)6, ((ac)4ad)2, (ac)2ad((ac)2d)2acad (91, 18, +) 720 ((ac)3ad)2, (ac)4(acd)6 (91, 12, +) 720 ((ac)2dacad)2, (ac)6(acd)4 (91, 8, +) 720 ((ac)2ad)2, (ac)45 (91, 4, −) 720 acadacdad, (ac)90 or ((ac)2ad)2, (ac)42(acd)3 Two actions (91, 4, +) 720 (acad)2, (ac)90 (91, 2, +) 720 (acad)2, (ac)89dacd (93, 4, +) 736 (acad)2, (ac)92 (93, 2, +) 736 (acad)2, (ac)91dacd (94, 4, −) 744 acadacdad, (ac)93 (95, 4, +) 752 (acad)2, (ac)94 (95, 2, +) 752 (acad)2, (ac)93dacd (97, 64, +) 768 (ac)6, (acad)4, ((ac)2ad)4 (97, 64, +) 768 (ac)6, (acd)6, ((ac)2ad)4 (97, 64, +) 768 (ac)6, (ac)3dacadacd(ac)2dacdad (97, 32, +) 768 (acd)6, (ac)4d(acad)2(ac)2d (97, 32, +) 768 (acad)4, (acd)6, (ac)12, ((ac)4adac)2 (97, 32, +) 768 (acad)4, (ac)4ad(acd)4ad, (ac)12 (97, 32, +) 768 (ac)4d(acad)2(ac)2d, (ac)4ad(acd)4ad (97, 32, +) 768 (ac)3adacd(ac)2dad, (ac)12 (97, 32, +) 768 (ac)2dacad(acd)3ad, (ac)12 (97, 32, +) 768 (acad)4, (ac)4ad(ac)2dacdacad, (ac)12 (97, 32, +) 768 ((ac)3ad)2, (ac)12, (ac)2dacdacad(acd)4ad

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 41

Bordered topological types (g, k, ε) with a group action of order 8(g−1) for 2≤g ≤101 (cont.) (g, k, ε) |G| Extra relations Comments (97, 16, −) 768 (ac)4d(ac)3adacad Two actions or ((ac)2d)3, ((ac)5ad)2 (97, 16, −) 768 (acad)4, (ac)3ad(ac)2dacad, (ac)24 Two actions or (acad)4, (ac)4ad(acd)4ad, (ac)8d((ac)2d)2 (97, 16, +) 768 (acad)4, (ac)3adacd(ac)2dad, (ac)24 (97, 16, +) 768 (acad)4, (ac)4adac(acd)2acad, ((ac)6d)2 (97, 16, +) 768 (ac)2dacad(acd)3ad, (ac)8dacd(ac)2dacd (97, 16, +) 768 (ac)3adacdacacdad, ((ac)6d)2 (97, 16, +) 768 ((ac)3ad)2, (acad)4, (ac)24 (97, 16, +) 768 ((ac)3ad)2,ac(acd)2acad(acd)4ad, (ac)6d((ac)2d)3 Two actions or ((ac)2dacad)2, (ac)4adacd(ac)3dad, (ac)6d(ac)3d(ac)2dacd (97, 16, +) 768 ((ac)2dacad)2, (ac)4adacd(ac)3dad, (ac)8(acd)4 (97, 8, +) 768 ((ac)2ad)2, (ac)48 (97, 8, +) 768 (ac)2ad(acd)2ad, (ac)48 (97, 8, +) 768 ((ac)3ad)2, (acad)4, (ac)20(acd)4 (97, 8, +) 768 (acad)4, (ac)3adacda(cad)2, (ac)21d(ac)3d (97, 4, −) 768 acadacdad, (ac)96 Two actions or ((ac)2ad)2, (ac)45(acd)3 (97, 4, +) 768 (acad)2, (ac)96 (97, 4, +) 768 (ac)2ad(acd)2ad, (ac)46d(ac)2d (97, 2, +) 768 (acad)2, (ac)95dacd (99, 98, +) 784 (ac)4, (acd)14 (99, 28, +) 784 (acd)4, (ac)14 (99, 4, +) 784 (acad)2, (ac)98 (99, 2, +) 784 (acad)2, (ac)97dacd (100, 4, −) 792 acadacdad, (ac)99 (101, 100, +) 800 (ac)4, (acad)10 (101, 20, +) 800 (acd)4, (ac)7ad(ac)3adacad(ac)5ad (101, 16, −) 800 (acad)4, (ac)3ad(acd)2adcad, (ac)25 (101, 4, +) 800 (acad)2, (ac)100 (101, 2, +) 800 (acad)2, (ac)99dacd

20 − 7. Finite group actions of order 3 (g 1)

Here the groups act with signature σ3 = (0; +; [−]; {(2, 2, 2, 5)}) and admit the partial presentation

a, c, d | a2,c2,d2, (cd)2, (ad)5,....

20 − ≤ ≤ Bordered topological types (g, k, ε) with a group action of order 3 (g 1) for 2 g 101 (g, k, ε) |G| Extra relations Comments 2 ∼ (4, 5, +) 20 (ac) G = C2 × D5 (4, 1, +) 20 (acd)2, (ac)4ad 3 ∼ (10, 10, −)60(ac) , acadadacadacdad G = A5 3 5 ∼ (10, 6, −)60(acd) , (ca) G = A5 (16, 5, +) 100 ((ac)2d)2 3 ∼ (19, 20, +) 120 (ac) G = C2 × A5 4 2 ∼ (19, 15, −) 120 (ac) , acacd(acad) G = S5 5 3 ∼ (19, 12, +) 120 (ac) , (acad) G = C2 × A5

42 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

20 − ≤ ≤ Bordered topological types (g, k, ε) with a group action of order 3 (g 1) for 2 g 101 (cont.) (g, k, ε) |G| Extra relations Comments 6 4 2 2 ∼ (19, 10, −) 120 (ac) , (acd) , (ac) (ad) acadacd G = S5 6 2 2 3 ∼ (19, 10, −) 120 (ac) , ((ac) ad) ,acad(acd) ad Two actions of G = C2 × A5 or (ac)6, ((ac)2ad)2, acadacdadacadad 3 3 ∼ (19, 6, −) 120 (acd) or (ac) adacdad Two actions of G = C2 × A5 (25, 20, −) 160 (ac)4, (acd)5 (25, 16, +) 160 (ac)5, (acd)4 (37, 30, +) 240 (ac)4, (acd)6 (37, 20, +) 240 (ac)6, (acd)4 (37, 20, +) 240 (ac)6, ((ac)2ad)2 (37, 12, +) 240 (ac)3dacad(ac)2d (49, 40, +) 320 (ac)4, (acad)4 (49, 32, +) 320 (ac)5, (ac)2(da)2cadacdacad (49, 20, −) 320 (acd)5, (ac)8, ((ac)3ad)2 Two actions or (ac)8, ((ac)3ad)2, (ac)2ad(acd)3ad (49, 16, +) 320 (ac)3ad(acd)2ad (49, 16, +) 320 (acd)4, ((ac)2d)2((ad)2c)2 (73, 40, +) 480 (ac)6, (ac)2d(ac)2(adac)2dad (73, 30, +) 480 (ac)8, ((ac)3ad)2, (acd)6 (73, 20, +) 480 ((ac)3d)2, (ac)2dacd(acad)3 (73, 12, +) 480 ((ac)2dacd)2, ((ac)2ad)3 (76, 25, +) 500 ((ac)2dacad)2, (ac)10, (ac)4adac(ad)2acad (91, 60, +) 600 (ca)5, (acd)6, ((ac)2dadacad)2 (91, 50, −) 600 (ac)6, (acd)5, ((ac)3dacad)2 (91, 30, −) 600 (ac)2(dac)2(acd)2, (ac)10, (ac)3ad(ac)2(adac)2d (91, 20, +) 600 ((ac)3d)2, (acad)5 (97, 80, +) 640 (ac)4, (acad(acd)2ad)2 (97, 40, +) 640 (ac)8, ((ac)3ad)2, (acad)4 (97, 40, +) 640 (ac)8, ((ac)3ad)2, ((ac)2d)2(acad)2 (97, 32, +) 640 (ac)10, ((ac)4ad)2, ((ac)2d)2((ad)2c)2 (97, 16, +) 640 (acd)4, ((ac)2adacad)2 Two actions or ((ac)3(ad)2)2, ((ac)2d)4, ((acad)2ad)2 (100, 66, −) 660 (ca)5, (acd)5 (100, 55, −) 660 (ac)6, (acd)6, (ac)3d((ac)2ad)2, (acad)5

8. Finite group actions of order 6(g − 1) There are four diﬀerent signatures with which these groups may act. If G acts with signature σ4 = (0; +; [−]; {(2, 2, 2, 6)} then G admits the partial presentation

a, c, d | a2,c2,d2, (cd)2, (ad)6,....

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (2, 2, 2, 6) for 2 ≤ g ≤ 101 (g, k, ε) |G| Extra relations Comments 2 3 ∼ (3, 3, −)12(ac) , (ad) cG= D6 3 ∼ (3, 2, +) 12 (ac) ,acadad G= D6 2 ∼ (5, 6, +) 24 (ac) G = C2 × D6 (5, 2, +) 24 (acd)2, (ac)6 (7, 6, +) 36 (ac)3,acd(ad)2cad (7, 3, −)36(acad)2, (acd)3

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 43

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (2, 2, 2, 6) for 2 ≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (9, 8, +) 48 (ac)3,acad(acd)2ad (9, 4, +) 48 (ac)2dadacd, (ac)6 (9, 6, −)48(ac)4, (acd)3 or (ac)4, (ac)2dacad Two actions (9, 6, +) 48 (ac)4, (acad)2 (9, 2, +) 48 (ac)2dacacd, (acdad)2, (ac)4(ad)2 (11, 3, −)60(acad)2, (ac)2(acd)3 (11, 2, +) 60 (acdad)2, (ac)4dacd (13, 9, −)72(ac)4, (acd)4, (acad)3c (13, 6, +) 72 (acad)2, (ac)6 or (acdad)2, (ac)6, (acacad)2 Three actions or (acacd)2, (ac)6 (15, 3, −)84(acad)2, (ac)4(acd)3 (15, 2, +) 84 (acdad)2, (acacad)2, (ac)5d(ac)2d (17, 12, +) 96 (ac)4, (ac(ad)2)2 (17, 8, +) 96 (ac)6, (acacad)2, (ac(ad)2)2 (17, 6, +) 96 (ac)3(da)2cd (17, 6, +) 96 (acad)2, (ac)8 (17, 4, +) 96 cacdacadacad (17, 2, +) 96 (acdad)2, (acacad)2, (ac)7dacd (19, 18, +) 108 (ac)3, (acd)6 (19, 9, −) 108 cacdacdad, (ac)6 (19, 6, +) 108 (acdad)2, (acacad)2, (ac)9 (19, 3, −) 108 (acad)2, (ac)6(acd)3 (21, 15, −) 120 (ac)4, (acad)3, (acd)5 (21, 12, −) 120 (ac)5, (acd)4, (ac)2(ad)2acdacad (21, 12, −) 120 (ac)5, (acadad)2, (ac)2(dac)3ad Two actions or (ac)5, (acadad)2, (ac)2ad(ac)2dacad (21, 10, −) 120 (ac)6, (ac)3d(ac)2ad, ac(ad)2acd(ad)2 (21, 6, +) 120 (acad)2, (ac)10 (21, 6, −) 120 (acadad)2, (ac)4d(ad)2c, (acacd)3 Two actions or (acadad)2, (ac)4d(ad)2c, (ac)2ad(ac)2dacad (21, 2, +) 120 (acdad)2, (acacad)2, (ac)8d(ac)2d (23, 3, −) 132 (acad)2, (ac)8(acd)3 (23, 2, +) 132 (acdad)2, (acacad)2, (ac)10dacd (25, 18, +) 144 (ac)4, (acd)4 (25, 6, +) 144 (acad)2, (ac)12 or (acdad)2, (acacad)2, (ac)12 Two actions (25, 6, +) 144 (acacd)2 (25, 24, +) 144 (ac)3, (acdad)4 (25, 6, −) 144 (acd)3, ((ac)3ad)2 (27, 3, −) 156 (acad)2, (ac)10(acd)3 (27, 2, +) 156 (acdad)2, (acacad)2, (ac)11d(ac)2d (29, 6, +) 168 (acad)2, (ac)14 (29, 2, +) 168 (acdad)2, (acacad)2, (ac)13dacd (31, 6, +) 180 (acdad)2, (acacad)2, (ac)15 (31, 3, −) 180 (acad)2, (ac)12(acd)3 (33, 32, +) 192 (ac)3, (acd)6adacad (33, 24, +) 192 (ac)4, (acad)4, ((acd)2ad)2 (33, 24, −) 192 (ac)4, (acdad)3 (33, 16, +) 192 (ac)6, (ac)2dacadacdad (33, 16, +) 192 (ac)6, (acad)3, (acd)4

44 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (2, 2, 2, 6) for 2 ≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (33, 12, +) 192 (ac)2(da)2c(ad)2c (33, 12, −) 192 (acd)3, (ac)8 (33, 12, +) 192 (acadad)2, (ac)8, ((ac)3ad)2 (33, 8, +) 192 (acadad)2, ((ac)3d)2 (33, 8, +) 192 (acacad)2, (acd)4 (33, 6, +) 192 (acad)2, (ac)16 (33, 2, +) 192 (acdad)2, (acacad)2, (ac)14d(ac)2d (35, 3, −) 204 (acad)2, (ac)14(acd)3 (35, 2, +) 204 (acdad)2, (acacad)2, (ac)16dacd (37, 27, −) 216 (ac)4, (ac)2d(acad)2 (37, 18, +) 216 (ac)6, (acadacd)2 (37, 18, +) 216 (ac)6, (ac)2adac(ad)2acad Three actions or (ac)6, (acacad)2, (ac(ad)2)2cacdad or (ac)6, (acadad)2, (ac)2ad(ac)2dad(ca)2d (37, 9, −) 216 acadadacdadad, (acacd)3 or (acd)4, (acacd)3 Two actions (37, 9, −) 216 (ac)2(ad)2(cad)2, (acacd)3 (37, 6, +) 216 (acad)2, (ac)18 or (acdad)2, (acacad)2, (ac)18 Two actions (39, 3, −) 228 (acad)2, (ac)16(acd)3 (39, 2, +) 228 (acdad)2, (acacad)2, (ac)17d(ac)2d (41, 30, +) 240 (ac)4, (acad)3 (41, 30, −) 240 (ac)4, (acd)5 or (ac)4, acadadacdadcadacd Two actions (41, 24, +) 240 (ac)5, (acd)4 (41, 24, +) 240 (ac)5, (ac(ad)2)2 (41, 20, +) 240 (ac)6, ((ac)2(ad)2)2, (ac)2dacdad(acd)2 (41, 20, −) 240 (ac)6, acadadacdadad or (ac)6, (ac)2acd(ac)2ad Two actions (41, 12, −) 240 (acd)4, (ac)2(ad)2acdacad or (acd)4, (acacd)2(ad)2c Two actions (41, 12, +) 240 (acadad)2, (ac)4d(ad)2c (41, 12, −) 240 (acadad)2, (acacd)3 or (acadad)2, (ac)2ad(acd)2(ac)2d Four actions or (acadad)2, (ac)2(dac)3ad or (acadad)2, (ac)2ad(ac)2dacad (41, 8, +) 240 (acacad)2, (ac)2dac(ad)2(ac)2d (41, 6, +) 240 (acad)2, (ac)20 (41, 6, −) 240 (ac)3d(ac)2(ad)2 (41, 2, +) 240 (acdad)2, (acacad)2, (ac)19dacd (43, 6, +) 252 (acdad)2, (acacad)2, (ac)21 (43, 3, −) 252 (acad)2, (ac)18(acd)3 (45, 6, +) 264 (acad)2, (ac)22 (45, 2, +) 264 (acdad)2, (acacad)2, (ac)20d(ac)2d (47, 3, −) 276 (acad)2, (ac)20(acd)3 (47, 2, +) 276 (acdad)2, (acacad)2, (ac)22dacd (49, 24, +) 288 (ac)6, (acadad)2, (acacd)4 or (ac)6, (acacad)2, (acdad)4 Two actions (49, 18, +) 288 (ac)2ad(acd)2ad, (ac)8 (49, 12, +) 288 (acadad)2, ((ac)4d)2, (ac)2ad(ac)2dad(ca)2d or ((ac)3ad)2, ((ac)2dacd)2, (acad)4 Two actions (49, 12, +) 288 (acadad)2, (ac)5dadacd, ((ac)3dacad)2 (49, 6, +) 288 (acad)2, (ac)24 or (acdad)2, (acacad)2, (ac)24 Two actions (49, 6, +) 288 (acadad)2, ((ac)4d)2, (acacad)3 (49, 6, +) 288 (acadacd)2, ((ac)4d)2

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 45

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (2, 2, 2, 6) for 2 ≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (51, 50, +) 300 (ac)3, (acd)8adacad (51, 15, −) 300 (acd)3, (ac)10 (51, 3, −) 300 (acad)2, (ac)22(acd)3 (51, 2, +) 300 (acdad)2, (acacad)2, (ac)23d(ac)2d (53, 6, +) 312 (acad)2, (ac)26 (53, 2, +) 312 (acdad)2, (acacad)2, (ac)25dacd (55, 54, +) 324 (ac)3, (acdad)6 (55, 27, −) 324 (ac)6, (ac)2(dac)2(acd)2, ((ac)2ad)3c Two actions or (ac)6, ((acd)2ad)2, (ac)3d((ac)2ad)2 (55, 18, +) 324 ((ac)3d)2, (ac)9, (acadacdad)2 Two actions or (acacad)2, (ac)9, (acadad)3 (55, 18, +) 324 (acacad)2, (ac)9, (acd)4adacad (55, 18, +) 324 (acacad)2, (ac)9, (acadad)2cacdad (55, 9, −) 324 (ac)2adaca(da)2cad, (ac)6(acd)3 (55, 9, −) 324 (acd)3, ((ac)5ad)2 (55, 9, −) 324 (ac)4adacdad (55, 6, +) 324 (acdad)2, (acacad)2, (ac)27 (55, 3, −) 324 (acad)2, (ac)24(acd)3 (57, 42, −) 336 (ac)4, (acad)4c, (acadad)3 (57, 28, −) 336 (ac)6, (acacd)3, (acdad)3 (57, 24, −) 336 ac(ad)3c(ad)2, (ac)7 (57, 21, −) 336 (acd)4, (ac)8, (ac)3(ad)2(ac)2(ad)2c, (ac)2(daca)2(da)2cad Two actions or (acad)3, (acacd)3 (57, 8, +) 336 (acacad)2, (ac)3(acd)4 (57, 6, −) 336 ((ac)3ad)2, (acad)4, ((acd)2ad)2, (ac)4(acd)3 (57, 6, +) 336 (acad)2, (ac)28 (57, 2, +) 336 (acdad)2, (acacad)2, (ac)26d(ac)2d (59, 3, −) 348 (acad)2, (ac)26(acd)3 (59, 2, +) 348 (acdad)2, (acacad)2, (ac)28dacd (61, 12, −) 360 (acacd)3, (ac)3(da)2c(ad)2c (61, 9, −) 360 (ac)2(ad)2(cad)2, (ac)4((ac)2d)3 (61, 6, +) 360 (acadacd)2, (ac)8d(ac)2d (61, 6, +) 360 (acad)2, (ac)30 or (acdad)2, (acacad)2, (ac)30 Two actions (63, 3, −) 372 (acad)2, (ac)28(acd)3 (63, 2, +) 372 (acdad)2, (acacad)2, (ac)29d(ac)2d (65, 48, +) 384 (ac)4, ((ac)2dacdad)2 (65, 48, +) 384 (ac)4, (acad)4, (acd)6 (65, 32, +) 384 (ac)6, (acd)4, (acacd)4 (65, 32, +) 384 (ac)6, (acacad)2, (acd)4adac(ad)2acad Two actions or (ac)6, (acadad)2, (ac)2ad(ac)2d(acd)3acad (65, 32, +) 384 (ac)6, (ac)2adac(da)3cacd Two actions or (ac)6, (acad)3, ((ac)2(da)2cd)2 (65, 24, +) 384 (acadad)2, (ac)8, (ac)2ad(ac)2dad(ca)2d Two actions or (ac)8, ((ac)2dacd)2, ((acd)2ad)2 (65, 24, +) 384 (acadad)2, (ac)8, (acacad)3 (65, 24, +) 384 (ac)8, ((ac)3ad)2, (ac)2(ad)2(acd)2(ad)2 (65, 24, +) 384 (ac)8, ((ac)3ad)2, (acad)4, ((acd)2ad)2 (65, 24, −) 384 (ac)8, ((ac)3ad)2, (acad)4, (acd)6, (ac)2(ad)2acdad(acd)2 Two actions or (acdad)3, (ac)8, ((ac)3ad)2

46 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (2, 2, 2, 6) for 2 ≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (65, 16, +) 384 (acadad)2, (ac)5dadacd, (ac)2ad(ac)2(dac)4ad (65, 16, +) 384 (ac)3adacad(acd)2 (65, 8, +) 384 (acacad)2, (acacd)4 (65, 8, +) 384 (acd)4, (ac)4(da)2c(ad)2c (65, 6, +) 384 (acad)2, (ac)32 (65, 2, +) 384 (acdad)2, (acacad)2, (ac)31dacd (67, 6, +) 396 (acdad)2, (acacad)2, (ac)33 (67, 3, −) 396 (acad)2, (ac)30(acd)3 (69, 6, +) 408 (acad)2, (ac)34 (69, 2, +) 408 (acdad)2, (acacad)2, (ac)32d(ac)2d (71, 3, −) 420 (acad)2, (ac)32(acd)3 (71, 2, +) 420 (acdad)2, (acacad)2, (ac)34dacd (73, 72, +) 432 (ac)3, (acd)10adacad (73, 54, +) 432 (ac)4, ((acad)2ad)2 (73, 54, +) 432 (ac)4, (acadacdad)2 (73, 24, +) 432 (acacad)2, (ac)9, (acdad)4 (73, 18, +) 432 ((ac)3ad)2, ((ac)2(ad)2)2 or (acd)4, ((ac)3ad)2 Two actions (73, 18, +) 432 (ac)2(ad)2(cad)2, (ac)12 (73, 18, +) 432 (acd)4, ((ac)4d)2 (73, 18, +) 432 (ac)4(da)2c(ad)2c, (ac)2(ad)3(cad)2ad, ((ac)2dacad)2 (73, 18, +) 432 (acadacd)2, (ac)12 (73, 18, +) 432 (ac)2(adacad)2, (ac)12 or (acacad)2, (acadad)2cacdad, (ac)12 Two actions (73, 18, −) 432 (acd)3, (ac)12 (73, 6, +) 432 (acad)2, (ac)36 or (acdad)2, (acacad)2, (ac)36 Two actions (73, 6, −) 432 ((ac)3ad)2, (acad)4, ((acd)2ad)2, ((ac)3d)3 (75, 3, −) 444 (acad)2, (ac)34(acd)3 (75, 2, +) 444 (acdad)2, (acacad)2, (ac)35d(ac)2d (77, 6, +) 456 (acad)2, (ac)38 (77, 2, +) 456 (acdad)2, (acacad)2, (ac)37dacd (79, 6, +) 468 (acdad)2, (acacad)2, (ac)39 (79, 3, −) 468 (acad)2, (ac)36(acd)3 (81, 60, +) 480 (ac)4, (acdad)4, ((ac)2dadacad)2 (81, 48, +) 480 (ac)5, (ac)2(daca)2(da)2dcad (81, 40, +) 480 (ac)6, ((ac)2(ad)2)2 (81, 30, −) 480 (ac)8, ((ac)3ad)2, (ac)2ad(acd)3ad Two actions or (acd)5, (ac)8, ((ac)3ad)2 (81, 24, +) 480 (ac)3ad(acd)2ad (81, 24, +) 480 (acd)4, (acad)4, (ac)10 (81, 24, +) 480 (acadad)2, (ac)10, ((ac)4ad)2 (81, 20, −) 480 ((ac)2(ad)2)2, (ac)4(da)2c(da)2cd Two actions or ac(ad)2acd(ad)2, ((ac)3dacd)2 (81, 12, +) 480 ((ac)3ad)2, (acad)4, ((acd)2ad)2, (ac)5d(ac)2(dac)2acd (81, 12, −) 480 (acacd)3, (ac)2(ad)2(acd)2(ad)2 Two actions or (ac)3d(ac)2adacad, (ac)2(ad)2(acd)2(ad)2 (81, 12, +) 480 (acadad)2, ((ac)3ad)2cacd (81, 8, +) 480 (acacad)2, (acdad)4, (ac)6(acd)4 (81, 6, +) 480 (acad)2, (ac)40 (81, 2, +) 480 (acdad)2, (acacad)2, (ac)38d(ac)2d

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 47

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (2, 2, 2, 6) for 2 ≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (83, 3, −) 492 (acad)2, (ac)38(acd)3 (83, 2, +) 492 (acdad)2, (acacad)2, (ac)40dacd (85, 9, −) 504 (ac)2(ad)2(cad)2, (ac)10(dacac)2d (85, 6, +) 504 (acdacad)2, (ac)10d(ac)4d (85, 6, +) 504 (acad)2, (ac)42 or (acdad)2, (acacad)2, (ac)42 Two actions (87, 3, −) 516 (acad)2, (ac)40(acd)3 (87, 2, +) 516 (acdad)2, (acacad)2, (ac)41d(ac)2d (89, 8, +) 528 (acacad)2, (acdad)4, (ac)3(acacd)4 (89, 6, −) 528 ((ac)3ad)2, (acad)4, ((acd)2ad)2, (ac)8(acd)3 (89, 6, +) 528 (acad)2, (ac)44 (89, 2, +) 528 (acdad)2, (acacad)2, (ac)43dacd (91, 18, +) 540 (acacad)2, (acadad)2cacdad, (ac)15 (91, 9, −) 540 (ac)2adac(ad)2acad, (ac)12(acd)3 (91, 6, +) 540 (acdad)2, (acacad)2, (ac)45 (91, 3, −) 540 (acad)2, (ac)42(acd)3 (93, 6, +) 552 (acad)2, (ac)46 (93, 2, +) 552 (acdad)2, (acacad)2, (ac)44d(ac)2d (95, 3, −) 564 (acad)2, (ac)44(acd)3 (95, 2, +) 564 (acdad)2, (acacad)2, (ac)46dacd (97, 96, +) 576 (ac)3, (acdad)8 (97, 48, +) 576 (ac)6, (acacad)3, (acadacdad)2, (acd)6 Two actions or (ac)6, ((ac)2dacad)2, (acadad)3, (acd)6 (97, 48, −) 576 (ac)6, (acad)4, (acd)6, (ac)3d(ac)2ad(ac)2ad (97, 48, +) 576 (ac)6, (ac)2(ad)2(ac)2(acd)2, (acad)2(ad)2c(ad)2c(ad)2 Two actions or (ac)6, (ac)2(ad)3(cad)2ad, ((ac)3dacad)2 (97, 24, +) 576 ((ac)3d)2, (ac)2(da)2c(daca)3d, (acad)2(ad)2c(ad)2c(ad)2 (97, 24, +) 576 (acacad)2, (acdad)4, (ac)12 Two actions or ((ac)3ad)2, (acad)4, ((acd)2ad)2, (ac)12 (97, 24, +) 576 (acadad)2, (ac)3ad(ac)2d(ac)2dacad (97, 24, +) 576 ((ac)4d)2, (ac)2(ad)3(cad)2ad, ((ac)2dacad)2 Two actions or (ac)4(da)2c(ad)2c, (ac)2(ad)3(cad)2ad, (acacd)4 (97, 18, +) 576 (ac)2(ad)2(cad)2, (ac)16 (97, 12, +) 576 ((ac)3ad)2, ((ac)2dacd)2 (97, 12, +) 576 ((ac)3ad)2, (ac)3dac(ad)2(ac)2d Two actions or ((ac)4d)2, (ac)2ad(ac)2dad(ca)2d (97, 12, +) 576 (acadad)2, ((ac)4d)2 (97, 12, −) 576 (acd)3, ((ac)7ad)2 (97, 6, +) 576 (acad)2, (ac)48 or (acdad)2, (acacad)2, (ac)48 Two actions (97, 6, +) 576 (acdacad)2, (ac)14d(ac)2d (99, 98, +) 588 (ac)3, (acd)12adacad (99, 21, −) 588 cacdacdad, (ac)14 (99, 3, −) 588 (acad)2, (ac)46(acd)3 (99, 2, +) 588 (acdad)2, (acacad)2, (ac)47d(ac)2d (101, 60, −) 600 (ac)5, (acd)5,ac(ad)3c(ad)2c(ad)2acad (101, 50, +) 600 (ac)6, (acadad)2, (ac)2ad(ac)2(dac)6ad Two actions or (ac)6, (acacad)2, (acd)5(adac)2(da)2cad (101, 30, +) 600 (ac)2(dac)2(acd)2, ((acd)2ad)2, (ac)10 Two actions or (acadad)2, (ac)2ad(ac)2(dac)2ad, (ac)10 (101, 6, +) 600 (acad)2, (ac)50 (101, 2, +) 600 (acdad)2, (acacad)2, (ac)49dacd

48 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

If G acts with signature σ5 =(0;[−]; {(2, 2, 3, 3)})thenG admits the partial presentation a, c, d | a2,c2,d2, (cd)3, (ad)3,....

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (2, 2, 3, 3) for 2 ≤ g ≤ 101 (g, k, ε) |G| Extra relations Comments ∼ (2, 3, +) 6 ac G = D3 ∼ (2, 1, +) 6 acad, acdc G = D3 (4, 3, +) 18 (ac)3, (acd)2 2 ∼ (5, 6, +) 24 (ac) G = S4 3 2 ∼ (5, 4, +) 24 (ac) , (acad) G = S4 (9, 6, +) 48 (ac)4, ((ac)2d)2 (9, 4, +) 48 acadcacd (10, 9, +) 54 (ac)3, (acad)3 3 ∼ (11, 6, −)60(acd) G = A5 (13, 12, +) 72 (ac)3, (acd)4 (13, 6, +) 72 ((ac)2d)2, (ac)6 (17, 16, +) 96 (ac)3, (acad)4 (17, 12, +) 96 (ac)4, (acad)3 (21, 12, +) 120 (ac)5, ((ac)2ad)2 (25, 12, +) 144 (ac)6, ((ac)3d)2, (ac)3(acd)4 (25, 6, +) 144 ((ac)2d)2 (26, 25, +) 150 (ac)3, (acad)5 (26, 15, +) 150 (ac)5, (acad)3 (28, 27, +) 162 (ac)3, (acd)6 (28, 9, +) 162 (acad)3, ((ac)3d)2 (28, 9, +) 162 acad(cacd)2 (29, 21, −) 168 (ac)4, ((ac)2d)3 (33, 24, +) 192 (ac)4,a(cad)2c(acd)2 (33, 16, +) 192 (ac)6, ((ac)3d)2, (acad)4 (37, 36, +) 216 (ac)3, (acad)6 (37, 18, +) 216 (ac)6, (acad)3 (41, 12, +) 240 (ac)2adc(ac)2d, (ac)2dcacdc(acad)2 (49, 48, +) 288 (ac)3, (acd)8 (49, 24, +) 288 (ac)6, (acd)4 (49, 12, +) 288 (acad)3, ((ac)4d)2 (50, 49, +) 294 (ac)3, (acad)7 (50, 21, +) 294 (acad)3, (ac)7 (57, 42, +) 336 (ac)4, (acad)4 (57, 42, +) 336 (ac)4, (acadcacd)2 (57, 24, +) 336 (ac)7, ((ac)3ad)2, (acad)4 (61, 45, −) 360 (ac)4, (acd)5 (61, 36, +) 360 (ac)5, (ac(dac)2)2 (61, 36, −) 360 (ac)5, (acd)5 (61, 12, +) 360 ((ac)2ad)2 or (ac)2adc(ac)2d, (acd)6 Two actions (65, 64, +) 384 (ac)3, (acad)8 (65, 24, +) 384 (acad)3, (ac)8 (65, 24, +) 384 a(cad)2c(acd)2, (ac)8, ((ac)4d)2 (65, 16, +) 384 ((ac)3d)2, (acad)2cacdcacd (73, 54, +) 432 (ac)4, ((ac)2dacad)2 (73, 36, +) 432 (ac)6, (ac)2ad(ca)2d(ac)2d

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 49

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (2, 2, 3, 3) for 2 ≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (73, 36, +) 432 (ac)6, ((ac)3d)2, (acad)3cadcdacdcacd (73, 18, +) 432 (ac)2adacda(cad)2, (ac)4ad(cacd)2 (76, 75, +) 450 (ac)3, (acd)10 (76, 15, +) 450 (acad)3, ((ac)5d)2 (82, 81, +) 486 (ac)3, (acad)9 (82, 27, +) 486 (acad)3, (ac)9 (82, 27, +) 486 ((ac)3d)2, (ac)9, (acd)6 (85, 36, −) 504 (ac)7, ((ac)3ad)2,acadcacd(cad)2acadc (85, 28, −) 504 (ac)2adcacd(cad)2c (97, 48, +) 576 (ac)6, (ac)2(adc)2(ac)2dacd, (acad(ca)2d)2 (97, 48, +) 576 (ac)6, ((ac)3d)2, (acd)8 (97, 24, +) 576 (acd)4 (97, 24, +) 576 (ac)2(adc)2(ac)2dacd, (acad(ca)2d)2, (ac)4ad(acd)3c (97, 24, +) 576 (ac)2adacda(cad)2, ((ac)4d)2 (101, 100, +) 600 (ac)3, (acad)10 (101, 30, +) 600 (acad)3, (ac)10

Every group G acting with signature σ6 = (0; +; [3]; {(2, 2)}) admits the partial presentation G = x, c | x3,c2,.... In order to describe the extra relations which make the group ﬁnite, it is more convenient to work with the generators u = xc and v = x2c, which gives

G = u, v | (vu−1)3, (vu−1v)2,....

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (0; +; [3]; {(2, 2)})for2≤ g ≤ 101 (g, k, ε) |G| Extra relations Comments ∼ (2, 3, +) 6 uv G = C6 2 ∼ (2, 1, +) 6 u G = D3 3 ∼ (3, 3, −)12u G = A4 (4, 3, +) 18 u2v2 2 ∼ (5, 6, +) 24 (uv) G = C2 × A4 4 ∼ (5, 4, +) 24 u G = S4 (8, 3, +) 42 u6,v2u2vu (9, 6, +) 48 u3v3 (9, 4, +) 48 (u2v)2 (10, 9, +) 54 u6, (uv)3 5 ∼ (11, 6, −)60u G = A5 (13, 6, +) 72 u6,u2vuv2uv (13, 12, +) 72 (uv)3,u4v4 (14, 3, +) 78 u6,v2u2vuvu (17, 16, +) 96 (uv)3,u8 (17, 12, +) 96 u6, (uv)4 (20, 3, +) 114 u6,u2vuv2uvuv 3 2 ∼ (21, 12, +) 120 (u v) G = C2 × A5 (22, 3, +) 126 u6,u2v2uvuvuv (25, 12, +) 144 u4v4 (25, 6, +) 144 u2vuv2uv (26, 25, +) 150 (uv)3,u10 (26, 15, +) 150 u6, (uv)5

50 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (0; +; [3]; {(2, 2)})for2≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (28, 27, +) 162 (uv)3,u6v6 (28, 9, +) 162 u6,u2vuvuv2uvuv (29, 21, −) 168 u7, (uv)4 (29, 6, +) 168 u6,u2v2u2vuvuv2 (31, 18, −) 180 (u2v)3,u5v5 (32, 3, +) 186 u6,u2v2uvuvuvuv (33, 24, +) 192 (uv)4, (u4v)2 (33, 16, +) 192 u8,u3vuv3uv (37, 36, +) 216 (uv)3,u12 (37, 18, +) 216 u6, (uv)6 (38, 3, +) 222 u6,u2v2u2vuuvuvuv (40, 3, +) 234 u6,u2vuv2uvuvuvuv (41, 12, +) 240 u5v5, (u2vuv)2 (44, 3, +) 258 u6,u2v2uvuvuvuvuv (49, 48, +) 288 (uv)3,v6u6v2u2 (49, 24, +) 288 (u2v2)2,u12 (49, 12, +) 288 u6,u2v2u2vuvuv2uvuv (50, 49, +) 294 (uv)3,u14 (50, 21, +) 294 u6, (uv)7 (50, 3, +) 294 u6,u2vuvuv2uvuvuvuv (53, 6, +) 312 u6,v2u2v2u2vuvuvuvu (55, 27, −) 324 u9,u2v2uv2u2v (57, 42, +) 336 (uv)4, (u5v)2 (57, 42, +) 336 u8, (uv)4 (57, 24, +) 336 u8,u3v2uv3uv2 (57, 6, +) 336 (u4v)2,v2u2v2u2vuvu (58, 3, +) 342 u6,u2v2uvuvuvuvuvuv (61, 36, +) 360 u5v5, (uv)5 (61, 12, +) 360 u10, (u2vuv)2 (62, 3, +) 366 u6,u2v2u2vuvuv2u2v2uv (64, 9, +) 378 u6,v2u2v2u2vuvuv2u2vu (65, 64, +) 384 (uv)3,u16 (65, 24, +) 384 u6, (uv)8 (65, 24, +) 384 u2vu2v2uv2,u6v6 (65, 16, +) 384 (u3v2)2, (u2v)4 (68, 3, +) 402 u6,u2vuv2uvuvuvuvuvuv (73, 36, +) 432 u8,u2vuuv2uv2uv (73, 36, +) 432 u3vuv3uv, v7u2vu2v2 (73, 18, +) 432 (u4v)2,u2v3u2vuvuvuv (74, 3, +) 438 u6,u2v2uvuvuvuvuvuvuv (76, 75, +) 450 (uv)3,u8v4u2v6 (76, 15, +) 450 u6,u2v2u2vuvuvuv2uvuvuv (77, 6, +) 456 u6,u2v2u2vuvuv2uvuvuvuv (80, 3, +) 474 u6,v2u2vuv2uvu2vuvuvuvu (82, 81, +) 486 (uv)3,u18 (82, 27, +) 486 u6, (uv)9 (82, 27, +) 486 u6v6, (uv3)3,u2vuvuv2uvuv (82, 27, +) 486 u4v4u2v2

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 51

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (0; +; [3]; {(2, 2)})for2≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (85, 63, −) 504 (uv)4, (u2v2)3,u3v3u3vu2v (85, 36, −) 504 u9, (u3vuv)2 (85, 28, −) 504 u7,u2vu2vuv2uv2uv (85, 12, +) 504 u12, (u3v)3,v3u3v2uvu2 (85, 6, +) 504 u6,u2v2u2v2uvuvuvuvuvuv (92, 3, +) 546 u6,u2v2u2vuvuv2u2v2uvuvuv (92, 3, +) 546 u6,v2u2vuvuvuvuvuvuvuvu (94, 3, +) 558 u6,v2u2vuvuv2u2vuv2uvu2vu (97, 48, +) 576 u3v2uv3u2v, (uv)6 (97, 48, +) 576 u3vuv3uv, u8v8 (97, 24, +) 576 (u4v)2,u2vuvuvuv2uvuvuv (97, 24, +) 576 (u2v2)2 (97, 24, +) 576 u12,u3v2uv3u2v (98, 3, +) 582 u6,u2v2u2vuv2uvuvuvuvuvuv (101, 100, +) 600 (uv)3,u20 (101, 30, +) 600 u6, (uv)10

A bordered smooth epimorphism θ :Λ→ G from an NEC group Λ with signature σ7 = (0; +; [2, 3]; {(−)}) onto a ﬁnite group G has to map the canonical reﬂection c0 onto the identity. Since c0 is the unique orientation reversing element amongst the canonical generators of Λ, it follows that the surface U/ker θ is orientable, see [9, Theorem 2.1.3]. In other words, groups associated to signature σ7 always act on bordered orientable surfaces. Every group G acting with signature σ7 admits the following partial presentation G = x, y | x2,y3,.... Taking the alternative generators u = xy and v = xy2, we have

G = u, v | (uv−1u)2, (u−1v)3,....

This presentation is more convenient to describe the extra relations which make the quotient ﬁnite.

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (0; +; [2, 3]; {(−)})for2≤ g ≤ 101 (g, k, ε) |G| Extra relations Comments 2 ∼ (2, 3, +) 6 u G = D3 ∼ (2, 1, +) 6 uv G = C6 3 ∼ (3, 4, +) 12 u G = A4 (4, 3, +) 18 u2v2 4 ∼ (5, 6, +) 24 u G = S4 2 ∼ (5, 4, +) 24 (uv) G = C2 × A4 (8, 7, +) 42 u6,u2v2uv (9, 6, +) 48 (u2v)2 (9, 4, +) 48 u3v3 (10, 9, +) 54 u6, (uv)3 5 ∼ (11, 12, +) 60 u G = A5 (13, 12, +) 72 u6,u2vuv2uv (13, 6, +) 72 (uv)3,u4v4 (14, 13, +) 78 u6,u2v2uvuv

52 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (0; +; [2, 3]; {(−)})for2≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (17, 16, +) 96 u6, (uv)4 (17, 12, +) 96 (uv)3,u8 (20, 19, +) 114 u6,u2vuvuv2uv 3 2 ∼ (21, 12, +) 120 (u v) G = C2 × A5 (22, 21, +) 126 u6,v2u2vuvuvu (25, 12, +) 144 u2vuv2uv (25, 6, +) 144 u4v4 (26, 25, +) 150 u6, (uv)5 (26, 15, +) 150 (uv)3,u10 (28, 27, +) 162 u6,u2vuvuv2uvuv (28, 9, +) 162 (uv)3,u6v6 (29, 28, +) 168 u6,u2v2u2v2uvuv (29, 24, +) 168 u7, (uv)4 (31, 12, +) 180 (u2v)3,u5v5 (32, 31, +) 186 u6,u2v2uvuvuvuv (33, 24, +) 192 u8,u3vuv3uv (33, 16, +) 192 (uv)4, (u4v)2 (37, 36, +) 216 u6, (uv)6 (37, 18, +) 216 (uv)3,u12 (38, 37, +) 222 u6,u2v2u2vuv2uvuv (40, 39, +) 234 u6,u2vuvuvuvuv2uv (41, 12, +) 240 u5v5, (u2vuv)2 (44, 43, +) 258 u6,v2u2vuvuvuvuvu (49, 48, +) 288 u6,u2v2uvuvu2vuvuv2 (49, 24, +) 288 (u2v2)2,u12 (49, 12, +) 288 (uv)3,u6v6u2v2 (50, 49, +) 294 u6, (uv)7 (50, 49, +) 294 u6,u2vuvuvuvuv2uvuv (50, 21, +) 294 (uv)3,u14 (53, 52, +) 312 u6,u2v2u2v2uvuvuvuv (55, 36, +) 324 u9,u2v2uv2u2v (57, 42, +) 336 u8, (uv)4 or u8,u3vu2v3u2v Two actions (57, 28, +) 336 (u4v)2,u2v2u2v2uvuv (57, 24, +) 336 (uv)4, (u5v)2 (58, 57, +) 342 u6,u2v2uvuvuvuvuvuv (61, 36, +) 360 u10, (u2vuv)2 (61, 12, +) 360 u5v5, (uv)5 (62, 61, +) 366 u6,u2v2uvuvu2vuvuvuv2 (64, 63, +) 378 u6,u2v2u2v2uvuvu2v2uv (65, 64, +) 384 u6, (uv)8 (65, 24, +) 384 (uv)3,u16 (65, 24, +) 384 (u3v2)2, (u2v)4 (65, 16, +) 384 u2v2uv2u2v, u6v6 (68, 67, +) 402 u6,u2vuvuvuvuvuvuv2uv (73, 54, +) 432 u8,u2v2uv2uvu2v (73, 36, +) 432 (u4v)2,u3v2uvuvuvuv2 (73, 18, +) 432 u3vuv3uv, u9v2uv2 (74, 73, +) 438 u6,v2u2vuvuvuvuvuvuvu (76, 75, +) 450 u6,u2v2u2vuvuvuv2uvuvuv (76, 15, +) 450 (uv)3,u8v6u2v4

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 53

Bordered topological types (g, k, ε) on which a group G of order 6(g − 1) acts with signature (0; +; [2, 3]; {(−)})for2≤ g ≤ 101 (cont.) (g, k, ε) |G| Extra relations Comments (77, 76, +) 456 u6,u2v2u2vuvuvuvuv2uvuv (80, 79, +) 474 u6,u2v2u2vuvuvuvuvuv2uv (82, 81, +) 486 u6, (uv)9 (82, 27, +) 486 u6v6, (u3v)3,u2vuvuv2uvuv (82, 27, +) 486 u4v4u2v2 (82, 27, +) 486 (uv)3,u18 (85, 72, +) 504 u7,u2vu2vuv2uv2uv (85, 56, +) 504 u9, (u3vuv)2 (85, 84, +) 504 u6,v2u2v2u2vuvuvuvuvuvu (85, 42, +) 504 u12, (u3v)3,u3v3u2vuv2 (85, 24, +) 504 (uv)4, (u2v2)3,u3v3u3vu2v (92, 91, +) 546 u6,u2v2u2vuvuvuv2u2v2uvuv (92, 91, +) 546 u6,u2v2uvuvuvuvuvuvuvuv (94, 93, +) 558 u6,u2v2u2vuvuvuvuvuv2uvuv (97, 48, +) 576 (u4v)2,u2vuvuvuv2uvuvuv (97, 48, +) 576 u12,u3v2uv3u2v (97, 24, +) 576 (u2v2)2 (97, 24, +) 576 u3v2uv3u2v, (uv)6 (97, 24, +) 576 u3vuv3uv, u8v8 (98, 97, +) 582 u6,u2v2uvu2v2uvuvu2v2uvuv (101, 100, +) 600 u6, (uv)10 (101, 30, +) 600 (uv)3,u20

9. Infinite families An interesting problem in group theory is to find families of M∗-groups and study their properties, see the survey [7]. Using the results in the previous sections, it is not hard to identify some common patterns in the list in Section 5 and find infinite families of M∗-groups to which the above ones belong. Their existence can be proved with the help of Reidemeister-Schreier theory (see [15]). For example, in the case of M*-groups, if we add the relation (ac)6 = 1 to the group presentation a, c, d | a2,c2,d2, (cd)2, (da)3 , we find easily that the normal subgroup of index 12 generated by the two commutators [cd, ad]and[cd, da]isfree abelian of rank 2. Hence adding the further relations [cd, ad]n =[cd, da]n =1(or indeed just one of these, since dc = dc), we obtain a quotient of order 12n2 which is an extension of Cn× Cn by D6. It follows that for every positive integer n,there exists an M*-group of order 12n2 acting on topological type (1+n2,n2, +); see also [21]. Other families can be found in the same way from the computational data. For instance, let us focus on families of groups of order 8(g−1). Here we consider smooth quotients of the extended Hecke group C2 ∗ C4, which has three generators a, c, d satisfying the relations a2 = c2 = d2 =(cd)2 =(da)4 =1. If we add the relation acadacdad = 1 then we find that in the resulting group, the subgroup generated by 3 (ac) is normal, of index 24, with quotient S4. Reidemeister-Schreier theory shows that this subgroup is free (of rank 1), and hence infinite cyclic. Adding the further relation (ac)3n = 1 therefore gives a quotient of order 24n, which is an extension of Cn by S4. It follows that for every positive integer n, there exists a group of order

54 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

24n acting on topological type (3n +1, 4, −). Adding this and other extra relations to a2 = c2 = d2 =(cd)2 =(da)4 = 1, we obtain the following families. Infinite families of groups of order 8(g − 1)acting on bordered topological type (g, k, ε) (g, k, ε) |G| Extra relations (2n +1, 4, +) 16n (acad)2, (ac)2n (2n +1, 2, +) 16n (acad)2, (ac)2n−1dacd (3n +1, 4, −)24n acadacdad, (ac)3n (6n +1, 8, +) 48n ((ac)2ad)2, (ac)3n (6n +1, 4, −)48n ((ac)2ad)2, (ac)3n−3(acd)3 (8n +1, 8, +) 64n (ac)2ad(acd)2ad, (ac)4n (8n +1, 4, +) 64n (ac)2ad(acd)2ad, (ac)4n−2d(ac)2d (16n +1, 16, +) 128n ((ac)3ad)2, (acad)4, (ac)4n (16n +1, 8, +) 128n ((ac)3ad)2, (acad)4, (ac)4n−4(acd)4 In contrast with M∗-groups, very few things are known about families of groups of order 8(g − 1). An important feature of these groups is the following, see [8]. Let S be a surface on which such a group G acts. Let us write S = Sc/τ where Sc is an unbordered orientable surface and τ : Sc → Sc is a symmetry. Then, except for a finite number of (known) surfaces S, the full group Aut (Sc) of conformal and anticonformal automorphisms of Sc viewed as a Riemann surface is isomorphic to the direct product G ×τ, where G is the group or order 8(g − 1) acting on S. This has important applications, as the computation of the symmetry type of the ∗ surfaces Sc (see [4] for the case where G is an M -group). Let ν(g) be the order of the largest (finite) group acting on a compact bordered surface of genus g ≥ 2. May showed in [18]thatν(g) ≥ 4(g − 1) for all g and that the equality holds for infinitely many values of g. The existence of groups of order 16n acting on bordered surfaces of genus g =2n + 1 for all values of n (see the above list) shows that the above value of ν(g)canbeimprovedtoν(g) ≥ 8(g − 1) for odd g. Infinite families of groups of order 6(g − 1) can also be found from the computational data. For instance, we have the following: Infinite families of groups of order 6(g − 1) acting on bordered topological type (g, k, ε) (g, k, ε) |G| Relations (4n +1, 6, +) 24na2,c2,d2, (cd)2, (da)6, (acad)2(ac)2k (4n +3, 3, −) 12(2n +1) a2,c2,d2, (cd)2, (da)6, (acad)2, (ac)2n−2(acd)3 (6n +1, 6, +) 36na2,c2,d2, (cd)2, (da)6, (acdad)2, (acacad)2, (ac)3n (6n − 1, 2, +) 12(3n − 1) a2,c2,d2, (cd)2, (da)6, (acdad)2, (acacad)2, (ac)3n−2dacd (6n − 3, 2, +) 12(3n − 2) a2,c2,d2, (cd)2, (da)6, (acdad)2, (acacad)2, (ac)3n−4d(ac)2d (3n2 +1, 3n2, +) 18n2 a2,c2,d2 =(cd)3, (da)3, (ac)3, (acd)2n (n2 +1,n2, +) 6n2 a2,c2,d2, (cd)3, (da)3, (ac)3, (acad)n (n2 + n +2, 3, +) 6(n2 +n+1) x3,c2, (xc)6, (xc)2(x2c)2(xcx2c)n−1 (n2 +n+2,n2 +n+1, +) 6(n2 +n+1) x2,y3, (xy)6, (xy)2(xy2)2(xyxy2)n−1

The first five groups act with signature σ4 = (0; +; [−]; {(2, 2, 2, 6)}), the next two with signature σ5 = (0; +; [−]; {(2, 2, 3, 3)}), and the last two act with signatures σ6 = (0; +; [3]; {(2, 2)})andσ7 = (0; +; [2, 3]; {(−)}) respectively. To finish, let us mention that another important feature of surfaces admitting group actions of large order is their relation with regular maps: there is a bijection between bordered surfaces of genus g with 12(g − 1) automorphisms and regular

FINITE GROUP ACTIONS OF LARGE ORDER ON COMPACT BORDERED SURFACES 55 maps of type {3,q}, and also between bordered surfaces of genus g with 8(g − 1) automorphisms and regular maps of type {4,q}, see [13, Section 6].

References [1] N. L. Alling, N. Greenleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Math., 219, Springer-Verlag, 1971. MR0333163 (48:11488) [2] W. Bosma, J. Cannon, C. Playoust, The Magma Algebra System I: The User Language, J. Symbolic Comput. 24 (1997), 235–265. MR1484478 [3]S.A.Broughton,Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69, (1991), 233–270. MR1090743 (92b:57021) [4]E.Bujalance,F.J.Cirre,M.D.E.Conder,Riemann surfaces with maximal real symmetry, in preparation. [5] E. Bujalance, F. J. Cirre, M. D. E. Conder, B. Szpietowski, Finite group actions on bordered surfaces of small genus, J. Pure Appl. Algebra 214 (2010), no. 12, 2165–2185. MR2660907 (2011i:30037) [6]E.Bujalance,F.J.Cirre,J.J.Etayo,G.Gromadzki,E.Mart´ınez, A survey on the minimum genus and maximum order problem for bordered Klein surfaces, Groups St Andrews 2009 in Bath, Vol 1, 161–182, London Math. Soc. Lecture Notes Series 387, Cambridge Univ. Press, 2011. [7] E. Bujalance, F. J. Cirre, P. Turbek, Groups acting on bordered Klein surfaces with maximal symmetry, Groups St. Andrews 2001 in Oxford. Vol. I, 50–58, London Math. Soc. Lecture Note Ser., 304, Cambridge Univ. Press, Cambridge, 2003. MR2051517 [8] E. Bujalance, A. F. Costa, G. Gromadzki, D. Singerman, Automorphism groups of complex doubles of Klein surfaces, Glasgow Math. J. 36 (1994), no. 3, 313–330. MR1295507 (95j:30040) [9] E. Bujalance, J. Etayo, J. M. Gamboa and G. Gromadzki, Automorphism groups of compact bordered Klein surfaces, Lecture Notes in Maths. 1439, Springer Verlag, Berlin 1990. MR1075411 (92a:14018) [10] E. Bujalance, E. Mart´ınez, A remark on NEC groups representing surfaces with boundary, Bull. London Math. Soc., 21, (1989), no. 3, 263–266. MR986369 (90a:20094) [11] M. Conder and P. Dobcsányi, Applications and adaptations of the low index subgroups procedure,Math.Comp.74, (2005) 485–497. MR2085903 (2005e:20046) [12] J. J. Etayo, Klein surfaces with maximal symmetry and their groups of automorphisms, Math. Ann., 268, (1984), 533–538. MR753412 (86g:30058) [13] N. Greenleaf, C. L. May, Bordered Klein surfaces with maximal symmetry, Trans. Amer. Math. Soc. 274 (1982), no. 1, 265–283. MR670931 (84f:14022) [14] A. Hurwitz, Uber¨ algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann., 41, (1893), 403–442. MR1510753 [15] R. C. Lyndon, P. E. Schupp, Paul E. Combinatorial group theory, Ergebnisse der Mathe- matik und ihrer Grenzgebiete, Band 89. Springer-Verlag, Berlin-New York, 1977. xiv+339 pp. MR0577064 (58:28182) [16] A. M. Macbeath, The classification of non-euclidean plane crystallographic groups, Canad. J. Math., 19, (1967), 1192–1205. MR0220838 (36:3890) [17] C. L. May, Automorphisms of compact Klein surfaces with boundary, Pacific J. Math. 59 (1975), no. 1, 199–210. MR0399451 (53:3295) [18] C. L. May, A bound for the number of automorphisms of a compact Klein surface with boundary, Proc. Amer. Math. Soc. 63 (1977), no. 2, 273–280. MR0435385 (55:8345) [19] C. L. May, Large automorphism groups of compact Klein surfaces with boundary I, Glasgow Math. J. 18 (1977), no. 1, 1–10. MR0425113 (54:13071) [20] C. L. May, The species of bordered Klein surfaces with maximal symmetry of low genus, Pacific J. Math. 111 (1984), no. 2, 371–394. MR734862 (85d:30067) [21] C. L. May, AfamilyofM ∗-groups, Canad. J. Math. 38 (1986), no. 5, 1094–1109. MR869715 (87m:20114) [22] H. A. Schwarz, Ueber diejenigen algebraischen Gleichungen zwischen zwei veränderlichen Grössen, welche eine Schaar rationaler eindeutig umkehrbarer Transformationen in sich selbst zulassen, J. reine und angew. Math. 87, (1897, 139–145.)

56 E. BUJALANCE, F. J. CIRRE, AND M. D. E. CONDER

Departamento de Matematicas´ Fundamentales, Facultad de Ciencias, UNED, c/ Senda del Rey, 9, 28040 Madrid, Spain E-mail address: [email protected] Departamento de Matematicas´ Fundamentales, Facultad de Ciencias, UNED, c/ Senda del Rey, 9, 28040 Madrid, Spain E-mail address: [email protected] Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11358

Surfaces of low degree containing a canonical curve

Izzet Coskun

Abstract. In this paper, we generalize a classical theorem of del Pezzo [D] and Fujita [F1] and a recent theorem of Casnati [Ca] concerning low degree surfaces containing a canonical curve Ccan. Every canonical curve of genus g is contained in a surface of degree less than or equal to 2g−3. We study canonical curves that are contained in a surface of degree smaller than 2g − 3. Fix an integer κ ≥−1. Our main theorem is that if g>3κ+12and(g, κ) =(10 , −1), then the minimal degree surface S containing Ccan has degree d = g + κ if and → κ+3 only if κ is odd and C is a double cover f : C B of a curve B of genus 2 . In this case, S is the image of X = P(f∗KC ) under the linear system |OX (1)|. Our methods also apply to curves embedded with complete linear systems of high degree.

1. Introduction Let C be a smooth, projective, non-hyperelliptic curve of genus g over the g−1 complex numbers C. The canonical linear system |KC | embeds C in P as a non-degenerate curve Ccan of degree 2g − 2. The extrinsic geometry of Ccan closely reﬂects the intrinsic geometry of C. In this paper, we explore the implications of the existence of a low degree surface containing Ccan for the intrinsic geometry of C.

Let p be a point on Ccan.LetS be the surface obtained by taking the cone over Ccan with vertex at p.ThenS has degree 2g − 3. Therefore, every canonical curve is contained in a surface of degree less than or equal to 2g − 3. Ciliberto and Harris [CH]provethatifg ≥ 23 and C is general in moduli, then a minimal degree surface X containing Ccan has degree 2g − 3andX is a cone over Ccan with vertex on Ccan. The purpose of this paper is to study the geometry of canonical curves that are contained in surfaces of degree less than 2g − 3. Let X be an irreducible, non-degenerate variety of degree d and dimension r in Pn. Then the invariants d, r and n satisfy the inequality d + r − 1 ≥ n. Varieties for which d + r − 1 − n is small have been classiﬁed by Bertini, Castelnuovo, del Pezzo, Fujita, Ionescu, Livorni among many others (see [Be], [D], [EH], [F1], [F2], [F3], [Io], [L1], [L2]). For example, a classical theorem of Bertini and del Pezzo

2010 Mathematics Subject Classiﬁcation. Primary 14N25; Secondary 14H51, 14H30, 14N05. Key words and phrases. Canonical curve, low degree surface. During the preparation of this article the author was partially supported by the NSF grant DMS-0737581, the NSF-CAREER grant DMS-0950951535, and an Alfred P. Sloan Foundation Fellowship.

c 2012 American Mathematical Society 57

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[EH] asserts that irreducible, non-degenerate varieties that satisfy d + r − 1=n are quadric hypersurfaces, the Veronese surface in P5, rational normal scrolls and cones over these varieties.

Since Ccan is non-degenerate, any surface containing Ccan is also non-degenerate and has degree at least g − 2. If Ccan iscontainedinasurfaceS of degree g − 2, then, by the classiﬁcation of varieties of minimal degree [EH], S is either a rational normal scroll or the Veronese surface in P5. Moreover, by the Babbage-Enriques- Petri Theorem ([ACGH, p.124, 131]), C is either trigonal or isomorphic to a plane quintic curve. In particular, if g>6, then C is trigonal.

If the minimal degree surface S containing Ccan has degree g − 1, then S is either a del Pezzo surface (more precisely, the anti-canonical image of a quadric surface in P3 or a blow-up of P2 in less than or equal to 5 possibly inﬁnitely near points) or a cone over an elliptic normal curve of degree g −1inPg−2 [F1]. If S is a del Pezzo surface, then C is birational to a complete intersection (2, 4) in P3 or to a planesexticcurvewith10− g possibly inﬁnitely near double points. Otherwise, C is bi-elliptic, that is, C admits a two-to-one map to an elliptic curve. In particular, if g>10, then C is bi-elliptic.

Casnati [Ca] proves that if the minimal degree surface S containing Ccan has degree g,then7≤ g ≤ 12. In these cases, S is a conic bundle over P1 whose fibers 1 cut out the unique g4 on C. In particular, if g>12, then Ccan is not contained in a surface of degree g. These examples suggest that if the genus is large enough, the existence of a surface of low degree containing Ccan implies that C is a small degree cover of a curve of low genus. Our first theorem makes this precise. Theorem 1.1. Fix κ ≥−1. Assume that the minimal degree surface S containing a canonical curve Ccan of genus g has degree d = g + κ<2g − 3 and that (g, κ) =(10 , −1). (1) If κ is even, then g ≤ 3κ +12. (2) If κ is odd, then either g ≤ 3κ +12 or C admits a two-to-one map f : → κ+3 C B to a curve B of genus 2 .Ifg>3κ +12,thenS is the image of the ruled surface X = P(f∗KC ) over B under the linear system |OX (1)|. κ+3 (3) Conversely, if C is a double cover of a curve of genus 2 ,thenCcan is contained in a surface of degree g + κ. Remark 1.2. When κ = −1, we recover the theorem of del Pezzo [D]andFujita [F1]thatacurveofg>10 which lies on a surface of degree g − 1 is bi-elliptic. When κ = 0, we recover Casnati’s bound that if the minimal degree surface containing Ccan has degree g,then7≤ g ≤ 12. Here the lower bound is trivial since curves of degree g ≤ 6 always lie on a surface of degree g − 1orless(see Example 1.7). Remark 1.3. After I wrote this paper, I became aware that Casnati in [Ca2], independently and using different techniques, proved the case κ =1ofTheorem 1.1. Remark 1.4. Let C be a smooth, sextic plane curve. Then the genus of C is 2 10 and Ccan is contained in the three-uple Veronese embedding of P .Thecurve C cannot be bi-elliptic since its gonality is five instead of four. Therefore, we conclude that there are canonical curves of genus 10 that are not bi-elliptic and

SURFACES OF LOW DEGREE CONTAINING A CANONICAL CURVE 59 lie on a surface of degree 9. This example explains the need to exclude the case (g, κ)=(10, −1) in Theorem 1.1. Remark 1.5. Let C be a curve of type (4,r+ 2), with r ≥ 2, on P1 × P1.Then C has genus 3r + 3. A general such curve has trivial automorphism group, hence it cannot be a double cover of a curve of lower genus. Consider the embedding 1 1 3r+2 φ : P × P → P given by the linear system |OP1×P1 (2,r)|. Then the image of φ is a surface of degree 4r.Furthermore,φ embeds C as the canonical curve Ccan. In this case, we have that κ =4r − (3r +3)=r − 3. Hence g =3r +3=3κ + 12. If C is general, then Ccan cannot be contained in a surface of degree less than 4r. Otherwise, by Theorem 1.1, C would be a double cover of a curve of genus less than r/2. We conclude that the bounds in Theorem 1.1 are sharp. We will say that a curve C ⊂ Pn is cut out by quadrics if the homogeneous ideal of C is generated by quadratic equations. Our proof of Theorem 1.1 applies more generally to curves C that are embedded by non-special complete linear systems and are cut out by quadrics. For example, these assumptions are satisﬁed when C is embedded by a complete linear system of degree s ≥ 2g +2[ACGH, p. 143]. For simplicity, we will restrict ourselves to this case. By taking a cone over C with avertexonC, it is clear that every curve of degree s is contained in a surface of degree s − 1. Our next theorem studies the curves that are contained in a surface of strictly smaller degree. Theorem 1.6. Fix κ ≥−1.LetC be a curve of genus g and degree s>2g +1 embedded in Ps−g by a complete linear system. Assume that the minimal degree surface containing C has degree d = s − g + κ

60 IZZET COSKUN

• A general curve C of genus 7 can be realized in P2 as a curve of degree 7 with eight nodes. The blow-up of P2 at the nodes of C embeds into P6 by the linear system of quartic curves vanishing at the nodes of C.The surface has degree 8 and contains Ccan [ACGH]. We, therefore, conclude that the minimal degree surface containing a canonical curve of genus 7 can have degree 5, 6, 7 or 8. The minimal degree surface containing Ccan has degree 5 if and only if C is trigonal. The minimal degree surface containing Ccan has degree 6 if and only if C can be realized as a plane sextic or C is bi-elliptic [ACGH]. If the minimal degree surface containing 1 Ccan has degree 7, then S is a conic bundle over P and C has a unique 1 g4 [Ca]. For the general canonical curve of genus 7, the minimal degree surface containing Ccan has degree 8. In the last section, using the classification of surfaces of low degree, we will make some remarks about curves C such that Ccan iscontainedinasurfaceof degree g + κ for small values of κ. The organization of this paper is as follows. In the next section, we will recall some basic facts concerning ruled surfaces and the Δ-genus that are used in the proof. In §3, we will prove Theorems 1.1 and 1.6. In the final section, we will study the geometry of surfaces of degree g + κ containing a canonical curve for small values of κ. Acknowledgements: I would like to thank Gianfranco Casnati, Lawrence Ein and Mihnea Popa for invaluable comments. This paper was inspired by the papers of Ciliberto and Harris [CH]andCasnati[Ca]. I would also like to thank the organizers of the AMS Special Session on Computational Algebraic and Analytic Geometry, Mika Seppälä, Tanush Shaska and Emil Volcheck, for giving me the opportunity to present my work at the Joint Meetings.

2. The background on ruled surfaces and Δ-genus In this section, we recall some basic facts concerning ruled surfaces and the Δ-genus. We refer the reader to [B, §III] and [H, V.2] for more details on ruled surfaces and to [F3], [F4]and[Ho] for more details on the Δ-genus. Let X be a smooth projective variety of dimension r.LetL be a base-point-free line bundle on X of degree d whose complete linear system gives rise to a birational n morphism ρL : X → P . Recall that the Δ-genus, first introduced by Fujita (see [F4]), is defined by Δ=Δ(X, L)=d + r − h0(X, L). To prove Theorems 1.1 and 1.6, we will use a rough classification of surfaces by Δ-genus due to Tony Horowitz. Recall that a surface S is birationally ruled if it is birational to B × P1 for some curve B.AsurfaceS is geometrically ruled if S admits a morphism π : S → B to a curve such that the fibers are all isomorphic to P1. A geometrically ruled surface is isomorphic to the projectivization of a rank 2 vector bundle E over B [H, V.2.2]. AsurfaceS is projectively ruled if S is the birational image in projective space of a geometrically ruled surface such that the fibers are mapped to lines. We will reduce the proofs of Theorems 1.1 and 1.6 to the following Theorem of Horowitz.

SURFACES OF LOW DEGREE CONTAINING A CANONICAL CURVE 61

Theorem A of [Ho]. Let X be a surface and let L be a line bundle on X as above. 2 0 If (X, L) =( P , OP2 (3)) and 3Δ(X, L)+6

(1) If n = g − 1 and φV restricted to C is the canonical embedding of C,then d − g +3 h = . 2 (2) If n = s−g and V restricted to C is a non-special, complete linear system on C of degree s,then d + g − s +1 h = . 2 Proof. 2 Let C0 be the section on S with C0 = e. The Neron-Severi space of S is generated by the numerical equivalence classes of C0 and a ﬁber F of π.By adjunction, the canonical class of S is numerically equivalent to

KS ≡−2C0 +(2h + e − 2)F.

By assumption, the image of φV is a geometrically ruled surface of degree d. Therefore, the degree of L on F is one. If we express the numerical equivalence 2 class of L ≡ C0 + mF ,writingL = e +2m = d,weseethatL is numerically equivalent to d − e L ≡ C + F. 0 2 On the other hand, since C is a bi-section of π, C is numerically equivalent to C ≡ 2C0 + rF. We can compute r in two diﬀerent ways. First, the degree of the curve is s. Hence, s = L · C =2e +(d − e)+r. We conclude that the numerical equivalence class of C is

C ≡ 2C0 +(s − d − e)F.

In particular, when φV restricts to the canonical linear system on C, C ≡ 2C0 + (2g − 2 − d − e)F. On the other hand, by the adjunction formula,

deg(KC )=(KS + C) · C. Hence, 2g − 2=2(2h − 2+s − d). We conclude that d + g − s +1 h = . 2

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In particular, when φV restricts to the canonical linear system on C,then d − g +3 h = . 2

Let f : C → B be a two-to-one morphism from a smooth, projective curve C of genus g to a smooth, projective curve B of genus h. Given a line bundle L of degree d on C, f∗L is a vector bundle of rank two on B. By the Riemann-Roch Theorem, the vector bundle f∗L has degree d +2h − g − 1. The surface P(f∗L)is a geometrically ruled surface over B. In this paper, we will be especially interested in the case when L is the canonical line bundle KC . The vector bundle f∗KC is a rank two bundle on B of degree ∗ 2h + g − 3. In this case, by duality, (f∗KC ) ⊗ KB = f∗OC . The natural inclusion of OB in f∗OC leads to an exact sequence

0 →OB → f∗OC →OB(−D) → 0, where D is half of the ramification divisor of f. For simplicity, set M = OB(D). −1 The norm map splits this sequence leading to the equality f∗OC = OB ⊕ M .By duality, we conclude that f∗KC = KB ⊕ (KB ⊗ M). Therefore, the ruled surface X = P(f∗KC )overB has invariant e =2h − g − 1. The curve C naturally embeds in X. The linear system |OX (1)| gives rise to the map φ : X → Pg−1. The image of φ is a surface of degree 2h + g − 3. The map φ restricts to the canonical map on C. The inclusion KB → f∗KC gives rise to a section B0 of X over B.The effective cone of X is generated by the class of B0 and the class of a fiber [H, V.2.20]. Therefore, when h>1, OX (1) is ample since it has positive degree on both generators of the effective cone. The map φ restricts to the canonical map on B0. Hence, OX (1) is not very ample if B is hyperelliptic. In fact, if B is hyperelliptic, the image of X under φ is not normal since φ maps B0 two-to-one onto a rational normal curve of degree 2h − 2. On the other hand, if g>2h +1 and B is not hyperelliptic, then OX (1)isveryampleandφ is an embedding. There are sections B1 of X that are disjoint from B0 induced by the inclusion KB ⊗ M→ f∗KC .If g−1 g>2h +1, φ embeds both B0 and B1 into P as curves with disjoint spans. The scroll over these two curves is smooth. Hence, it follows that φ is an embedding.

3. Minimal degree surfaces containing a canonical curve In this section, we prove Theorem 1.1 and Theorem 1.6.

Proof of Theorem 1.1. Let κ ≥−1. In Example 1.7, we saw that the canonical image of every non-hyperelliptic curve of genus g ≤ 6 is contained in a surface of degree g − 1org − 2. We can, therefore, assume that g>6. Let S be g−1 asurfaceinP of degree d = g + κ containing Ccan. Then, by the Babbage- Enriques-Petri Theorem, Ccan is cut out by quadrics unless C is trigonal [ACGH]. If C is trigonal, then Ccan lies on a rational normal surface scroll of degree g − 2. Since by assumption the degree of the minimal surface containing Ccan has degree g + κ>g− 2, we conclude that Ccan is cut out by quadrics.

Take a quadric Q containing Ccan but not S. By Bezout’s Theorem, S ∩ Q has degree 2d.Sinced<2g −3, S cannot be everywhere singular along Ccan.LetX be

SURFACES OF LOW DEGREE CONTAINING A CANONICAL CURVE 63 the minimal desingularization of S.LetL be the pull-back of OS(1) to X.Denote the proper transform of C in X again by C. We ﬁrst bound the Δ-genus of the pair (X, L). Δ(X, L)=d +2− h0(X, L) ≤ d − g +2=κ +2. If g>3κ + 12, then 3Δ + 6 = 3(κ +2)+6=3κ +123κ+12 and (X, L) =( P , OP2 (3)), then S is projectively ruled. Hence π : X → B is a geometrically ruled surface. From now on we assume that g>3κ+12and(g, κ) =(10 , −1). We may, therefore, assume that S is projectively ruled.

Since Ccan is cut out by quadrics, Ccan is contained in the intersection of S by a quadric. Consequently, C is either a bi-section or a section of π : X → B.IfC is a bi-section, then by Lemma 2.1, the genus h of the curve B is related to g and d by the formula d − g +3 κ +3 h = = . 2 2 Since the genus is an integer, κ must be odd. If C is in a section class, then B is isomorphic to C. Take a general irreducible hyperplane section H of S.ThecurveH has degree d and spans a projective linear space of dimension g − 2 and the normalization of H is isomorphic to C.LetD be the hyperplane divisor on the normalization of H. Cliﬀord’s Theorem says that h0(D) − 1 ≤ deg(D)/2 for any special divisor on a curve Y of genus g with equality O 1 when D = Y or D = KY or when Y is a hyperelliptic curve and D is mg2 for 0 d/2. Hence, the hyperplane divisor is non-special. However, by the Riemann-Roch Theorem, h0(D)=d − g +13κ + 12, leads to the inequality −4 >κ, which is a contradiction. We conclude that C cannot be in a section class on X. Therefore, it must be in a bi-section class. By Lemma 2.1, we conclude that C admits a two-to-one map to B,acurveof genus (κ +3)/2. As observed above, since the genus is an integer, κ must be odd. This concludes the proof of Part (1) and the ﬁrst statement in Part (2) of Theorem 1.1. Conversely, suppose C admits a two-to-one map f : C → B to a curve of genus h =(κ +3)/2. Then, as observed in §2, E = f∗KC is a rank two vector bundle on B of degree g +2h−3. The curve C embeds into the geometrically ruled surface PE g−1 over B. The line bundle OPE(1) gives a map from PE → P onto a projectively ruled surface of degree g +2h − 3=g + κ. Furthermore, the restriction of OPE(1) to C is the canonical linear series on C. Therefore, if C is a double cover of a curve κ+3 of genus 2 ,thenCcan iscontainedinasurfaceofdegreeg + κ.Asanaside, observe that whether B is hyperelliptic is determined from the singularities of the surface. The map f : C → B determines the surface S.ThecurveC is a bi-section −1 of the projectively ruled surface. If q1,q2 are the two points in f (p)forp ∈ B,

64 IZZET COSKUN then S contains the line q1q2 spanned by q1 and q2. As the point p varies over B, the lines spanned by the points in f −1(p) sweep out the surface S. Hence, S can g−1 be recovered from the map f. The image of X = P(f∗KC )inP under the map φ defined by the linear system |OX (1)| is a projectively ruled surface swept out by the lines spanned by the pairs of points on Ccan that are mapped to the same point on B by f. Hence, S is the image of X under φ. This concludes the proof of Theorem 1.1. Proof of Theorem 1.6. The proof of Theorem 1.6 requires only minor mod- ification. Let S be a surface of degree d = s − g + κ in Ps−g containing C.LetX be the minimal desingularization of S and let L be the pull-back of OPs−g (1) on X. The Δ-genus of (X, L) can be calculated as follows. Δ=Δ(X, L) ≤ s − g + κ +2− (s − g +1)=κ +1. Hence, if s−g +1 > 3κ+9, then 3Δ+6 ≤ 3κ+9 1is3h − 3+2g − 4h +2=2g − h − 1. Since the dimension of Mg is 3g − 3, the codimension of this locus is g + h − 2. When h = 0 or 1, this calculation has to be modified because curves of genus 0 and 1 have positive dimensional automorphism groups. A simple calculation shows that the dimension

SURFACES OF LOW DEGREE CONTAINING A CANONICAL CURVE 65 of the locus of hyperelliptic curves is 2g − 1. Hence, the codimension is g − 2. Similarly, the dimension of the locus of bi-elliptic curves is 2g − 2. Hence, the codimension is g − 1.

Corollary 3.2. Let κ ≥−1 be an odd integer. Let g>max(10, 3κ + 12). Then the codimension of the locus of curves C in Mg such that the minimal degree Pg−1 κ−1 surface in containing Ccan has degree g + κ is g + 2 . Proof. By Part (2) of Theorem 1.1, C has to be a double cover of a curve of κ+3 κ+3 genus 2 . By Part (3) of Theorem 1.1, every double cover of a curve of genus 2 iscontainedinasurfaceofdegreeg + κ. Hence, by Lemma 3.1, the codimension of the locus of curves in Mg such that the minimal degree surface containing Ccan κ+3 − κ−1 has degree g + κ is g + 2 2=g + 2 .

4. Low degree examples Smooth surfaces that have small sectional genus have been classified by Ionescu [Io], Livorni [L1], [L2] and in positive characteristic by Andreatta and Ballico [AB]. In this section, using the classification of surfaces with small sectional genus, we give some examples of canonical curves of genus g contained in surfaces of degree g + κ when g ≤ 3κ + 12. We first specialize [CH, Lemma 1.3] to our case. Lemma 4.1. Let −1 ≤ κ ≤ 5.LetS be the minimal degree surface with degree g +κ<2g −3 containing a canonical curve Ccan of genus g.ThenS is birationally ruled. Proof. First, suppose that g + κ<2g − 4. Then, by Clifford’s Theorem, OH (1) is non-special on a general hyperplane section H of S.Leth be the genus of H. Then, by the Riemann-Roch Theorem, g + κ − h ≥ g − 2. Therefore, by the genus formula, 2 KS · H ≤ 2h − 2 − H =2h − 2 − g − κ ≤ κ − g − 2 ≤−2.

Since H is ample, no multiple of KS can have a section. By Enriques’ Theorem [B], S is birationally ruled. If g + κ =2g − 4, then κ = g − 4. Hence, if −1 ≤ κ ≤ 5, then 3 ≤ g ≤ 9, respectively. In Example 1.7, we saw that every canonical curve of genus 3, 4, 5, 6 or 7 is contained in a surface of degree less than or equal to 1, 2, 4, 5, 8, respectively. A general curve of genus 8 can be realized as a (5, 5) curve on P1 × P1 with eight nodes. The linear system of (3, 3) curves vanishing on the nodes of C maps the 7 surface to P as a surface of degree 10 containing Ccan [ACGH]. Hence, every canonical curve of genus 8 is contained in a surface of degree less than or equal to 10. Similarly, a general curve C of genus 9 can be realized as a plane degree eight curve with 12 nodes. The linear system of quintic curves vanishing at the nodes embeds the blow-up of P2 at the nodes as a surface of degree 13 in P8 containing Ccan. Hence, every canonical curve of genus 9 is contained in a surface of degree at most 13. Therefore, in all these cases the minimal degree surface containing Ccan has degree strictly less than 2g − 4. This concludes the proof of the Lemma.

Remark 4.2. When κ ≥ 6, in addition to birationally ruled surfaces, we would need to allow K3 surfaces [CH].

66 IZZET COSKUN

For the rest of this section, let −1 ≤ κ ≤ 3andg ≤ 3κ + 12. Suppose that the minimal degree surface S containing Ccan has degree g +κ<2g −3. For simplicity, we will assume that S is smooth. The smoothness assumption is for convenience and can be removed. If S is a singular surface of degree d in Pn, then the projection of S from a singular point is either a curve, in which case S is a cone, or is a surface of degree less than or equal to d−2inPn−1. Successively projecting S from singular points leads to an analysis of the singular case as well. We leave this analysis to the interested reader. 1 Let Fr, r ≥ 0, denote the ruled surface P(OP1 ⊕OP1 (r)) over P .LetE denote the class of the curve with minimal self-intersection. Let F denote the class of a fiber of the projective bundle. • κ =0. Suppose that the minimal degree surface S containing Ccan is smooth and has degree g. By Clifford’s Theorem, the hyperplane section of S can have genus at most 2. In fact, by del Pezzo’s classification [L1, Theorem 0.2], the hyperplane section must have genus 2. By the classification of surfaces whose hyperplane sections have genus 2 [L1, Table] and [AB, Table], we conclude that ≤ ≤ ≤ if S is not a scroll, then S must be a blow-up of Fr with 0 r 2inm 7 P11−m | − m | points embedded in by the linear system 2E +(r +3)F i=1 Ei ,where Ei denote the classes of the exceptional divisors. If we take a curve C on S with − m − class 4E +(2r +5)F i=1 2Ei,thenC is a curve of genus 12 m that embeds in P11−m as a canonical curve. In particular, note that the projection from S to P1 4 defines a g1 on C. This classification agrees with the one given in [Ca]. • κ =1. Suppose that the minimal degree surface S containing Ccan is smooth and has degree g + 1. Then, by Clifford’s Theorem, a hyperplane section of S can have genus at most 3. Since in Example 1.7 we have analyzed canonical curves of genus g ≤ 7, we may assume that 15 ≥ g ≥ 8. If the hyperplane section of S has genus 2, then by the classification of surfaces whose hyperplane sections have genus 2[L1,Table]and[AB, Table], S has to be a scroll. In the proof of Theorem 1.1, we have seen that C has to be a double cover of a curve B of genus 2 and S is the image of the surface X = P(f∗KC ) under the linear system |OX (1)|.Infact,since B is hyperelliptic, S isnotsmoothinthiscase. We may assume that hyperplane sections of S have genus 3 and that S is not a scroll. By the classification of surfaces whose hyperplane sections have genus 3 [L1, Table] and [AB, Table], there are two possibilities. First, S may be the blow-up of P2 − Pg−1 | − 15−g | in 15 g points embedded in by the linear system 4H i=1 Ei ,where P2 H denotes the pull-back of the hyperplane class of and Ei denote the classes of − 15−g the exceptional divisors of the blow-up. In this case C has class 7H 2 i=1 Ei and embeds in Pg−1 as a canonical curve of genus g. ≤ ≤ − Second, S may be a blow-up of Fr with 0 r 3in15 g points embedded Pg−1 | − 15−g | in by the linear system 2E +(r +4)F i=1 Ei . Curves in the class − 15−g 4E +(2r +6)F 2 i=1 Ei are mapped to canonical curves of genus g under this linear system. Note that both types of curves are very special in moduli. For 2 example, curves of the first type admit a g7 and curves of the second type admit a 1 g4. Remark 4.3. After I wrote this paper, I became aware that Casnati in [Ca2], independently and using different techniques, classified canonical curves of genus g that are contained in surfaces of degree g + 1. Unlike here, Casnati does a very careful analysis of the singular surfaces as well. The classification remains

SURFACES OF LOW DEGREE CONTAINING A CANONICAL CURVE 67 essentially the same. Assuming that g ≤ 15 and the surface is not a scroll, the curve is either birational to a plane septic with 15 − g possibly inﬁnitely near double points as in our ﬁrst case or it lies in a conic bundle over P1 and hence 1 admits a g4 as in our second case (see Theorem D [Ca2]).

• κ =2. Suppose that the minimal degree surface S containing Ccan is smooth and has degree g + 2. A general curve of genus 8 can be realized as a (5, 5) curve on P1 × P1 with eight nodes. The linear system of (3, 3) curves vanishing on the nodes 7 of C maps the surface to P as a surface of degree 10 containing Ccan [ACGH]. Hence, every canonical curve of genus 8 is contained in a surface of degree at most 10. We may, therefore, assume that 18 ≥ g>8. By Clifford’s Theorem, a hyperplane section of S can have genus at most 4. Since in this case the surface cannot be a scroll, by the classification of surfaces whose hyperplane sections have genus at most 4 [L1, Table] and [AB, Table], we conclude that the hyperplane section of S must have genus 4. The following can be deduced from [L1, Table] and [AB, Table]. ≤ ≤ − First, S can be the blow-up of Fr,with0 r 4, in 18 g points embedded Pg−1 | − 18−g | in by the linear system 2E +(r +5)F i=1 Ei . Curves in the class − 18−g 4E +(2r +7)F 2 i=1 Ei are mapped to canonical curves of genus g under this linear system. If g>16, then this is the only other possibility. 1 1 g−1 Second, S can be the blow-up ofP × P in 16 − g points embedded in P |O − 16−g | O − by the linear system P1×P1 (3, 3) i=1 Ei . Curves in the class P1×P1 (5, 5) 16−g 2 i=1 Ei map to canonical curves of genus g. This is the only other possibility if g>10. Third, if g = 10 (respectively, 9), S may be the two-uple Veronese embedding of a cubic surface (respectively, of the blow-up of a cubic surface in a point p)inP3. Under this embedding complete intersections of two cubic surfaces (respectively, those that are double at p)inP3 map to canonical curves of genus 10 (respectively, 9). Note that each of these curves are very special in moduli. The curves of the 1 1 first type admit a g4. The curves of the second type admit a g5. Finally, curves of the third type are complete intersections of two cubic surfaces in P3.

• κ =3. Suppose the minimal degree surface S containing Ccan is smooth and has degree g + 3. For simplicity, we will assume that 21 ≥ g>9. By Clifford’s Theorem a hyperplane section of S has genus at most 5. If the genus is 3, then S is a scroll over a curve B of genus 3 and C has to be a double cover of B.Thiscase has been studied in the proof of Theorem 1.1, so we may assume that S is not a scroll. Then, by the classification in [L1]and[AB], we conclude that a hyperplane section of S has genus 5. The possibilities can be read off from these tables. ≤ ≤ − First, S may be the blow-up of Fr with 0 r 5in21 g points, embedded Pg−1 | − 21−g | in by the linear system 2E +(r +6)F i=1 Ei . Curves in the class − 21−g 4E +(2r +8)F 2 i=1 Ei are mapped to canonical curves of genus g under this linear system. If g>18, then this is the only possibility. − Pg−1 Second, S may be the blow-up of F1 in 18 g points, embedded in under | − 18−g | | − 18−g | the linear system 3E +5F i=1 Ei . Curves in the class 5E +8F 2 i=1 Ei map to canonical curves of genus g under this embedding. This is the only other possibility if g>13.

68 IZZET COSKUN

Let D be the del Pezzo surface obtained by blowing up P2 in 5 points. Denote the classes of the exceptional divisors by Ai.Third,S may be the blow-up of D5 in − Pg−1 | − 5 − 13−g | 13 g points embedded in by the linear system 6H 2 i=1 Ai i=1 Ei . − 5 − 13−g Curves in the class 9H 3 i=1 Ai 2 i=1 Ei map to canonical curves of genus g. These are the only possibilities for g>9. Observe that these curves are very special. In the first case, the curves admit 1 1 a g4. In the second case, the curves admit a g5. In the final case, the curve admits 2 a g9 with 5 triple points. Remark 4.4. In these examples, the assumption that S is smooth rules out certain singular conic bundles over P1 or over an elliptic curve that contain canonical curves. Most importantly, when κ =2andg ≤ 10 or κ =3andg ≤ 13, there are canonical curves that are triple covers of elliptic curves that lie on singular conic bundles over the elliptic curve. See Remark 4.7 for more details on these surfaces. This concludes the classification of canonical curves of genus g that are contained in a smooth surface of degree g + κ for −2 ≤ κ ≤ 3. This classification can be carried out for several more values of κ. However, the list of possibilities grows rapidly and quickly becomes unwieldy. Since these examples already illustrate the technique, we conclude the discussion here. The reader should observe that, just like in Theorem 1.1, if we assume that the genus is relatively large compared to κ, then the number of possibilities is small. The following proposition makes the next case after Theorem 1.1 more precise. Proposition 4.5. Let 0 <κmax(2κ +12, 3κ +4) and (g, κ) =(15 , 2), (21, 4).

If the minimal degree surface S containing a canonical curve Ccan of genus g has degree g + κ and S is not projectively ruled, then S is a conic bundle over P1 and 1 C admits a g4. Proof. Let X be the minimal desingularization of S and let L be the pull-back of OPg−1 (1) on X. We compute the Δ-invariant of X as follows Δ=Δ(X, L)=g + κ +2− g = κ +2. Hence, by assumption 1 1 (h0(X, L) − 8) ≥ (g − 8) >κ+2=Δ. 2 2 By [Ho,TheoremB],S is either projectively ruled or ruled by conics. The state- 2 ment of Theorem B in [Ho] forgets to omit the two exceptions (X, L)=(P , OP2 (4)) 2 and (P , OP2 (5)). Plane curves of degree 7 (respectively, 8) are mapped to canonical curves under the linear system |OP2 (4)| (respectively, |OP2 (5)|). These canonical curves are contained in the four-uple, respectively, ﬁve-uple Veronese embedding of P2. This explains the need to exclude the cases (g, κ)=(15, 2) and (21, 4) in the statement of the proposition. Since by assumption, S is not projectively ruled, we conclude that S is a conic bundle over a curve. In order to reach this conclusion, we did not need to assume that g>3κ +4.By[Ho, Corollary 1.8 (2)], we have that 0 1 1 g ≤ h (X, L) ≤ 3Δ + 6 − 8h (X, OX )=3κ +12− 8h (X, OX ).

SURFACES OF LOW DEGREE CONTAINING A CANONICAL CURVE 69

1 If g>3κ+4, then h (X, OX )=0andX and S are rational surfaces. In particular, P1 P1 1 S is a conic bundle over and the projection of Ccan to gives a g4 on C.Observe that since the degree of the minimal surface containing Ccan is greater than g − 2, C cannot be trigonal. This concludes the proof of the proposition. Remark 4.6. More generally, by the same argument, if 3κ+12 ≥ g>max(2κ+ 12, 3κ +4− 8i) and the minimal degree surface S containing Ccan has degree g + κ and is not projectively ruled, then S is a conic bundle over a curve B of genus at most i. The projection of S to B deﬁnes a map of degree at most 4 from C to B. Remark 4.7. The bounds in Proposition 4.5 are sharp. Assume that κ ≥ 2is |O κ | even. Let C be a curve in the linear system P1×P1 (5, 2 +4).ThenC is a curve |O κ | P1 × P1 of genus 2κ + 12. The linear system P1×P1 (3, 2 +2) embeds as a surface S of degree 3κ + 12 and restricts to the canonical embedding on C.Inparticular, S does not contain any conics and a general curve C in the linear system does not 1 admit a g4. To get examples when κ is odd, one can repeat the construction with 1 1 the surface F1 instead of P × P . Let S be a cone over an elliptic normal curve of degree r in Pr.LetC be the intersection of S with a general cubic hypersurface. Then C has genus 3r +1. The two-uple Veronese embedding of S is a surface S in P3r of degree 4r.This embedding restricts to the canonical embedding on C. Therefore, κ = r − 1. In particular, g =3r +1 = 3κ + 4. We conclude that S is a conic bundle over an elliptic curve B and C is a triple cover of B.

References [AB] M. Andreatta and E. Ballico. Classification of projective surfaces with small sectional genus: char p>0, Rend. Sem. Mat. Univ. Padova, 84 (1990), 175–193. MR1101291 (92c:14030) [ACGH] E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris. Geometry of Algebraic curves, vol. 1, Grundlehren der mathematischen Wissenschaften 267, Springer-Verlag, 1984. [B] A. Beauville. Complex algebraic surfaces. London Math. Soc. Student Texts 34. Cambridge University Press, 1996. MR1406314 (97e:14045) [Be] E. Bertini. Introduzione alla geometria proiettiva degli iperspazi Enrico Spoerri, Pisa, 1907. [Ca] G. Casnati. Canonical curves on surfaces of very low degree. Proc.Amer.Math.Soc.140 (2012), 1185–1197. MR2869104 [Ca2] G. Casnati. Curves of genus g whose canonical model lies on a surface of degree g +1. to appear Proc. Amer. Math. Soc. MR2869104 [CH] C. Ciliberto and J. Harris. Surfaces of low degree containing a general canonical curve. Comm. Algebra 27 (1999), 1127–1140. MR1669124 (2000c:14051) [D] Del Pezzo. Sulle superficie di ordine n immerse nello spazio di n + 1 dimensioni, Rend. Circ. Mat. Palermo, 1, 1886. [EH] D. Eisenbud, and J. Harris. On varieties of minimal degree (a centennial account) Algebraic geometry, Bowdoin 1985 Proc. Sympos. Pure Math. 46 no. 1, Amer. Math. Soc., Providence, RI, 1987. MR927946 (89f:14042) [F1] T. Fujita. Classification of projective varieties of Δ-genus one. Proc. Japan Acad. Ser. A 58 (1982), 113–116. MR664549 (83g:14003) [F2] T. Fujita. On polarized manifolds of Δ-genus two. J.Math.Soc.Japan36 (1984), 709–730. MR759426 (85m:14015) [F3] T. Fujita. Classification theories of polarized varieties, London Math. Soc. Lecture Notes Series 155, Cambridge University Press, 1990. MR1162108 (93e:14009) [F4] T. Fujita. On the structure of polarized varieties with Δ-genus zero, J. Fac. Sci. Univ. of Tokyo, 22 (1975), 103–115. MR0369363 (51:5596) [H] R. Hartshorne. Algebraic geometry. Springer, 1977. MR0463157 (57:3116) [Ho] T. Horowitz. Varieties of low Δ-genus. Duke Math. J. 50 no. 3 (1983), 667–683. MR714823 (85d:14022)

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[Io] P. Ionescu. Embedded projective varieties of small invariants. Algebraic geometry, Bucharest 1982. Lecture Notes in Math. 1056, Springer, Berlin, 1984. MR749942 (85m:14024) [L1] E.L. Livorni, Classiﬁcation of algebraic surfaces with sectional genus less than or equal to six.I: Rational surfaces. Pac. J. Math. 113 no. 1 (1984), 93–114. MR745598 (85j:14068) [L2] E.L. Livorni, Classiﬁcation of algebraic non-ruled surfaces with sectional genus less than or equal to six. Nagoya Math. J. 100 (1985), 1–9. MR818155 (87c:14043)

University of Illinois at Chicago, Department of Mathematics, Statistics and Com- puter Science, Chicago, Illinois 60607 E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11374

Ideals of curves given by points

E. Fortuna, P. Gianni, and B. Trager

Abstract. Let C be an irreducible projective curve of degree d in Pn(K), where K is an algebraically closed field, and let I be the associated homogeneous prime ideal. We wish to compute generators for I, assuming we are given sufficiently many points on the curve C.InparticularifI can be generated by polynomials of degree at most m and we are given md +1 pointson C, then we can find a set of generators for I. We will show that a minimal set of generators of I can be constructed in polynomial time. Our constructions are completely independent of any notion of term ordering; this allows us the maximal freedom in performing our constructions in order to improve the numerical stability. We also summarize some classical results on bounds for the degrees of the generators of our ideal in terms of the degree and genus of the curve.

1. Introduction Let C be an irreducible projective curve of degree d in Pn(K), where K is an algebraically closed field, and let I = I(C) be the associated homogeneous prime ideal of P = K[x0,...,xn] consisting of all the polynomials vanishing on C.We wish to compute generators for I, assuming we are given sufficiently many points on the curve C. In particular if I can be generated by polynomials of degree at most m and we are given at least md+1 points on C, then we can find a set of generators for I. It is a simple consequence of Bezout’s theorem that any polynomial of degree k which vanishes on more than kd points of C must be contained in I. Although the number of monomials in n + 1 variables of degree at most m is not polynomial in both n and m, we will present a process which constructs generators degree by degree and results in a polynomial time algorithm for computing generators for I. Polynomial time algorithms for computing Gröbner bases of ideals of affine points were presented in [MB], and then extended to minimal generators of ideals of projective points in [MMM]. These algorithms require exact arithmetic, and assume a term ordering is given. We present new algorithms which are completely independent of any notion of term ordering. We believe that this flexibility is necessary when working with approximate coefficients. Given a homogeneous ideal I, there are many different choices for monomials representing cosets of P/I, i.e. for a complement of I. It is well known that the

2010 Mathematics Subject Classiﬁcation. Primary 14H50, Secondary 13P10. Key words and phrases. Algebraic curves, border bases, interpolation. This research was partially supported by M.I.U.R. and by G.N.S.A.G.A.

c 2012 American Mathematical Society 71

72 E. FORTUNA, P. GIANNI, AND B. TRAGER natural coset representatives associated with Gröbner bases do not remain stable with respect to small coefficient perturbations of the ideal generators. Border bases were introduced to help overcome this problem ([KR]). Given a fixed choice of complement for a zero-dimensional ideal, its border basis is uniquely determined in contrast with Gröbner bases where the complement is uniquely determined by the given term ordering. Border bases are usually defined for zero-dimensional ideals, which guarantees a finite basis. We extend the definition to homogeneous ideals of any dimension, but bound the degree in order to preserve finiteness. In much of the literature on border bases, the complements are required to be closed under division by variables. This makes complements of border bases more similar to complements of Gröbner bases which also have this property. In particular this is done in [HKPP], therefore their algorithms need to explicitly decide whether or not candidate leading monomials have coefficients which are so small that they should be treated as zero. As suggested by Mourrain and Trébuchet ([MT]), we only require the complement to be connected to 1. This means we require that each complement monomial of degree i is a multiple of some complement monomial of degree i − 1. This extra flexibility in the choice of complement monomials means that we can use standard numerical software like the QR algorithm with column pivoting (QRP) ([GVL]) to choose our complement in each degree. One could define a complement to be any set of representatives for P/I, but with this definition we would not be able to obtain algorithms which are polynomial in both the degree of the curve and the number of variables. In particular, requiring the complement to be connected to 1 implies a strong condition on the syzygy module. We show that the syzygy module for a vector space basis of a homogeneous ideal whose complement is connected to 1 is generated by vectors whose entries have degree at most one, generalizing the result of Mourrain and Trébuchet for border bases of zero-dimensional ideals. These special generators of the syzygy module can be used to obtain a polynomial time algorithm for constructing minimal generators for our ideals. The algorithms developed by Cioffi [C] have a similar complexity in the case of exact coefficients but her use of Gröbner bases requires a term ordering which determines a unique complement which may be numerically unstable when working with approximate points. The motivation for this paper came from the desire to be able to compute the generators of space curves starting from numerical software which generates points on curves, in particular computing points on canonical or bicanonical models for Riemann surfaces presented as Fuchsian groups ([GSST]). Our intended applications differ from that of [HKPP]and[AFT] since we assume that the points defining our curve are generated by a numerical algorithm whose errors tend to be very small as opposed to empirical measurements whose errors could be much larger. Thus our goal is to present algorithms based on numerically stable constructions like SVD for computing ideal generators and QRP for deciding which monomials represent the complement of our ideal. In section 2 of this paper we derive some general properties of border bases and complements for general homogeneous ideals. Using these properties we present a polynomial time algorithm for finding a minimal basis for a homogeneous ideal. In section 3 we specialize to ideals of curves given by points, and use point evaluation matrices to complete the task of computing border bases for homogeneous ideals of curves. Assuming exact arithmetic of unit cost, we also provide an overall

IDEALS OF CURVES GIVEN BY POINTS 73 complexity analysis of the border basis algorithm and the minimal basis algorithm. In section 4 we consider the situation of approximate points and show how our algorithms can be adapted to use standard numerical software, where we allow numerical algorithms like the QRP to choose the complement monomials in order to help improve the numerical stability. In general the stability also strongly depends on how the points are distributed, and we feel this is an interesting problem for future research, along with the possibility of using other approaches such as interval arithmetic. In order to use Bezout’s theorem, we assume we have at least a bound on the degree of our curve. In the last section we summarize some classical results on bounds for the degrees of generators of our ideal in terms of the degree and genus of the curve. In particular, as shown by Petri ([P]), a non-hyperelliptic canonical curve of genus g ≥ 4 can be generated in degrees 2 and 3. We also recall the completely general result of Gruson-Lazarsfeld-Peskine ([GLP]) which shows that any non-degenerate curve of degree d in Pn(K) can be generated in degree d−n+2. Sometimes we have additional information about the nature of the curve whose points we are given. For instance, if we know the Hilbert function, the rank of our point evaluation matrices is explicitly given instead of being determined by examining its singular value spectrum.

2. Border bases for homogeneous ideals

Let K be an algebraically closed ﬁeld. For any s ∈ N,letTs be the set of all P K P P terms of degree s in = [x0,...,xn] and let s be the vector subspace of T P n+s P ⊕s P generated by s. Recall that dim s = s . We will also set ≤s = i=0 i. Notation 2.1. We will use the following notation: (1) For any subset Y ⊂ Pn(K)denotebyI(Y ) the radical homogeneous ideal of P consisting of all the polynomials vanishing on Y . (2) For any homogeneous ideal I in P,letIs = I ∩Ps (so that I = ⊕s≥0Is) and let I≤s = I ∩P≤s. (3) For any S ⊂P,denotebyI(S) the ideal generated by S. ⊂P P (4) For any S s,denotebyS the vector subspace of s generated by S. ⊂P + n ⊂P (5) For any S s,letS = j=0(xj S) s+1. h (6) If a =(a1,...,ah) ∈ K and F =[F1,...,Fh] is a list of polynomials, we set a ·F = a1F1 + ...+ ahFh. (7) For any ﬁnite set A,wedenoteby|A| its cardinality. Definition 2.2. Let J be a proper homogeneous ideal in P and s ∈ N.Let N0 = {1} and, for each k =1,...,s, assume that Nk is a set of monomials in Tk such that N ⊂N+ P ⊕N k k−1 and k = Jk k . We call N = {N0,...,Ns} a complement of the ideal J up to degree s.

Remark 2.3. Let N = {N0,...,Ns} be a complement of a proper homogeneous ideal J up to degree s. Then: N ⊂N+ N (1) the condition k k−1 implies that is connected to 1, i.e. for each ∈N · · m there exist variables xi1 ,...,xik such that m = xi1 ... xik and · · ∈N xi1 ... xij for each j

74 E. FORTUNA, P. GIANNI, AND B. TRAGER

(2) from the deﬁnition it follows that, for k =1,...,s, N + ∩N+ ⊕N k−1 =(Jk k−1 ) k .

Lemma 2.4. Let J be a proper homogeneous ideal of P. Assume that Nk−1 ⊆ Tk−1 and Nk ⊆Tk are sets of monomials such that

(a) Pk−1 = Jk−1 ⊕Nk−1 N + ∩N+ ⊕N (b) k−1 =(Jk k−1 ) k . Then + ∩N+ (1) Jk = Jk−1 +(Jk k−1 ) (2) Pk = Jk ⊕Nk. Proof. By hypothesis (a), we have P P+ + N + k = k−1 = Jk−1 + k−1 ⊇ + and, since Jk Jk−1 , we have also P ∩ + ∩N+ Jk = k Jk = Jk−1 +(Jk k−1 ), which proves (1). Hence, by the previous relations and hypothesis (b), we have P + N + + ∩N+ N N k = Jk−1 + k−1 = Jk−1 +(Jk k−1 )+ k = Jk + k . N ⊆N+ On the other hand, again by hypothesis (b) we have that k k−1 and hence ∩N ⊆ ∩N+ ∩N { } Jk k Jk k−1 k = 0 , which completes the proof of (2). Remark 2.5. It is always possible to choose a complement up to any fixed degree for any proper homogeneous ideal J of P incrementally. Namely, if N = {N0,...,Nk−1} is a complement of J up to degree k − 1, it is sufficient to choose N ⊆T N + ∩N+ ⊕N aset k k such that k−1 =(Jk k−1 ) k : then Lemma 2.4 assures that Pk = Jk ⊕Nk and hence that N = {N0,...,Nk} is a complement of the ideal J up to degree k. Moreover, again by Lemma 2.4, once one has a complement N of J up to any fixed degree s, one can get a set of generators of the ideal I(J1,...,Js) as the union ∩N+ of sets of generators of Jk k−1 for k =1,...,s. Definition 2.6. Let J be a proper homogeneous ideal in P and assume that N = {N0,...,Ns} is a complement of J up to degree s. N N + \N N (1) For all k =1,...,s let (∂ )k = k−1 k; the elements in (∂ )k will be called border monomials in degree k. (2) For each m ∈ (∂N )k let ψ(m) be the unique polynomial in Nk such that m + ψ(m) ∈ Jk; we will call the homogeneous polynomial m + ψ(m)the border polynomial associated to m. (3) If Bk denotes the set of all border polynomials of degree k,thesetB = B1 ∪ ...∪Bs is called the border basis of J up to degree s associated to N . Remark 2.7. This notion of bounded degree border basis applies to arbitrary homogeneous ideals and coincides with the classical notion of border basis ([KR]) in the case of homogeneous zero-dimensional ideals, provided we choose s to be larger than the maximal degree of any monomial in the (finite) complement N of J.

IDEALS OF CURVES GIVEN BY POINTS 75

Proposition 2.8. Assume that N is a complement of a proper homogeneous ideal J up to degree s and let B = B1 ∪ ...∪Bs be the associated border basis. Then B ∩N+ (1) k is a basis of the vector space Jk k−1 for each k =1,...,s, (2) B is a set of generators of the ideal I(J1,...,Js). Proof. ∈ ∩N+ N + N ∪N (1) Let b Jk k−1 .Since k−1 =(∂ )k k, there exist ∈ K ai,bi such that b = aimi + bimi. ∈N mi∈(∂N )k mi k

For each mi ∈ (∂N )k let ψ(mi) be the unique polynomial in Nk such that mi + ψ(m ) ∈ J . Then we can write b = u + v with i k u = ai(mi + ψ(mi)),v= bimi − aiψ(mi). ∈N mi∈(∂N )k mi k mi∈(∂N )k

Note that u ∈ Jk ∩Bk and v ∈Nk.Sinceb ∈ Jk,thenwehavethatv = b − u ∈ Jk ∩Nk = {0}.Thusb = u and hence b ∈Bk. (2) follows immediately from (1) and Remark 2.5. We now suggest a simple method to construct recursively both a complement and a border basis of a homogeneous ideal J up to any fixed degree. Even if the ideal may not have an explicit representation, if we assume the capability of computing a basis of its intersection with a vector subspace generated by a finite set N of monomials of the same degree, we will be able to compute a border basis for J up to any fixed degree. We will denote this condition by saying that the ideal is represented by the function ComputeBasisJ ,which,foranysuchN, returns a basis for the intersection J ∩N. In the next section we will see that such a function can be easily computed for ideals of points. In the description of the algorithms we will use the following notations:

a. If v is a polynomial and S = {n1,...,nt} is a set of monomials, then coeffs(v, S) will denote the vector (a1,...,at) of the coefficients of the monomials of S in v. b. Given a matrix A, we will denote RRE(A)=(E,Σ) where - E is the completely reduced row echelon form of A (i.e. each pivot is equal to 1, and in each of the columns containing a pivot all the elements different from the pivot are zero) - Σ is the set consisting of the indexes of the columns containing the pivots of E . Algorithm BorderBasisWithComplement Input:

- a function ComputeBasisJ representing a homogeneous ideal J - s ∈ N Output:

- {N0,...,Ns} a complement of J up to degree s - {B1,...,Bs} the associated border basis. Procedure:

- N0 = {1} -fork =1..s repeat

76 E. FORTUNA, P. GIANNI, AND B. TRAGER

N + { } — construct the set of distinct monomials k−1 = m1,...,mt N + { | ≤ ≤ ∈N } k−1 := xim 1 i n, m k−1 V N + k := ComputeBasisJ ( k−1) q := |Vk| |N + | t := k−1 × N + ∈V — compute the q t matrix with rows coeﬀs(v, k−1)forv k N + | ∈V A := matrix(coeﬀs(v, k−1) v k) (E,Σ) := RRE(A) — the monomial with index not in Σ are put in Nk Nk := {mj | j/∈ Σ} B — every row represents an element of k B { | } k := j ei,j mj i =1,...,q Proposition 2.9. Given a proper homogeneous ideal J in P represented by a function ComputeBasisJ and s ∈ N, the algorithm BorderBasisWithComplement constructs a complement and a border basis for J up to degree s.

Proof. At each step, the rows of E correspond to polynomials of the form ∈N ∈N ∩N+ m + ψ(m), with m k and ψ(m) k, which are a basis of Jk k−1 . Moreover , since the monomials in Nk correspond to the non-pivot positions, by N + ∩N+ ⊕N construction we have that k−1 =(Jk k−1 ) k . Then by Lemma 2.4 we get that Pk = Jk ⊕Nk; hence {N0,...,Nk} is a complement of J up to degree k and B1 ∪ ...∪Bk is the associated border basis up to degree k.

By Proposition 2.8 a border basis B of J up to degree s is a set of generators of the ideal I(J1,...,Js), but in general it is not minimal. We will see how one can eliminate redundant polynomials in B so as to obtain a minimal set of generators of that ideal.

Proposition 2.10. Let J be a proper homogeneous ideal in P. Assume that N = {N0,...,Ns} is a complement of J up to degree s and let B = B1 ∪ ...∪Bs be the associated border basis. Then: B N + B∪ (1) The ideal L = I( , s ) is homogeneous and zero-dimensional, and N + N s is the border basis of L associated to its complement . B N + (2) The module Syz( , s ) is generated by vectors whose entries have degree at most 1.

Proof. (1) Since Ls = Js,wehavethatPs = Ls ⊕Ns.Moreover,since N + ⊆ N ∅ N N s L, choosing s+1 = we see that the triple L, s, s+1 satisﬁes the hypotheses of Lemma 2.4 (we set ∅ = {0}). Thus we get that Ps+1 = Ls+1;sothe ideal L is zero-dimensional. If we set Nj = ∅ for all j ≥ s +1,thenN = {Nj }j∈N N N + B N + is a complement of L.Moreover,since(∂ )s+1 = s ,ifwelet s+1 = s ,then B∪N+ s is a border basis of L. N ⊆N+ N { } N (2) Since k+1 k for each k and 0 = 1 ,wehavethat is connected to 1 (see Remark 2.3). Thus it is possible to apply Theorem 4.3 in [MT], which implies that the syzygies among the elements of a border basis of a zero-dimensional ideal with a complement connected to 1 can be generated by syzygies whose coeﬃcients have degree at most 1.

IDEALS OF CURVES GIVEN BY POINTS 77

By changing bases in each constant degree subspace we obtain the following more general result: Proposition 2.11. Let J be a proper homogeneous ideal in P. Assume that N = {N0,...,Ns} is a complement of J up to degree s.LetV = V1 ∪...∪Vs where V ∩N+ V N + k is a basis of Jk k−1 for each k =1,...,s. Then the module Syz( , s )is generated by vectors whose entries have degree at most 1.

The following corollary shows that the redundant elements in Vs can be expressed as a combination of elements in Vs−1 and the other elements in Vs:

Corollary 2.12. Under the hypotheses of Proposition 2.11, let f ∈Vs and W V \{ } ∈ V V W ∈V+ W denote s = s f .Iff I( 1,..., s−1, s), then f s−1, s .

Proof. By hypothesis there exists a syzygy among the elements of V = V1 ∪ ...∪Vs such that the coefficient of f is a non-zero constant. Hence by Proposition V N + 2.11, there exists a homogeneous generator of Syz( , s ) whose entries have degree at most 1 and where the coefficient of f is a non-zero constant. Since deg f = k and the coefficient of f is constant, in this generating syzygy only the elements of Vk−1 ∪Vk can have non-zero coefficients. We now describe two methods to construct a minimal set of generators of the ideal I(J1,...,Js) depending on whether we start with a set of generators which form a border basis or not.

Proposition 2.13. Let s ∈ N and let N = {N0,...,Ns} be a complement up to degree s of a proper homogeneous ideal J in P.Given{V1,...,Vs} where Vk is ∩N+ a basis of J k−1 ,thenforeachk =1,...,s it is possible to construct a set of polynomials Gk ⊆Vk such that G1 ∪ ...∪Gs is a minimal set of generators of the ideal I(J1,...,Js).

Proof. Let G1 = V1 and assume that G1 ∪ ... ∪Gk−1 is a minimal set of generators of I(J1,...,Jk−1) with Gi ⊆Vi for i =1,...,k− 1. Note that a polynomial f ∈Vk is redundant w.r.t. G1 ∪ ...∪Gk−1 ∪Vk if and only if it is redundant w.r.t. V1 ∪...∪Vk. Thus, by Corollary 2.12 it suffices to look V+ ∪V G ⊆V for linear relations among the elements of k−1 k and for a set k k such that V+ V V+ ⊕G V+ ∩V k−1 + k = k−1 k . Hence it is sufficient to find a basis of k−1 k , extend it with elements w1,...,wt to a basis of Vk and define Gk = {w1,...,wt}. V+ ∩V In order to compute the intersection k−1 k consider the monomial basis ∪ ∪ V+ V N + \N+ N S = S1 S2 S3 of k−1 + k ,whereS1 =(∂ )k−1 k−1, S2 =(∂ )k and N N + N ∪N V+ ⊂ N + N + S3 = k.Observethat k−1 =(∂ )k k and k−1 (∂ )k−1 + k−1 .Let | | |V+ | |V | | N | si = Si ,fori =1, 2, 3, and l = k−1 ; with this notation k = (∂ )k = s2. Let U be the matrix whose columns contain the coefficients of the polynomials V+ ∪V × of k−1 k with respect to S.ThusU is a (s1 + s2 + s3) (l + s2) block matrix of the form ⎛ ⎞ U1 0 ⎝ ⎠ U = U2 U3 U4 U5 l s2 s2 and, if we denote by π2 : K × K → K the projection on the last s2 coordinates, V V+ ∩ the vectors of π2(Ker U) are the coordinates (w.r.t. k) of the vectors of k−1 Vk.

78 E. FORTUNA, P. GIANNI, AND B. TRAGER

In order to compute Ker U we can reduce ourselves to consider the matrix U 0 U = 1 . U2 U3 Namely, since S3 = Nk,ifUv =0,thenUv ∈ Jk ∩Nk = {0}, hence Ker U = Ker U. In order to ﬁnish the construction it is then suﬃcient to reduce to echelon form the matrix whose rows are generators of π2(Ker U): the indexes of the columns without pivots correspond to the elements in Vk to select for constructing Gk.

The proof of the previous proposition guarantees the correctness of the following: Algorithm MinimalBasis Input: - s ∈ N - {N0,...,Ns} a complement of a homogeneous ideal J up to degree s {V V } V ∩N+ - 1,..., s where k is a basis for J k−1 Output: {G1,...,Gs} where:

— Gk ⊂Vk for each k ∈{1,...,s} — the polynomials in G1 ∪...∪Gs are a minimal set of generators of the ideal I(J1, .., Js) Procedure:

- G1 = V1 -fork =2..s repeat N + \N+ S1 := (∂ )k−1 k−1 S2 := (∂N )k |V+ | l := k−1 s1 := |S1| s2 := |S2| —constructthe(s1 + s2) × (l + s2) matrix with columns the s1 + s2 ∪ ∈V+ ∪V — coeﬃcients w.r.t. S1 S2 of the polynomials v k−1 k. ∪ | ∈V+ ∪V U := matrix (coeﬀs(v, S1 S2) v k−1 k) + ∩V — compute the intersection Vk−1 k K := Ker(U) dk := |K| —constructthedk × s2 matrix with rows the last s2 entries of the — vectors in K. MK := matrix(π2(v) |v ∈ K) (RMK, Σ) := RRE(MK) — the polynomials in Vk with index not in Σ are put in Gk Gk := {vj ∈Vk |j ∈ Σ}

In the case when we start with a border basis, we can improve our previous + ∩V construction. The computation of the generators of the intersection Vk−1 k can then be accomplished with only some column subtractions, and the construction of each level of the minimal basis can be completed with one column echelon reduction of an s2 × (l − s1) matrix. The new algorithm is based on the following

IDEALS OF CURVES GIVEN BY POINTS 79

Proposition whose proof is an immediate consequence of the properties of a border basis.

Proposition 2.14. Let s ∈ N and let J be a proper homogeneous ideal in P. Assume that N = {N0,...,Ns} is a complement of J up to degree s and let B = B1 ∪ ...∪B s be the associated border basis. With the notation of the previous U 0 proof, if U = 1 we have: U2 U3

(i) after reordering the elements of Bk we can assume that the s2 × s2 block

U3 is the identity matrix Is2 (ii) each element in U1 is either 0 or 1; more precisely each row in U1 contains at least one element equal to 1 and each column in U1 contains at most one element equal to 1 (iii) by means of ﬁnitely many subtractions performed on the left l columns of U we can reduce U to the form I 0 0 U = s1 P1 P2 Is2 (iv) the set of the columns of the s2×(l−s1)matrixP2 is a basis of π2(Ker U)= π2(Ker U) (v) U has full row-rank and so dim Ker U = dim Ker U = l − s1 (vi) reducing P2 to echelon form by column operations, the pivots indicate the redundant elements in Bk.

3. Curves given by points A natural application of the results of the previous section is the construction of the ideal of an irreducible projective curve starting from the knowledge of a ﬁnite setofpointsonit.Letusrecallthefollowingclassicresult:

Proposition 3.1. Assume that C is an irreducible projective curve in Pn(K) of degree d.LetR = {R1,...,Rh} be a set of points on C.

(1) For all s ∈ N such that h>sd,wehaveI(C)≤s = I(R)≤s. (2) If I(C) can be generated by polynomials of degree at most m and h>md, then I(C)=P·I(R)≤m

Proof. (1) It suﬃces to prove that I(R)k ⊆I(C)k for all k ≤ s.Iff ∈I(R)k, the polynomial f vanishes on h>sd≥ kd =degf · deg C points. Since C is irreducible, by B´ezout’s Theorem the hypersurface V (f) contains C, i.e. f ∈I(C)k. (2) By hypothesis I(C)=P·I(C)≤m, thus the result follows immediately from (1).

The previous result allows us to reduce the construction of I(C) to the computation of a set of generators for the ideal J = I(R)whereR = {R1,...,Rh} is a set of h points in Pn(K). By Proposition 2.8 this can be done by computing a border ∩N+ basis of J. In the case of an ideal of points, we are able to compute Jk k−1 using the point evaluation maps.

80 E. FORTUNA, P. GIANNI, AND B. TRAGER

N N + { }⊆P Assume that we have computed k−1 and let k−1 = m1,...,mt k. Consider the h × t evaluation matrix ⎛ ⎞ m1(R1) ... mt(R1) ⎜ . . ⎟ MR = ⎝ . . ⎠ m1(Rh) ... mt(Rh) where, if Ri =[ri,0,...,ri,n], by mj (Ri)wemeanmj(ri,0,...,ri,n). Note that the rank of the matrix MR and its null-space Ker MR does not depend on the chosen representation of the points in the projective space. Note also that each vector in ∩N+ Ker MR is the vector of the coordinates of a polynomial in Jk k−1 w.r.t. the { } ∩N+ basis m1,...,mt . In particular dim Ker MR = dim(Jk k−1 ). Performing Gaussian elimination by rows, followed if necessary by a permuta- { } tion of the columns (which corresponds to a permutation of the basis m1,...,m t ), Ir A −A we can assume that MR = . In this way the columns of are 0 0 It−r a basis of the null-space Ker MR.IfwechooseasNk the first r monomials in the permuted basis, we have the null-space in border form, which gives us Bk. We now want to estimate the complexity of our procedure to compute a minimal set of generators up to degree s of the ideal J of h distinct points. Our algorithm first computes a border basis up to degree s, then minimizes this basis removing redundant elements. The basic tool is Gaussian elimination; recall that the complexity of Gaussian elimination performed on a m × p matrix is O(mp min(m, p)). As for the first phase to compute the border basis, the k-th step of the recursive procedure described above to compute Nk and Bk requires to perform Gauss- ian elimination on the h × t matrix MR. As already observed, dim Ker MR = ∩N+ − N dim(Jk k−1 ); hence, by Remark 2.3, dim Ker MR = t dim k .Inparticu- lar dimNk = |Nk| =rkMR ≤ h.Thusforeachi =1,...,swe have that |Ni|≤h |N +|≤ and hence i (n +1)h. Therefore the complexity of each step of the algorithm is O(nh3). As a consequence the complexity of the recursive algorithm in s steps 3 to compute a border basis of I(J1,...,Js)isO(snh ). As for the minimizing phase outlined in Proposition 2.14, note that for each k |B | | N |≤ |B+ |≤ 2 we have that k = (∂ )k (n +1)h, k−1 (n +1) h and hence |B| |B+ ∪B |≤ k = k−1 k (n +1)(n +2)h. B+ ∪B Moreover the distinct monomials appearing in the polynomials of k−1 k form N ++ 2 a subset of k−2 and therefore they are at most (n +1) h. |B+ |≤ 2 Since the number of the left columns in U is k−1 (n+1) h, when we reduce the matrix U to the form U by means of subtractions among the left columns, we need at most (n +1)2h column subtractions. The length of each of these columns is at most (n +1)2h, thus the complexity of the reduction of U to U is O(n4h2). The last part of the minimizing phase requires to reduce the matrix P2 to echelon form by column operations. The number of rows of P2 is equal to |(∂N )k|≤ (n +1)h, while the number of its columns is ≤ (n +1)2h.Thusthecomplexityof 4 3 the algorithm to reduce P2 is O(n h ). Hence the complexity of the algorithm outlined in Proposition 2.14 to compute 4 3 a minimal set of generators for J≤s is O(sn h ).

IDEALS OF CURVES GIVEN BY POINTS 81

4. Curves in Pn(C) given by approximate points

In this section we consider the situation where the points R = {R1,...,Rh} on the irreducible projective curve C in Pn(C) are given only approximately. In this case we can perform the computations needed for the described procedure by replacing Gaussian elimination by more suitable and numerically stable tools. As observed in Section 2, if one is only interested in computing a set of generators of I(J1,...,Js) (not necessarily a border basis up to degree s), it is sufficient ∩N+ N to compute a basis of Jk k−1 and compute a complement k for each k (see Remark 2.5). The former task corresponds to computing a basis of the null-space of the h × t matrix ⎛ ⎞ m1(R1) ... mt(R1) ⎜ . . ⎟ MR = ⎝ . . ⎠ m1(Rh) ... mt(Rh) { } N + where m1....,mt = k−1. A numerically stable way to compute both the rank and an orthogonal basis of Ker MR is the SVD-algorithm which assures that one can find a unitary h×h matrix t U, a unitary t × t matrix V and an h × t real matrix Σ such that MR = UΣV ;the elements σij of the matrix Σ are zero whenever i = j and for i =1,...,l=min{h, t} we have σ1,1 ≥ ...≥ σl,l ≥ 0. Either we know the rank of MR (see for instance Proposition 5.8) or we can examine the singular values σi of Σ in order to obtain a rank determination as in [GVL]. In any case if rk MR = r, then by the properties of the SVD decomposition the last t − r columns of V are an orthogonal basis of Ker MR. In order to compute Nk and continue to the next step, we take advantage of the stability properties of the QRP-algorithm which, given a matrix M,constructs a unitary matrix Q, a permutation matrix P and an upper-triangular matrix R = R1 R2 such that S = QRP . The permutation matrix P exchanges columns in order to improve the condition number of the matrix R1.IfM has full row-rank, then R1 is invertible; otherwise it is possible to use the diagonal elements of R1 to make a rank determination of M. In our case we apply the QRP-algorithm to the (t − r) × t matrix S whose rows − N + are the last t r columns of V ; recall that the columns of S are indexed by k−1. In the decomposition S = QRP the columns of R are a permutation of the columns of S and the monomials corresponding to the columns of R1 will be chosen to be border monomials, while the monomials corresponding to the columns of R2 will be chosen as the complement Nk. Observe that the rows of R correspond to a new ∩N+ basis for Jk k−1 . If we want to compute a border basis of I(J1,...,Js), we can compute the −1 −1 B matrix R1 R = I R1 R2 whose rows correspond to a basis k of Ker MR consisting of border polynomials. Otherwise, if we want to compute a minimal set of generators of I(J1,...,Js), we can proceed as described in the algorithm MinimalBasis using SVD to compute kernels and QRP to select stable pivot columns. Using the notation of Proposition 2.13, the first step is to compute a basis of Ker U. In order to do this, we use an SVD construction taking into account that, by Proposition 2.14 (v) dim Ker U = dim Ker U = l−s1. We then apply the QRP-algorithm to the matrix N whose rows

82 E. FORTUNA, P. GIANNI, AND B. TRAGER contain the projection by π2 of a set of generators of Ker U and whose columns ∩N+ are indexed by the generators of Jk k−1 .WethusobtainN = Q R P . Examining the diagonal elements of R we can determine its rank r; the columns of ∩N+ R correspond to a permuted basis of Jk k−1 and the generators corresponding to the ﬁrst r columns are redundant and can be discarded. Exact Gaussian elimination, SVD and QRP-algorithm applied to an m × n matrix all have the same complexity O(mn min(m, n)). In the approximate algorithm the computation of the null-space using SVD has a complexity O(nh3)andis followed by a QRP-algorithm which has a complexity O(n3h3). Thus the complex- 3 3 ity to compute a set of generators or a border basis of I(J1,...,Js)isO(sn h ), while the complexity to compute a minimal set of generators of I(J1,...,Js)bythe algorithm MinimalBasis is O(sn6h3). Alternatively, after computing a border basis, we can give a numerical algorithm based on Proposition 2.14, which would reduce the complexity to O(sn4h3)witha slight loss of numerical precision. Example 4.1. We implemented our algorithm in Octave. Here is the result we obtained when we tested it on the following parametric sextic space curve C taken from [JWG]: x =3s4t2 − 9s3t3 − 3s2t4 +12st5 +6t6 y = −3s6 +18s5t − 27s4t2 − 12s3t3 +33s2t4 +6st5 − 6t6 z = s6 − 6s5t +13s4t2 − 16s3t3 +9s2t4 +14st5 − 6t6 w = −2s4t2 +8s3t3 − 14s2t4 +20st5 − 6t6. By Theorem 5.1 the ideal of this curve of degree 6 in P3(C) can be generated by polynomials of degree at most 5. We chose 31 > 6 · 5 points using roots of unity of the following form: s =1; t = exp(2πik/31) k ∈{1,...,31}. Running the algorithm BorderBasisWithComplement, we obtained no polynomials of degree 1 (showing that the ideal is not contained in any hyperplane), no polynomials of degree 2, 4 polynomials of degree 3, 11 of degree 4 and 22 of degree 5, yielding a set of generators for the ideal in .08 seconds. Among the 20 monomials of degree 3 the QRP-algorithm chose z2x, yxw, y2w, z2w as border monomials and the remaining 16 as generators of a complement. We obtained a minimal basis consisting of only the 4 polynomials of degree 3 in an additional time of .02 seconds: 2 2 2 f1 = z x +0.06666666666 z y +0.06805555555 zy − 0.0361111111 zyx −0.2833333333 zyw − 0.55 zx2 − 1.066666667 zxw +0.01527777778 y3 −0.09166666666 y2x +0.3055555555 yx2 +0.1833333334 x2w,

2 2 f2 = yxw +0.2 z y +0.1416666667 zy − 0.4833333333 zyx − 0.1 zyw −0.9 zx2 − 0.2000000001 zxw +0.025 y3 − 0.15 y2x +0.5 yx2 +0.3000000001 x2w,

2 2 2 f3 = y w − 0.8 z y − 0.3166666667 zy − 0.5666666667 zyx +0.4 zyw +0.6000000001 zx2 +0.8000000002 zxw − 0.01666666666 y3 +0.09999999999 y2x − 0.3333333333 yx2 − 0.2000000002 x2w,

IDEALS OF CURVES GIVEN BY POINTS 83

2 3 2 2 f4 = z w − 0.6666666667 z − 0.162962963 z y +0.06049382717 zy −0.03209876545 zyx +0.9703703704 zyw − 0.4888888888 zx2 +0.3851851853 zxw − 0.1666666667 zw2 +0.01358024691 y3 −0.08148148149 y2x +0.2716049383 yx2 − 0.9444444445 yw2 +0.1629629629 x2w − 0.2222222224 xw2

Using continued fractions, we then attempted to convert the coeﬃcients from ﬂoating point numbers to rational numbers, obtaining the following polynomials:

2 1 2 49 2 11 3 − 13 − 11 2 − 11 2 f1 = z x + 15 z y + 720 zy + 720 y 360 zyx 120 y x 20 zx 11 2 − 17 − 16 11 2 + 36 yx 60 zyw 15 zxw + 60 x w,

1 2 17 2 1 3 − 29 − 3 2 − 9 2 1 2 f2 = yxw + 5 z y + 120 zy + 40 y 60 zyx 20 y x 10 zx + 2 yx − 1 − 1 3 2 10 zyw 5 zxw + 10 x w,

2 − 4 2 − 19 2 − 1 3 − 17 1 2 3 2 − 1 2 f3 = y w 5 z y 60 zy 60 y 30 zyx + 10 y x + 5 zx 3 yx 2 4 − 1 2 + 5 zyw + 5 zxw 5 x w,

2 − 2 3 − 22 2 49 2 11 3 − 13 − 11 2 f4 = z w 3 z 135 z y + 810 zy + 810 y 405 zyx 135 y x − 22 2 22 2 131 52 22 2 − 1 2 − 17 2 45 zx + 81 yx + 135 zyw + 135 zxw + 135 x w 6 zw 18 yw − 2 2 9 xw . The floating point coefficients were sufficiently accurate to recover the exact rational coefficients and the previous polynomials generate the exact ideal of the curve over Q.

5. Degree bounds for ideal generators Let C be an irreducible projective curve (seen as a set of points in Pn(K)) and let I = I(C) ⊂P= K[x0,...,xn] be the prime homogeneous ideal consisting of all the polynomials vanishing on C. In Section 3 we saw that the computation of I can be reduced to the computation of the ideal of suﬃciently many points on the curve (see Proposition 3.1). In order to be sure that one has enough points, it is necessary to bound the degrees of generators of the ideal I. Such a bound can be obtained from the regularity of the curve. Recall that if 0 → ... → F1 → F0 → J → 0 is a graded free resolution of a polynomial ideal J ,wesaythatJ is k-regular if each Fj can be generated by polynomials of degree ≤ k + j.Thenwecallregularity of J the integer reg(J)= min{k | J is k-regular}. If reg(C)=reg(I(C)) = m, then (see for instance [E]) I can be generated by homogeneous polynomials of degree at most m; moreover the Hilbert function HFI (s) and the Hilbert polynomial HPI (s)ofP/I take the same value for all integers s ≥ m. The next result gives information about the regularity of the curve as a function of the curve degree and of the dimension of the embedding projective space: Theorem 5.1 (Gruson-Lazarsfeld-Peskine [GLP]). Let D⊂Pn(K) be a reduced and irreducible curve of degree d not contained in any hyperplane, with K algebraically closed and n ≥ 3.Thenreg(D) ≤ d − n +2. Moreover, if D has genus g>1 then reg(D) ≤ d − n +1.

84 E. FORTUNA, P. GIANNI, AND B. TRAGER

The previous result and many of the degree bounds apply to curves not contained in any hyperplane, i.e. non-degenerate, which happens if and only if I1 = n I(C)1 = (0). On the other hand a degenerate curve in P (K) is a non-degenerate curve in the projective subspace V (I1) having dimension n − dim I1. So the bounds of this section apply to degenerate curves if we replace n by n − dim I1. Remark 5.2. A parametrization of a curve C ⊂ Pn(K) can be regarded as a machine delivering as many points on the curve as needed. Moreover if 1 n ψ : P (K) → P (K) is a rational map whose image is the curve C and ψ([t0,t1]) = [F0(t0,t1),...,Fn(t0,t1)] with Fi(t0,t1) homogeneous polynomials of degree s,then deg C ≤ s. Using this upper bound for the curve degree, via Theorem 5.1 we obtain an upper bound for the degrees of generators of C. This allows us to use our procedure to compute generators of the ideal I of the curve and in particular to compute an implicit representation of C starting from a parametric one. Sharper bounds for the degrees of generators can be obtained for certain curves obtained as the image of the embedding given by a complete linear series. A ﬁrst result in this direction concerns canonical curves (see for instance [SD2]): Theorem 5.3 (Petri [P]). The ideal of a non-hyperelliptic canonical curve of genus g ≥ 4 can be generated by polynomials of degree 2 and of degree 3. Proposition 5.4. Let C ⊂ Pn(K) be a non-degenerate irreducible curve of geometric genus g ≥ 4 and of degree d =2g − 2, with K algebraically closed and n = g − 1. Then the ideal I = I(C) can be generated by polynomials of degree 2 and of degree 3. Proof. By Theorem VI.6.10 in [W] any non-degenerate curve of genus g and degree 2g − 2 embedded in Pg−1(K) is a smooth non-hyperelliptic canonical curve. Then the conclusion follows from Theorem 5.3. The following result of Saint-Donat gives bounds for the degrees of generators of curves of genus g which are the image of an embedding given by a complete linear series of suﬃciently high degree: Theorem 5.5. ([SD1])LetC ⊂ Pn(K) be an irreducible nonsingular curve of genus g embedded by a complete linear series of degree d. (1) If d ≥ 2g +1, then the ideal I of C can be generated by polynomials of degree 2 and of degree 3. (2) If d ≥ 2g +2,thenI can be generated by polynomials of degree 2. The following result shows that we can remove the condition of being embedded by a complete linear series: Proposition 5.6. Let C ⊂ Pn(K) be a non-degenerate irreducible curve of geometric genus g and degree d, with K algebraically closed and n = d − g.Then (1) If d ≥ 2g + 1, then the ideal I = I(C) can be generated by polynomials of degree 2 and of degree 3. (2) If d ≥ 2g +2,thenI = I(C) can be generated by polynomials of degree 2. Proof. Recall that any curve C of degree d and genus g in Pn(K) can be seen as the image of the map given by a linear series contained in L(D) for some divisor D of degree d on a non-singular model C of C.

IDEALS OF CURVES GIVEN BY POINTS 85

For any divisor D on C,denotebyl(D) the dimension of the complete linear series L(D). We also denote by K a canonical divisor. If d ≥ 2g +1,then l(K − D) = 0 and hence by Riemann-Roch Theorem l(D)=d − g +1;moreover(see[F]) the map induced by D is an embedding. Thus we can see C as the image of an embedding of C in Pd−g(K). So, since C is non-degenerate, C is embedded by a complete linear series if and only if n = d − g. Moreover, if d ≥ 2g+1 and n = d−g,thecurveC is the image of a non-singular curve through a map which is an embedding, therefore C is non-singular. Hence the hypotheses of Theorem 5.5 are fulﬁlled and it implies our result. Example 1. Assume that C has genus g ≥ 3 and is embedded by the bicanonical map. Since deg(2K)=4g − 4 ≥ 2g + 2, by Theorem 5.5 I can be generated by quadratic polynomials. Example 2. If C has genus g = 2 and is embedded by the tricanonical map (which gives an embedding because deg(3K)=6g − 6=6≥ 2g +1),since deg(3K) ≥ 2g + 2 again by Theorem 5.5 I can be generated by polynomials of degree 2. Other results giving bounds for the degrees of generators of the curve ideal in diﬀerent situations are available in the literature; the following one (see [A]) gives results in the hyperelliptic case: Proposition 5.7 (Akahori). Let C ⊂ Pn(K) be an irreducible non-degenerate and non-singular hyperelliptic curve of genus g and degree d. (1) If d =2g, then the ideal I of C can be generated by polynomials of degrees 2, 3 and 4. (2) If d =2g − 1, then I can be generated by polynomials of degrees 2, 3, 4 and 5.

When we need to compute the null-space of the evaluation matrix MR for approximate points (see Section 4), we can try to determine its rank by inspecting its singular value spectrum. Since this is not guaranteed to work, it is useful to predict what the rank should be. Since the rank of MR equals |Nk| =dimPk − dim Ik, the following proposition gives a situation where we can predict this value. Recall that if g0,...,gn is a basis of the complete linear series L(D)ofdimension l(D)=n +1,thenIk =Kerϕk where ϕk : Pk →L(kD) deﬁned by ϕk(xi)=gi.

Proposition 5.8. If ϕk is surjective and k · deg(D) ≥ 2g − 1, then

dim Pk − dim Ik = k · deg(D) − g +1.

Proof. If ϕk is surjective, then dim Ik = dim Ker ϕk =dimPk − dim L(kD). Since deg(kD)=k · deg(D) ≥ 2g − 1, then i(kD) = 0. Hence by Riemann-Roch Theorem we get l(kD)=deg(kD) − g +1=k · deg(D) − g + 1 and the conclusion follows. Examples. 1. Assume that C is non-hyperelliptic of genus g ≥ 4andtake the canonical divisor D = K; in particular deg(D)=2g − 2andl(D)=g (i.e. in this case n = g − 1). By Theorem 5.3 we already know that I can be generated by polynomials of degree 2 and of degree 3, and so it suﬃces to know dim I2 and dim I3. Then by Noether’s Theorem the map ϕk : Pk →L(kD) is surjective for all k. Furthermore for all k ≥ 2wehavethatk · deg(D) ≥ 2g − 1, hence by Proposition

86 E. FORTUNA, P. GIANNI, AND B. TRAGER

5.8 we have dim Pk − dim Ik = k(2g − 2) − g +1. In particular

|N2| =dimP2 − dim I2 =2(2g − 2) − g +1=3g − 3

|N3| =dimP3 − dim I3 =3(2g − 2) − g +1=5g − 5.

2. If we choose a divisor D such that deg(D) ≥ 2g +1,thenϕk : Pk →L(kD) is surjective (see [M]); moreover for all k ≥ 1wehavethatk · deg(D) ≥ 2g − 1. Then we can compute dim Ik by Proposition 5.8. In particular if D =2K and g ≥ 3, then we know that I can be generated by quadratic polynomials. Since i(2K) = 0, by Riemann-Roch Theorem l(2K)= 3g − 3, i.e. n =3g − 4. Then

|N2| =dimP2 − dim I2 =2(4g − 4) − g +1=7g − 7. If g = 2 and we choose D =3K, we know that I can be generated by quadratic polynomials. Since i(3K) = 0, by Riemann-Roch Theorem l(3K)=5g − 5 = 5, i.e. n =4.Then

|N2| =dimP2 − dim I2 =2deg(3K) − g +1=11.

Remark 5.9. When the ideal can be generated in degree 2 and we know |N2| as above, then the algorithm to compute a minimal set of generators can be considerably simpliﬁed. If MR is the point evaluation matrix for all monomials of degree 2, since we know that rk MR = |N2|, a single application of the SVD-algorithm with this imposed rank computes a basis for the null-space of MR which directly yields a minimal set of generators of the ideal.

Acknowledgments We wish to thank Mika Sepp¨al¨a for proposing this problem to us and a referee for many helpful comments.

References [AFT] Abbott, J., Fassino, C., Torrente, M., Stable border bases for ideals of points,J.Symbolic Comput. 43 (2008), 883–894. MR2472538 (2010a:13043) [A] Akahori, K., The intersection of quadrics and defining equations of a projective curve, Tsukuba J. Math. 20 n. 2 (1996), 413–424. MR1422630 (97k:14029) [ACOR] Albano, G., Cioffi, F., Orecchia, F., Ramella, I., Minimally generating ideals of rational parametric curves in polynomial time, J. Symbolic Comput. vol. 30 n. 2 (2000), 137–149. MR1777168 (2001h:13036) [C] Cioffi, F., Minimally generating ideals of points in polynomial time using linear algebra, Ricerche Mat. vol. 48 n. 1 (1999), 55–63. MR1757287 (2001d:13031) [E] Eisenbud, D., The geometry of syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics 229 Springer-Verlag, New York, 2005. MR2103875 (2005h:13021) [F] Fulton, W., Algebraic curves. An introduction to algebraic geometry, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR1042981 (90k:14023) [GSST] Gianni, P., Seppälä, M., Silhol, R., Trager, B., Riemann Surfaces, Plane Algebraic Curves and Their Period Matrices,J.SymbolicComput.12 (1998), 789–803. MR1662036 (99m:14055) [GVL] Golub, G. H., Van Loan, C. F., Matrix computations, Johns Hopkins University Press, Baltimore, MD, 1996. MR1417720 (97g:65006) [GLP] Gruson, L., Lazarsfeld, R., Peskine, C., On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 n. 3 (1983), 491–506. MR704401 (85g:14033)

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[HKPP] Heldt, D., Kreuzer, M., Pokutta, S., Poulisse, H., Approximate computation of zero- dimensional polynomial ideals, J. Symbolic Comput. 44 (2009), 1566–1591. MR2561289 (2011b:13090) [JWG] Jia, X., Wang, H., Goldman, R., Set-theoretic generators of rational space curves,J. Symbolic Comput. vol. 45 n. 4 (2010), 414–433. MR2599820 (2011c:14096) [KR] Kreuzer, M., Robbiano, L., Computational commutative algebra 2, Springer-Verlag, Berlin, 2005. MR2159476 (2006h:13036) [MMM] Marinari, M. G., Möller, H. M., Mora, T., Gröbner Bases of Ideals Defined by Functionals with an Application to Ideals of Projective Points, Appl. Algebra Eng. Commun. Comput. 4 (1993), 103–145. MR1223853 (94g:13019) [MB] Möller, H. M., Buchberger, B., The Construction of Multivariate Polynomials with Preas- signed Zeros, Proc. EUROCAM 82, L.N.C.S 144 (1982), 24–31. MR680050 (84b:12003) [MT] Mourrain, B., Trébuchet, P., Stable normal forms for polynomial system solving,Theoret. Comput. Sci. 409 n. 2 (2008), 229–240. MR2474338 (2009m:13036) [M] Mumford, D., Varieties defined by quadratic equations, In: Questions on Algebraic Varieties (Corso C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Roma, (1970) pp. 29–100. MR0282975 (44:209) [P] Petri, K., Uber¨ die invariante Darstellung algebraischer Funktionen einer Veränderlichen, Math. Ann. 88 n. 3-4 (1923), 242–289. MR1512130 [SD1] Saint-Donat, B., Sur les équations définissant une courbe algébrique,C.R.Acad.Sci.Paris Sér. A-B 274 (1972), A324–A327. MR0289517 (44:6705b) [SD2] Saint-Donat, B., On Petri’s analysis of the linear system of quadrics through a canonical curve, Math. Ann. 206 (1973), 157–175. MR0337983 (49:2752) [W] Walker, R. J., Algebraic curves, Dover Publications Inc., New York, 1962. MR0144897 (26:2438)

Dipartimento di Matematica, Universita` di Pisa, Largo B. Pontecorvo 5, I-56127 Pisa, Italy E-mail address: [email protected] Dipartimento di Matematica, Universita` di Pisa, Largo B. Pontecorvo 5, I-56127 Pisa, Italy E-mail address: [email protected] IBM T.J.Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, New York 10598, USA E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11362

Non-genera of curves with automorphisms in characteristic p

Darren Glass

Abstract. We consider which integers g and σ can occur respectively as the genus and p-rank of a curve deﬁned over a ﬁeld of odd characteristic p which admits an automorphism of degree p.

1. Introduction This paper is intended to serve as a characteristic p analog to the paper by O’Sullivan and Weaver [6]. In that paper, the authors consider for which genera g there is a Riemann surface of genus g which admits an automorphism of order n for various choices of n. In this note, I consider the same question where we are instead working over an algebraically closed field of characteristic p and looking at curves admitting a Z/pZ-action. We determine which genera g can occur for such curves. Recall that the p-rank of a curve defined over a field k of characteristic p is the integer σ such that the cardinality of Jac(X)[p](k)ispσ.Itiswellknown that 0 ≤ σ ≤ g and in this note we establish conditions on pairs (g, σ) so that there exist curves of genus g and p-rank σ which admit a Z/pZ-action. In the case p = 2, Zhu has shown in [9] that all pairs (g, σ) with g ≥ σ ≥ 0 occur as the genus and 2-rank of curves over F2, even for hyperelliptic curves with automorphism group exactly Z/2Z.In[4], the author considers curves admitting a Z/2mZ-action in characteristic 2 for all odd m. In light of these results, we restrict our attention to the situation where our field has odd characteristic in this note. In particular, if we let a, b be the submonoid of Z generated by a and b (i.e. a, b = {ax + by|x, y ∈ Z≥0}) then we show in Sections 2 and 3 the following necessary conditions for such a curve to exist. Theorem 1.1. Let X be a curve of genus g and p-rank σ which admits a Z/pZ-action. Then we have the following conditions on g and σ. • ∈ p−1 ≡ Either g p, 2 or g 1 (mod p). • Either σ ∈p, p − 1 or σ ≡ 1 (mod p) • − ∈ p−1 g σ p, 2 These conditions are not sufficient for such a curve to exist; the difficulty comes because it is not possible to construct functions on arbitrary curves with arbitrary numbers of branch points and ramification degrees. Sections 2 and 3 prove that under additional hypotheses we can get sufficiency. One example of such a result

2010 Mathematics Subject Classiﬁcation. Primary 14H37, 14H40; Secondary 11G20.

c 2012 American Mathematical Society 89

90 DARREN GLASS is Theorem 2.4 which gives precise conditions under which a curve of genus g with a Z/pZ-action exists. Another example is the following: Corollary 1.2. Let σ ≥ (p − 1)(p +2). Then there exist curves of genus g Z Z − ∈ p−1 and p-rank σ admitting a /p -action if and only if g σ p, 2 . We note that our results generalize the results in [7, Lemma 2.7], in which the authors considered the possible p-ranks of Artin-Schreier curves. Our results allow the quotient curve to have genus gY > 0, which allows for more possible values of gX and σ. The main approach in our investigation will be to assume that X admits a Z/nZ-action with quotient Y , and consider the cover X → Y . We then use the Riemann-Hurwitz formula to compare the genera of X and Y and the Deuring-Shafarevich formula to compare their p-ranks. We also use results about the Frobenius Problem (also known as the coin problem or the conductor problem), which asks what numbers are representable as nonnegative integral linear combinations of ﬁxed integers. In particular, we recall the following theorem due to Sylvester [8], which is standard in any undergraduate number theory text: Theorem 1.3. Let a and b be ﬁxed coprime integers. Then any integer d> ab − a − b can be expressed as a linear combination d = ax + by where x, y ∈ Z≥0. Moreover, ab−a−b ∈a, b and exactly half of the integers between 1 and ab−a−b+1 are in a, b.

More generally, we will consider the sets a1,...ak of integers which can be expressed as the linear combination a1x1 + ...akxk for nonnegative choices of xi. While Sylvester’s theorem gives us a description of these sets in the case where k = 2, the question becomes more difficult in the case where k ≥ 3. In particular, while it is known that Z≥0 −a1,...ak is a finite set, when k ≥ 3 even finding the largest number in this set is NP-hard [2]. The author would like to thank Tony Weaver, Rachel Pries, and the referees for many valuable suggestions.

2. Non-genera for Z/pZ-actions

Let us begin by considering what genera occur as gX for some cover X → Y whose degree is p when working over a field of odd characteristic p. We recall that a Z/pZ-cover X → Y is defined by an equation T p − T = F where F is a function on the curve Y . Moreover, if the function F has poles of order ni all of which are relatively prime to p, then the Riemann-Hurwitz formula in characteristic p tells − p−1 us that gX = pgY p +1+ 2 (ni + 1). Throughout this paper, we will define the ramification type of a function with m poles of orders ni to be the m-tuple (n1,...,nm). To illustrate our method, we begin by considering some examples. Example . 2.1 Let p =3. The Riemann-Hurwitz formula implies that gX = 3gY − 2+ (ni +1). Let us consider the case where gY =0, and consider curves ramified at two points, so that gX = n1 + n2. We note that the only restriction on the values of ni is that they cannot be multiples of 3. In particular, one can obtain all values of gX ≥ 2 by setting either n1 =1and n2 = gX − 1 or n1 =2 and n2 = gX − 2. Moreover, one can construct a curve with gX =0(resp. 1) by looking at the cover X → Y ramified at a single point with ramification degree 1 (resp. 2). This implies that every gX occurs as the genus of an Artin-Schreier curve in characteristic 3 ramifiedinatmosttwopoints.

NON-GENERA OF CURVES WITH AUTOMORPHISMS IN CHARACTERISTIC p 91

Example 2.2. Letp =5. In this case, the Riemann-Hurwitz formula implies that gX =5gY − 4+2 (ni +1).WeagainsetgY =0and allow our cover to have two ramification points, so that gX =2(n1 +n2). Moreover, all even numbers gX ≥ 4 can occur in this case, again by choosing n1 =1or 2. Furthermore, an Artin- Schreier curve of genus 0 (resp. 2) can be constructed with a single ramification point. While parity restrictions mean that we are unable to construct covers over P1 of odd genus in this case, we may be able to construct curves X of odd genus that are covers of elliptic curves. The situation here is slightly more complicated, however, as to do so one must construct functions on curves of genus 1 with prescribed ramification divisors. For example, there are no functions on elliptic curves which have a single pole of order one. As we will see below, however, the restrictions are not as severe as they may initially seem. We note the following result is true regardless of the characteristic: Lemma 2.3. Let Y be a curve of genus 0. Then for any nonnegative integer R | except R =1there exists a function F on Y with poles of order ni so that p ni for all i and (ni +1)=R. Let Y be a hyperelliptic curve of genus gY > 0. Then for any nonnegative | integer R =1, 2 there exists a function F on Y with poles of order ni so that p ni for all i and (ni +1)=R. No such function exists for R =1or 2 Proof. On any curve Y there exist constant functions. These have no poles and therefore give the existence of functions where (ni +1)=0. On a curve of genus zero, there exists a function with a single pole of order one, and therefore the appropriate power of this function will have ramification type (R − 1) as long as R ≡ 1(modp). If R ≡ 1modp we can construct a function that has one pole of order R − 3 (which will not be a multiple of p as p>2) and a second pole of order 1. These two examples prove the first part of the lemma. To prove the second part of the lemma, we note that hyperelliptic curves automatically come equipped with functions that have ramification type (2) and (1, 1) and in particular there are many of the latter type of function. It is therefore possible to consider linear combinations of these functions that will have ramification type (2k), (k, k), (2k, 1, 1) and (k, k, 1, 1) for all k>0andp |k. In particular, this allows us to get values of R of the form 2k +1, 2k +2, 2k +5and2k +6forany p |k. Every positive integer other than 1 and 2 takes one of these forms. Note that R = 1 is impossible as, if a function has a pole at a point then that order must be at least one and vice versa. Moreover, the only way to obtain R =2wouldbeto have a single simple pole, which is impossible on curves of genus g ≥ 1 (See, for example, [3, §8.2, Prop 4]). Applying this lemma to our previous example, we are able to construct curves of all odd genera other than gX =3or5asZ/5Z-covers of curves of genus one. More generally, we note that computing the set of values gX that can occur has now been reduced to something that is very similar to the two-dimensional Frobenius p−1 Problem connected to the coprime pair of numbers p and 2 .Inparticular,we can apply Theorem 1.3 to learn about the nonnegative linear combinations of p p−1 and 2 and then remove those entries where b =0anda>0 (all of which are multiples of p) and add in the entries where b = −2 (all of which are congruent to one mod p).

92 DARREN GLASS

Theorem 2.4. Define the set p − 1 p − 1 G = p, − kp|0 ≤ k< ∪{kp +1|k ∈ Z≥ }. 2 2 0 Then there exists a curve of genus g defined over an algebraically closed field of characteristic p and admitting a Z/pZ-action if and only if g ∈ G. Moreover, there p2−4p+3 p2−3p are exactly 4 nonnegative integers not in G, the largest of which is 2 .

Proof. In order to construct a curve X of genus gX which admits a Z/pZ- action, it suffices to construct a curve Y of genus gY and a function on Y so that − p−1 (ni +1)=R where gX = pgY p +1+ 2 R. Lemma 2.3 tells us that for most choices of nonnegative integers gY and R we can do this. By also allowing the case of unramified covers, it follows that gX canbeexpressedasalinearcombination p−1 − ap + b 2 where a = gY is a nonnegative integer and b = (ni +1) 2 is either equal to −2 or is a positive integer. Additionally, if a =0thenb is allowed to be 0 as well. p−1 p2−4p+3 Theorem 1.3 tells us that p, 2 consists of all but 4 nonnegative in- p2−4p+1 tegers, and that the largest integer not contained in this set is 2 .Wemust eliminate all of the genera that arise in the Frobenius problem with b =0and a>0. In particular, one cannot have curves whose genus is a multiple of p less · p−1 Z Z p−3 than p 2 admitting a /p -action, so we ‘lose’ 2 possible genera. Moreover, · p−3 the largest such number is p 2 , which is larger than the largest number not lying p−1 in p, 2 . On the other hand, if b = −2 then we have the equation g =(a − 1)p +1 where a ∈ N. Because we are only interested in the case where g ≥ 0, this tells us that g ≡ 1modp. and that any such genus can be obtained as an unramified ≡ p2−4p+1 cover. We also note that if g 1modp and g< 2 then g cannot be p−1 representable as a nonnegative linear combination of p and 2 . In particular, if p−1 ≡− ≥ − g = ap + b 2 then b 2modp and therefore b p 2.Butthisimpliesthat ≥ (p−2)(p−1) p2−4p+1 p−3 g 2 > 2 . Therefore, all of the 2 genera which are congruent to one are in fact new examples and exactly offset those genera lost in the previous paragraph. This proves the theorem.

We conclude this section by listing the values of gX that do not occur as genera of a curve admitting a Z/pZ-action for some small values of p. p non-genera 3 none 5 3, 5 7 2, 4, 5, 7, 11, 14 11 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 17, 18, 19, 22, 24, 28, 29, 33, 39, 44

3. p-ranks In this section, we consider the p-ranks which can occur for curves of various genera that are deﬁned over an algebraically closed ﬁeld of characteristic p and admit a Z/pZ-action. Our main tool will be the following fact, which follows from the Deuring-Shafarevich formula [1]:

NON-GENERA OF CURVES WITH AUTOMORPHISMS IN CHARACTERISTIC p 93

Lemma 3.1. Let X → Y be a Z/pZ-cover of curves ramified at n points, where X has p-rank σX and Y has p-rank σY .Thep-ranks are related by the formula σX = pσY +(p − 1)(n − 1). It follows from the lemma that if X is a curve admitting a Z/pZ-action ramified in at least one point then its p-rank σX is representable as a nonnegative linear combination of p and p − 1 and if the action is unramified then σX is congruent to 1modp.Moreover,anysuchp-rank can be obtained by choosing an appropriate Y and n.Thevaluesofσ that do not satisfy these conditions are given by 2, 3,...,p− 2,p+2,...,2p − 3, 2p +2,...,3p − 4,...,(p − 4)p +2

p2−p−4 andinparticular,thereare 2 non-p-ranks, as described by the following theorem. Theorem 3.2. Let σ = kp − s where 0 ≤ s

Theorem 3.3. Let X be a curve of genus gX and p-rank σX which admits a Z Z − ∈ p−1 /p -action. Then gX σX p, 2 .

Proof. Let X be a curve of genus gX and p-rank σX which admits a Z/pZ- action ramiﬁed at n points and let Y be the quotient of the curve X by the Z/pZ- action. One can compute from the Riemann-Hurwitz formula that gX = pgY −(p− p−1 1) + 2 R where R is the degree of the ramiﬁcation divisor and in particular must be at least 2n. Setting a = gY − σY ≥ 0andb = R − 2n ≥ 0, we compute: p − 1 g = pg − (p − 1) + R X Y 2 p − 1 p − 1 = p(g − σ )+pσ − (p − 1) + (R − 2n)+ (2n) Y Y Y 2 2 p − 1 = pa + σ − (n − 1)(p − 1) − (p − 1) + b +(p − 1)n X 2 p − 1 = pa + b + σ 2 X p − 1 ∈p, + σ 2 X

In order to prove conditions which are suﬃcient in addition to being necessary, we need to show when there exists a function that has prescribed choices of R and n. The following lemma will give some existence results in this direction. Lemma 3.4. Let Y be a hyperelliptic curve and let R and m be integers such that R ≥ 2m>0 and R ≡ m (mod 2). Additionally, if m =1assume that R ≡ 1 (mod p)andifm =2assume that R ≡ 2 (mod p). Then there exists a function F on Y which has m poles of orders n1,...nm so that p |ni for all i and (ni +1)=R.

94 DARREN GLASS

Proof. As in the proof of Lemma 2.3 we begin by noting that hyperelliptic curves admit a function with a single pole of order 2 and they admit many functions that admit two simple poles. In particular, we can look at combinations of such functions to obtain functions with m poles that have the following ramiﬁcation types with the associated conditions on m and k Ram. Type (ni +1) m k (k, k, 1,...,1) 2k +2m − 2 m ≥ 2 & even p |k (k − 1,k− 1, 2, 2, 1,...,1) 2k +2m − 2 m ≥ 4 & even p |(k − 1) (2k, 1,...,1) 2k +2m − 1 m ≥ 1&odd p |k (2k − 2, 2, 2, 1,...,1) 2k +2m − 1 m ≥ 3&odd p |(k − 1) This proves the lemma.

Theorem 3.5. Let σ = rp + s(p − 1) with r ≥ 0 and s ≥ 2.Let =1if s is − − p−1 ∈ − even and =0if s is odd. If g σ 2 p, p 1 then there exists a curve of genus g and p-rank σ admitting a Z/pZ-action. We note that if σ is sufficiently large then one can express it in the desired form foreitherevenoroddchoicesofs. Explicitly, if r ≥ p − 1thenrp + s(p − 1) = (r − p +1)p +(s + p)(p − 1) and s + p will have opposite parity as s. Corollary 1.2 is an immediate consequence of Theorem 3.5. Proof. − − p−1 By the hypotheses, we can write gX = ap+b(p 1)+rp+s(p 1)+ 2 for some a, b ≥ 0. We wish to construct a curve with genus gX and p-rank σX which has a Z/pZ-action. In order to do this, we define σY = r and gY = a + r. It follows from [5] that there exist hyperelliptic curves of genus gY and p-rank σY ;letY be one such curve. It follows from Lemma 3.4 that as long as s ≥ 2 there exists a function F on Y which is ramified at s + 1 points with ramification degree n1,...,ns+1 so that p (ni +1) = 2s+2b+2+.WeletX be the curve defined by the cover T −T = F . It follows from the Riemann-Hurwitz and Deuring-Shafarevich formulae that: p − 1 genus(X)=pg − (p − 1) + ( (n +1)) Y 2 i p − 1 = ap + rp − p +1+ (2s +2b +2+) 2 p − 1 = ap + b(p − 1) + rp + s(p − 1) + 2 = gX and

p-rank of X = pσy +(p − 1)(n − 1) = pr +(p − 1)s

= σX as desired.

We note that allowing s =0ands = 1 would allow us to choose smaller values of σX and therefore somewhat increase our range of allowable p-ranks. However, this would add a congruence restriction on the ramiﬁcation divisor and therefore on the possible genera. We leave the details to the reader.

NON-GENERA OF CURVES WITH AUTOMORPHISMS IN CHARACTERISTIC p 95

Throughout this section, we have assumed that our base field is algebraically closed. However, we note that the construction we give proving existence will work over any field K of characteristic p so that there exists a hyperelliptic curve of genus gY and p-rank σY with an appropriate number of points defined over K. In general, the question about the minimal field over which such a curve will exist is open – in particular, it is not even known whether curves of general genus and p-rank exist over Fp, even without a restriction on the number of rational points. References 1. Richard M. Crew, Etale p-covers in characteristic p, Compositio Math. 52 (1984), no. 1, 31–45. MR742696 (85f:14011) 2. Frank Curtis, On formulas for the Frobenius number of a numerical semigroup, Math. Scand. 67 (1990), no. 2, 190–192. MR1096454 (92e:11019) 3. William Fulton, Algebraic curves. An introduction to algebraic geometry,W.A.Benjamin, Inc., New York-Amsterdam, 1969, Notes written with the collaboration of Richard Weiss, Mathematics Lecture Notes Series. MR0313252 (47:1807) 4. Darren Glass, The 2-ranks of hyperelliptic curves with extra automorphisms,Int.J.Number Theory 5 (2009), no. 5, 897–910. MR2553515 (2010h:11100) 5. Darren Glass and Rachel Pries, Hyperelliptic curves with prescribed p-torsion,Manuscripta Math. 117 (2005), no. 3, 299–317. MR2154252 (2006e:14039) 6. Cormac O’Sullivan and Anthony Weaver, A Diophantine Frobenius problem related to Riemann surfaces,Glasg.Math.J.53 (2011), no. 3, 501–522. MR2822795 7. Rachel Pries and Hui June Zhu, The p-rank stratification of Artin-Schreier curves, Annales de l’Institut Fourier, to appear. 8. J. J. Sylvester, Question 7382, Mathematical Questions from the Educational Times 41 (1884). 9. Hui June Zhu, Hyperelliptic curves over F2 of every 2-rank without extra automorphisms,Proc. Amer. Math. Soc. 134 (2006), no. 2, 323–331. MR2175998 (2006h:11074)

Department of Mathematics, Gettysburg College, 300 N. Washington Street, Get- tysburg, Pennsylvania 17325 E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11363

Numerical Schottky uniformizations of certain cyclic L-gonal curves

Rubén A. Hidalgo and Mika Seppälä

Abstract. In this paper we generalize Myrberg’s algorithm to provide numerical Schottky uniformizations of algebraic curves of the form r L L/n L−L/n y = (x − aj ) j (x − bj ) j , j=1

where nj ≥ 2 are integers, L =lcm(n1, ..., nr), where lcm stands for “lowest common multiple” and a1,..., ar, b1,..., br are pairwise diﬀerent points in the complex plane C. This will be a consequence of a numerical algorithm that permits to approximate certain type of uniformizations, called Whittaker uniformizations, of Riemann orbifolds with signatures of the form (0; n1,n1,n2,n2, ..., nr,nr).

1. Introduction A non-singular projective irreducible complex curve S (or closed Riemann surface) of genus g ≥ 2 is called a cyclic L-gonal curve, where L ≥ 2 is an integer, if there is a regular branched cover f : S → C of degree L, whose deck group is a cyclic group generated by a conformal automorphism τ : S → S of order L.Bycomposingf at the left by a suitable M¨obius transformation, if necessary, we may assume the branch values of f are given by the complex numbers c1 =0,c2 =1,c3 =2,c4, , ..., cn ∈ C. Then, S can be described by an algebraic curve of the form n L mj (1) y = (x − cj ) , j=1 where, for each j =1, ..., n, it holds that 2 ≤ mj ≤ L − 1andm1 + ···+ mn ≡ 0 mod L.Inthiscase,f(x, y)=x and τ(x, y)=(x, e2πi/Ly). By the Koebe uniformization theorem [16], the cyclic L-gonal curve S can be uniformized by a torsion free co-compact Fuchsian group F ,thatisS = H2/F , where H2 denotes the hyperbolic plane. By lifting τ to the universal cover space H2, we obtain a co-compact Fuchsian group F containing F as a normal subgroup so that τ = F/F .

2010 Mathematics Subject Classiﬁcation. Primary 30F10, 30F40. Key words and phrases. Kleinian groups, Riemann orbifolds, Numerical Uniformization. Partially supported by project Fondecyt 1110001 and UTFSM 12.11.01.

c 2012 American Mathematical Society 97

98 RUBEN´ A. HIDALGO AND MIKA SEPPAL¨ A¨

By the retrosection theorem [3, 16], the cyclic L-gonal curve S can also be uniformized by a Schottky group Γ, that is, Γ is a purely loxodromic Kleinian group, isomorphic to a free group (of rank equal to the genus of S) and with non-empty region of discontinuity Ω so that there is a regular holomorphic cover p :Ω→ S with Γ as its deck group. Unfortunately, it may be that the automorphism τ cannot be lifted to such a Schottky unifomization of S, that is, it may not be a conformal automorphism τ :Ω→ Ω (necessarily the restriction of a Möbius transformation) such that p ◦ τ = τ ◦ p. Necessary and sufficient conditions for τ to lift to a Schottky uniformization of S can be found in [12]. In our case, those conditions are equivalent to having a partition of the points c1,..., cn into a collection of pairwise disjoint pairs, say {a1,b1}, ..., {ar,br} (so n =2r), so that (i) the branch order of aj isthesameasfor bj and (ii) for any simple loop δj ⊂ C, surrounding a disc containing in its interior the points aj and bj and in the complement disc the rest of the cone points, lifts to exactly L loops on S. This obligates to consider cyclic L-gonal curves of the form r L L/nj L−L/nj (2) S : y = (x − aj ) (x − bj ) j=1 where nj ≥ 2 are integers, L =lcm(n1, ..., nr), where lcm stands for “lowest common multiple” and a1,..., ar, b1,..., br are pairwise different points in the complex plane C. The Riemann surface S, described by the equation (2), has genus r − − −1 (3) g =1+L(r 1 nj ), j=1 so that the quotient orbifold O = S/τ has signature (0; n1,n1,n2,n2, ..., nr,nr). If in (2) we set n1 = ···= nr = 2, then we obtain the hyperelliptic curve r 2 (4) S : y = (x − aj)(x − bj ), j=1 in which case τ corresponds to the hyperelliptic involution. To obtain a Fuchsian group F or a Schottky group Γ in terms of the values cj and mj is a difficult task (maybe except for the case when in Aut(S)there is a subgroup K so that τ is a normal subgroup of K and S/K has signature (0; a, b, c) for the case of Fuchsian groups). In general, one is left with the search of a numerical algorithm that permits to approximate either a Fuchsian group or a Schottky group. In the particular case n1 = ···= nr = 2 (that is, for hyperelliptic Riemann surfaces) Myrberg [21] proposed such an algorithm, but the convergence of it was not provided. In [24] Seppälä proved the convergence in the case that the hyperelliptic Riemann surface is real and, in [14], the authors provided the convergence of Myrberg’s algorithm for every hyperelliptic Riemann surface. In this paper, we generalize Myrberg’s algorithm to each curve as in (2), and prove its convergence, to provide a numerically approximation to a suitable Schot- tky group uniformizing the given curve. This paper is organized as follows. In Section 2 we recall some basic definitions and some preliminary facts on Schottky and Whittaker uniformizations. In Section 3 we provide some properties on Whittaker groups we will need later. In Section

NUMERICAL SCHOTTKY UNIFORMIZATION 99

4 we recall the opening arc process, deﬁne sequences of subgroups of Whittaker groups we will need in the construction of the generalized Myrberg’s algorithm and we prove some convergence process. In Section 5 we describe the generalized Myrberg’s algorithm and provide its convergence. In Section 6 we explain how to use the generalized Myrberg’s algorithm to provide numerical Schottky uniformizations of algebraic curves as in (2).

2. Preliminaries 2.1. Riemann orbifolds. Let S be a closed Riemann surface of genus γ, let x1, ..., xq ∈ S, all of them different, and integers 2 ≤ m1, ..., mq. The tuple O =(S, {(x1,m1), ..., (xq,mq)}) is called a Riemann orbifold [22, 27]. In this case, S is called the underlying Riemann surface,thepointsx1,..., xq the conical points and mj the conical order of xj , of the Riemann orbifold O.Thesignature of O is given by the tuple (γ; m1, ..., mq). A good orbifold is one with signature different from (0; n)or(0;n, m) with n = m. In this paper we are concerned with the class of Riemann orbifolds, called Whittaker orbifolds, which are those with signature of the form (0; n1,n1,n2,n2, ..., nr,nr). 2.2. Kleinian groups. A Kleinian group [18] is a discrete subgroup of PSL(2, C), which we may identify as the group of conformal automorphisms of the Riemann sphere C. Every Kleinian group K decomposes C into two disjoint sets; the limit set Λ(K), and its complement, the region of discontinuity (or regular set) Ω(K). A function group is a pair (Δ,K), where K is a finitely generated Kleinian group and Δ is an invariant connected component of Ω(K). If, moreover, K acts freely on Δ, then we say that (Δ,K)isafreely acting function group. ··· 2.3. Schottky groups. Let Ck,Ck, k =1, ,g,be2g Jordan curves on the Riemann sphere C such that they are mutually disjoint and bound a 2g-connected domain, say D. Suppose that for each k there exists a fractional linear transforma- ∈ C D ∩D ∅ tion Ak PSL(2, )sothat(i)Ak(Ck)=Ck and (ii) Ak( ) = .LetG be the group generated by all these transformations. As consequence of Klein-Maskit’s combination theorems, G is a Kleinian group, all its non-trivial elements are loxodromic and a fundamental domain for it is given by D. The group G is called a Schottky group of rank g, the set of generators A1,..., Ag is called a Schottky system of generators and the collection of loops C1, C1,..., Cg, Cg, is called a fundamental set of loops respect to these generators.In[20] is proved that a Schottky group of rank g is equivalent to a Kleinian group, with non-empty region of discontinuity, which is purely loxodromic and isomorphic to a free group of rank g.Thatevery set of g generators of G is always a Schottky system of generators is due to V. Chuckrow [8]. If Ω is the region of discontinuity of G, then it is known that Ω is connected and that Ω/G is a closed Riemann surface of genus g.

2.4. Whittaker groups. A Whittaker group of type (n1, ..., nr), where nj ≥ 2 are integers, is a Kleinian group K generated by r elliptic elements, say E1,...., Er, so that (i) Ej has order nj , (ii) there is a collection of pairwise disjoint simple closed curves, say C1,...., Cr, so that all of them bound a common domain D of connectivity r and (iii) Ej (D) ∩D= ∅ (see Figure 1). The set of generators E1,..., Er is called a set of Whittaker generators and the collection of loops C1,...., Cr is called a fundamental set of loops with respect to these Whittaker generators.The

100 RUBEN´ A. HIDALGO AND MIKA SEPPAL¨ A¨

C 12 C r2 C 11 C r1

E r E1

Figure 1

set D is a fundamental domain for K. Both fixed points of Ej belong to Cj and they divide Cj into two simple arcs, say Cj1 and Cj2,sothatEj(Cj1)=Cj2 (see Figure 1). As a consequence of Klein-Maskit’s combination theorems [18], K is a geometrically finite Kleinian group with a connected region of discontinuity Ω, every elliptic element of K is conjugate to a power of some Ej in K, it contains ∼ Z ∗···∗Z O no parabolic transformations and K = n1 nr .Thequotient =Ω/K is a Riemann orbifold of signature (0; n1,n1,n2,n2, ..., nr,nr), that is, a Whittaker orbifold. It is clear from the definition that two Whittaker groups of different types cannot be isomorphic, so they cannot be topologically equivalent, and that two Whittaker groups of same type are quasiconformally equivalent.

2.5. Uniformizations. Let (Δ,K) be a function group and let P :Δ→ Δ/K be the canonical projection map. A Riemann orbifold O is associated to (Δ,K) as follows. Let Δ∗ =Δ− F (K)whereF (K) denotes those points in Δ with non- trivial stabilizer in K.Inthiscase,Δ∗/K turns out to be the complement of a finite number of points in a closed Riemann surface S;wesetS as the underlying Riemann surface of O. The finite set P (F (K)) ⊂ S defines the conical points (these are those points over which P :Δ→ S fails to be a covering map). The order of a conical point x ∈ Δ/K is the maximal order of an elliptic element in K that fixes a point t ∈ Δ with P (t)=x. The tuple (Δ,K,P :Δ→O)is called an uniformization of the Riemann orbifold O. Existence of uniformizations of a good Riemann orbifold is a consequence of the Poincaré-Koebe Uniformization Theorem [10]. Uniformizations of a good Riemann orbifold O are naturally partially ordered in the sense that (Ω1,K1,P1 :Ω1 →O) ≤ (Ω2,K2,P2 :Ω2 →O)ifthere is a covering map Q :Ω2 → Ω1 so that P2 = P1Q. Highest uniformizations (Ω,K,P :Ω→O) are provided by universal covering spaces, that is, when Ω is simply-connected, for instance, if Δ is the unit disc and K is a Fuchsian group; in which case we talk of Fuchsian uniformizations.

2.6. Lowest uniformizations. We proceed to discuss lowest uniformizations for the cases of closed Riemann surfaces and Whittaker orbifolds. 2.6.1. Closed Riemann surfaces. Let S be a closed Riemann surface. If (Ω,G,P : Ω → S) is an uniformization of S, then we say that it is a Schottky uniformization of S if G is a Schottky group; we say that S is uniformized by the Schottky group G. Retrosection theorem (see [3] for a modern proof) asserts that every closed

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Riemann surface has a Schottky uniformization. As a consequence of the results in [19], the lowest uniformizations of closed Riemann surfaces are exactly the Schottky ones. 2.6.2. Whittaker orbifolds. If K is a Whittaker group, with region of discontinuity Ω, O =Ω/K and P :Ω→Ois the natural quotient map, then the uniformization (Ω,K,P :Ω→O) is called a Whittaker uniformization of the Whittaker orbifold O;wealsosaythatO is uniformized by K. A simple consequence of quasiconformal deformation theory asserts that every Whittaker orbifold admits a Whittaker uniformization. In [13] it was seen that Whittaker uniformizations are lowest uniformizations of Whittaker orbifolds. This fact follows from (i) the observation that each Whittaker group is defined by the conical orders and by a maximal collection of pairwise disjoint simple loops on the complement of these conical points, and (ii) Maskit’s classification of regular planar coverings [19]. Opposite to the case of closed Riemann surfaces, there are lowest uniformizations of Whittaker orbifolds which are not topologically equivalent to the Whittaker ones; see Example 1 below. In fact, Whittaker uniformizations correspond to those lowest uniformizations of Whittaker orbifolds whose group of covering transformations admits a Schottky group as a finite index normal subgroup with cyclic quotient (see Lemma 2, Lemma 3 and Remark 4).

Example 1. Let O be a Whittaker orbifold with signature (0; 2, 2, 3, 3). Exam- ples of lowest uniformizations for O,byLemma3below,aregivenby(Ωj ,Kj ,Pj : Ωj →O)(j =1, 2), where 2 ∗ 3 ∼ Z ∗ Z K1 = E1 : E1 =1 E2 : E2 =1 = 2 3 and 3 2 3 2 2 ∼ ∗ K2 = A, B, C : A = B = C =(AB) =(CB) =1 = D3 Z2 D3

Clearly, K1 is a Whittaker group of type (2, 3), but K2 is not a Whittaker group. If 6 φ : K1 → Z6 = u : u 3 2 2 ψ : K2 → D3 = v, w : v = w =(vw) =1 are deﬁned by 3 2 φ(E1)=u ,φ(E2)=u ψ(A)=ψ(C)=v, ψ(B)=w then −1 −1 N1 = Ker(φ)=K1 = E1E2E1E2 ,E1E2 E1E2 −1 −1 N2 = Ker(ψ)=CA ,C A Both groups N are Schottky groups of rank 2, so the Riemann surfaces, S = j ∼j Ω /N , are both of genus 2. Moreover, S (respectively, S ) admits a group H = Z j j ∼ 1 2 1 6 (respectively, H2 = D3) as the group of conformal automorphisms so that Sj /Hj = O.

102 RUBEN´ A. HIDALGO AND MIKA SEPPAL¨ A¨

2.7. Numerical uniformization. It seems that Burnside [5] provided the first explicit Fuchsian uniformization in a special case. Unfortunately, as already said in the introduction, the construction of explicit uniformizations, in the general situation, is not possible. It is for this that one may try to find a numerical algorithm that permits to approximate them. More precisely, let O be a given Riemann orbifold and assume we know the existence of an uniformization (Δ,K,P :Δ→O)for O but where we don’t know explicitly the group K.TheNumerical uniformization problem asks for an algorithm that permits to construct explicit uniformizations (Δn,Kn,Pn :Δn →On), each of them topologically equivalent to the given one, which approximate (Δ,K,P :Δ→O), that is, On is topologically equivalent to O and converges geometrically to O and the sequence of Kleinian groups Kn converges algebraically to K. Buser-Silhol [7](seealso[11, 26]) have developed numerical algorithms that allow one to approximate a Fuchsian uniformization of a given real hyperelliptic Riemann surface. They are able to compute equations for certain surfaces having given geometric properties. Buser has further studied differential geometric methods to calculate lengths of closed geodesic curves on a given hyperelliptic Riemann surface. In this paper we are interested in Whittaker orbifolds O and uniformizations (Ω,K,P :Ω→O)whereK is a Whittaker group. In Section 5 we will state an algorithm that permits to numerically approximate the Whittaker uniformization. This will provide an algorithm to numerically approximation a Schottky uniformization of cyclic L-gonal Riemann surfaces with quotient being the Whittaker orbifold O.

3. Basic facts on Whittaker groups 3.1. Schottky subgroups of Whittaker groups. As Whittaker groups provide lowest uniformizations of Whittaker orbifolds and Schottky groups provide the lowest uniformizations of closed Riemann surfaces, one may wonder if this lowest uniformization condition relates these two classes of Kleinian groups. By Selberg’s lemma [23], every finitely generated Kleinian group admits a torsion free normal subgroup of finite index. In some cases such a normal subgroup is provided by a Schottky group, but in general that is not the situation. Lemma 2 below asserts that Whittaker groups have Schottky groups as finite index normal subgroups and Lemma 3 provides the lowest index for such Schottky groups.

Lemma 2. Let K be a Whittaker group. Then every torsion free ﬁnite index subgroup of K is a Schottky group. In particular, K contains a Schottky group as a ﬁnite index normal subgroup.

Proof. Let K be a Whittaker group and let Ω be its region of discontinuity. If G is a torsion free finite index subgroup of K, then the finite index property ensures that G has also Ω as its region of discontinuity. As G is torsion free, G only contains loxodromic transformations besides the identity. Since subgroups of free products are again free products, all the above ensures that G is a Schottky group. This proves the first assertion. Now, as a consequence of Selberg’s lemma [23], K has a torsion free normal subgroup of finite index, so a Schottky group by the previous.

NUMERICAL SCHOTTKY UNIFORMIZATION 103

Lemma 2 asserts the existence of Schottky groups as normal subgroups of ﬁnite index in Whittaker groups. The following statement is about the minimal index Schottky normal subgroups of Whittaker groups.

Lemma 3. Let K be a Whittaker group of type (n1, ..., nr).IfL =lcm(n1, ..., nr), then K does not contain a Schottky group as a normal subgroup of index less than L and it does contain a Schottky group as a normal subgroup of index L. Proof. Let Γ be a Schottky group which is a normal subgroup of ﬁnite index N of the Whittaker group K and let ϕ : K → K/Γ be the canonical quotient surjective homomorphism. Let E1,..., Er be a set of Whittaker generators of K,whereEj has order nj . As Γ is the kernel of ϕ and it is torsion free, then ϕ(Ej) has order nj . It follows that nj divides the order N of K/Γ, for every j =1, ..., r.Inparticular, N is divisible by L. We have obtained that the index of any Schottky normal subgroup of K is at least L =lcm(n1, ..., nr). Now, let us consider the surjective L kj L/nj homomorphism φ : K → ZL = u : u =1, deﬁned by φ(Er)=U ,where kj ∈{1, 2, ..., nj − 1} is relatively prime to nj. By Lemma 2 and the fact that the kernel G of φ is a torsion free normal subgroup of index L,soaSchottkygroupas desired by Lemma 2.

Remark 4. As a consequence of Lemma 3, every Whittaker group K = E1, ..., Er of type (n1, ..., nr), has a Schottky group as a normal index L subgroup, where L =lcm(n1, ..., nr), and such index is the minimal possible with that property. By Riemann-Hurwitz formula, such a minimal index Schottky sub- − − r −1 group has rank g =1+L(r 1 j=1 nj ). Such a minimal index Schottky subgroup is not, in general, unique. For instance, if r =2andnj =3,then 3 3 K = E1,E2 : E1 = E2 =1, L =3andg = 2. In this case, the are exactly 3 4 surjective homomorphisms φ : K → Z3 = u : u =1, but their respective −1 −1 kernels are one of exactly two Schottky groups, these being G1 = E2 E1,E2E1 −1 −1 and G2 = E2E1,E2 E1 . The automorphism ρ of the group K, defined by −1 ρ(E1)=E1 and ρ(E2)=E2)satisfiesthatρ(G1)=G2. The non-uniqueness of the minimal index Schottky subgroup of a Whittaker group is a consequence of the fact that there are examples of different closed Riemann surfaces, of the same genus, both admitting a cyclic group of order L, so that the corresponding quotient Whittaker orbifods are the same. If nj =2,thenL = 2 and there is exactly one Schottky subgroup of index two, this being a hyperelliptic Schottky group [15], which uniformizes a hyperelliptic Riemann surface.

3.2. Geometry of Whittaker uniformizations. Let us consider a Whit- taker uniformization (Ω,K,Q :Ω→O), where K is a Whittaker group of type (n1, ..., nr) with region of discontinuity Ω. Let E1,..., Er be a set of Whittaker generators of K,whereEj has order nj , and let C1,..., Cr be a collection of fundamental loops for K with respect to these Whittaker generators. Denote by pj , qj ∈ Cj both ﬁxed points of Ej,forj =1, ..., r. There is a collection of pairwise disjoint simple arcs γj = Q(Cj )=Q(Cj1)=Q(Cj2)onO,whereCj1 and Cj,2 are as deﬁned in Section 2.4, connecting the two conical points pj = Q(pj )andqj = Q(qj ), for

104 RUBEN´ A. HIDALGO AND MIKA SEPPAL¨ A¨ j =1, .., r. We say that the collection of arcs γ1,..., γr is a set of Whittaker fundamental arcs for K respect to the Whittaker uniformization (Ω,K,Q:Ω→O).

Lemma 5. Let O be a Whittaker orbifold with signature (0; n1,n1, ..., nr,nr). Let pj,qj ∈O, j =1, ..., r, be the conical points so that both pj and qj have the same order nj.Letγ1,..., γr be a set of pairwise disjoint simple arcs so that γj connects pj with qj . Then there is a Whittaker uniformization (Ω,K,Q :Ω→O) so that the loops γ1,..., γr form a set of Whittaker fundamental arcs respect to it. Proof. This is consequence of the previous observation and quasiconformal deformation theory [3] (this was also observed by L. Keen in [15]fornj =2).

Lemma 6. Let O be a Whittaker orbifold with signature (0; n1,n1, ..., nr,nr). Let us consider two Whittaker uniformizations, say (Ω1,K1,Q1 :Ω1 →O) and (Ω2,K2,Q2 :Ω2 →O), both of them defining the same set of Whittaker fundamental −1 arcs on O.Then,thereisaMöbius transformation A so that K2 = AK1A and −1 Q2 = Q1A . Proof. j The hypothesis asserts that there is a fundamental set of loops C1 ,..., j j j Cr , with respect to the fundamental set of generators E1,..., Er for Kj ,forj =1, 2, 1 2 so that Q1(Ck )=Q2(Ck), for k =1, ..., r. We may construct a homeomorphism → 1 2 −1 A :Ω1 Ω2 satisfying AEk = EkA,fork =1, ..., r,andQ2 = Q1A .AsQj is locally conformal homeomorphism (except at the fixed points of the elliptic transformations), we also have that A is conformal homeomorphism. As the region of discontinuity of a Schottky group is of type OAD (that is, it admits no holomorphic function with finite Dirichlet norm (see [1, p. 241]).), then the region of discontinuity of K1 is also of type OAD (by Lemma 2). It follows from this (see [1, p. 200]) that every conformal map from Ω1 into the extended complex plane is fractional linear, in particular, A is a Möbius transformation.

4. Basic objects 4.1. Opening Arcs. We ﬁrst describe the opening arc process, that general- izes the opening arc process due to Myrberg, which is the basic tool in the algorithm. 4.1.1. Some preliminary facts. Let us start with the following simple observation.

Lemma 7. Let us consider any two branched regular coverings π1,π2 : C → C, both with covering group K = E,whereE is a M¨obius transformation of order n ≥ 2.Then,thereisaM¨obius transformation A so that π2 = Aπ1.

Proof. Let p, q be the fixed points of E.Seta = π1(p), b = π1(q), c = π2(p) and d = π2(q). We may define A(a)=c and A(b)=d.Ifz/∈{a, b},then −1 { n−1 } π1 (z)= u1,E(u1), ..., E (u1) . In this way, we may define A(z)=π2(u1). It is clear that A is a well defined homeomorphism of the Riemann sphere so that Aπ1 = π2.Sinceπj are locally conformal homeomorphisms in the complement of {p, q}, A is a Möbius transformation.

NUMERICAL SCHOTTKY UNIFORMIZATION 105

4.1.2. Opening arc. By a simple arc we mean the image of a homeomorphic embedding α :[0, 1] → C. Let Σ be a simple arc, say parametrized by the homeomorphic embedding α :[0, 1] → C,thatis,Σ=α([0, 1]), and fix an integer n ≥ 2. Set p = α(0) and q = α(1). Let us fix three different points a, b, c ∈ C −Σ. As consequence of Lemma 5, there is a Whittaker uniformization (C,K = EΣ,QΣ : C →O), where O is the orbifold whose underlying Riemann surface is the Riemann sphere C and whose conical points are p and q (both of order n ≥ 2) and there is a simple loop CΣ, containing both fixed points of the elliptic transformation EΣ, bounding two discs, ∗ ⊂ ∗ say DΣ and DΣ,sothatEΣ(DΣ) DΣ. By composing QΣ at the right by a suitable Möbius transformation, we may also assume {a, b, c}⊂DΣ and that QΣ(x)=x for x ∈{a, b, c}.Theabove Whittaker uniformization is uniquely determined by this normalization for QΣ. −1 C − → The conformal homeomorphism ΦΣ = QΣ : Σ DΣ satisfies the following properties:

(i) ΦΣ(a)=a, ΦΣ(b)=b and ΦΣ(c)=c; −1 → C − ∪ → C (ii) ΦΣ : DΣ Σ extends continuously to QΣ : DΣ = DΣ CΣ ±1 with QΣ(x)=QΣ(y) if and only if y = EΣ (x) for every pair of points x, y ∈ CΣ,and −1 (iii) ΦΣ (CΣ)=Σ.

Lemma 8. The map ΦΣ is uniquely determined by the normalization (i). Proof. Assume we have another conformal homeomorphism Ψ : C − Σ → D, where D is a Jordan disc on the Riemann sphere (say bounded by the simple loop Υ) with {a, b, c}⊂D,sothatΨ(x)=x for x ∈{a, b, c}, and that there is an elliptic Möbius transformation F of order n so that F (D) ⊂ D∗,where D∗ is the other disc bounded by Υ, such that Ψ−1(x)=Ψ−1(y) if and only if ±1 −1 y = F (x) for every pair of points x, y ∈ Υ. The map η =ΦΣΨ : D → DΣ is a conformal homeomorphism that fixes three different points and which can be extended to a conformal homeomorphism of the Riemann sphere satisfying the −1 condition EΣ ηF = η. Clearly, η is the restriction of a Möbius transformation that fixes three different points, so it is the identity.

We call ΦΣ the opening map of Σ normalized at the points a, b and c or just opening map of Σ if the choice of the points a, b and c is clear.

Remark 9. As a consequence of the proof of Lemma 8, DΣ and K = EΣ are also uniquely determined by the arc Σ and the normalization (i). Note that ±1 we have determined uniquely the two geometric generators EΣ , but we have not determined which one of the two generators to use. In practice, we must should a choice of one of these two geometric generators.

4.1.3. Explicit form of the opening map ΦΣ. LetusconsideraM¨obius transformation T so that T (p)=0andT (q)=∞ (for instance T (z)=(z − p)/(z − q)if p, q ∈ C) and the simple arc T (Σ) connecting 0 with ∞.Letπ : C → C be deﬁned by π(z)=zn and E(z)=e2πi/nz. By lifting the simple arc T (Σ), under π,we

106 RUBEN´ A. HIDALGO AND MIKA SEPPAL¨ A¨ obtain n simple arcs, each one connecting 0 with ∞. These arcs divide C into n pairwise disjoint discs. Let D beoneofthem.Choosethetwoarcs,sayΣ1 and Σ2, so that Σ=Σ1 ∪ Σ2 is the simple closed loop boundary of D,andE(Σ1)=Σ2. √ ThechoiceofD permits us to fix a branch of n z : C → D. Using this branch, n n n we compute T (a), T (b), T (c) ∈ D.LetM be the Möbius transformation satisfying that M( n T (a)) = a, M( n T (b)) = b and M( n T (c)) = c. Since, by Lemma 8, ΦΣ is unique under the normalization of fixing a, b and c we have that n ΦΣ(z)=M T (z) , that is AF (z)+B Φ (z)= , Σ CF(z)+D where A = ab (F (a) − F (b)) + bc (F (b) − F (c)) + ac (F (c) − F (a))

B = abF (c)(F (b) − F (a)) + bcF (a)(F (c) − F (b)) + acF (b)(F (a) − F (c))

C = a (F (c) − F (b)) + b (F (a) − F (c)) + c (F (b) − F (a))

D = aF (a)(F (b) − F (c)) + bF (b)(F (c) − F (a)) + cF (c)(F (a) − F (b))

z − p F (z)= n z − q

4.2. Special chain of subgroups. Let K be a Whittaker group of type (n1, ..., nr) with set of Whittaker generators E1,..., Er. We call E1,..., Er the ﬁrst generation elements of K.Foreachk ∈{1, 2, ...} and i1, ..., ik,ik+1 ∈{1, 2, ..., r} so that i1 = i2 = ··· = ik−1 = ik = ik+1,wecall any of the elements of the form

s − − −s − − − Es1 Es2 ···E k 1 Esk E E sk E k 1 ···E s2 E s1 i1 i2 ik−1 ik ik+1 ik ik−1 i2 i1 an (k +1)-generation element of K,wheresj ∈{1, ..., nj − 1}. A sequence of subgroups Kn

(i) K0 = K; (ii) Kj+1 ¡ Kj of ﬁnite index; (iii) Kj+1 is obtained by elimination of an element of Kj of lowest generation and adjoining conjugates of the other generators of Kj by all powers of the deleted one. Note that the above properties provide explicitly the way to construct admissible sequence of subgroups (see Example 10). Also, by the deﬁnition, each of these admissible groups is in fact a Whittaker group whose type is of the form ∈{ } (m1, ...., msn ), where mj n1, ..., nr . Moreover, the index of Kj+1 in Kj is equal to the order of the deleted generator of Kj used in (iii). Also, it follows that ∩∞ { } j=0Kj = I .

NUMERICAL SCHOTTKY UNIFORMIZATION 107

Example 10. This example explains the process of constructing admissible sequence of groups, at least the ﬁrst ones. Let r =3,n1 = n2 = n3 =2and K = E1,E2,E3. The ﬁrst four terms for an admissible sequence are the following ones. K0 = E1,E2,E3

K1 = E2,E3,E1E2E1,E1E3E1

K2 = E3,E1E2E1,E1E3E1,E2E3E2,E2E1E2E1E2,E2E1E3E1E2

K3 = E1E2E1,E1E3E1,E2E3E2,E2E1E2E1E2,E2E1E3E1E2,E3E1E2E1E3,

E3E1E3E1E3,E3E2E3E2E3,E3E2E1E2E1E2E3,E3E2E1E3E1E2E3

In this example, Kj+1 has index 2 in Kj , K1 is obtained by deletion of E1 from K0 in the process (iii), K2 is obtained by deletion of E2 from K1 in the process (iii), and K3 is obtained by deletion of E3 from K2 in the process (iii).

4.3. A convergence fact. Let K be a Whittaker group of type (n1, ..., nr), let E1,..., Er be a set of Whittaker generators of K (Ej of order nj ), and let C1, ..., Cr ⊂ Ω be a corresponding set of fundamental loops, all of them bounding a common domain D⊂Ω of connectivity 2r,sothatEj (D) ∩D= ∅.LetQ :Ω→ C be a regular branched cover with K as its deck group. We assume that we have fixed three different points on D,saya, b and c and that Q(a)=a, Q(b)=b and Q(c)=c.WesetQ(Cj)=Lj , for all j =1, .., r and S = C − (L1 ∪···∪Lr). Next, we consider the conformal homeomorphism Ψ = Q−1 : S →D. Let us consider an admissible sequence of subgroups {Kj } of K = K0,theneach of them a Whittaker group of some type. Note that it is possible to consider some of the K-translates of the original loops C1,..., Cr in order to obtain a collection of fundamental loops for Kj .Byconstruction,Kj has finite index in K,inparticular, each Kj has the same region of discontinuity Ω. It follows the existence of a regular branched covering map ηj :Ω→Oj , where Oj is the Riemann sphere with a finite collection of conical points, whose covering group is Kj with ηj (a)=a, ηj (b)=b, ηj (c)=c and a (not necessarily regular) covering Qj : C →O= O0,sothatQ = Qj ηj (see Figure 2). In particular, Qj (a)=a, Qj (b)=b and Qj (c)=c. The projection, under ηj , of the collection of fundamental loops for Kj is a collection of Whittaker fundamental arcs for the Whittaker uniformization (Ω,Kj,ηj :Ω→Oj ). The conical points of Oj are exactly the end points of these arcs; moreover, two of the conical points of the same arc have the same order. Set Dj = ηj (D). It follows from the construction that Qj : Dj → S is a conformal homeomorphism. We denote by ψj : S →Dj its inverse (notice that η0 = Q and that ψ0 =Ψ).

Theorem 11. The sequence ηj converges locally uniformly to the identity map and the sequence ψj converges locally uniformly to Ψ:S →D. Proof. Normality of the family.LetR ⊂ Ω be the union of the K-orbits of the points a, b and c.SetΩR =Ω− R and let us consider the collection of holomorphic maps ηj restricted to ΩR. The images of ΩR under each ηj misses the three points a, b and c. By Montel’s theorem ηj :ΩR → C is a normal family. Unfortunately, this is not enough to ensure ηj :Ω→ C to be a normal family.

108 RUBEN´ A. HIDALGO AND MIKA SEPPAL¨ A¨

E E1 2

C 1 C 2

C 3 η 2

E 3 η Q 1 2 2 2 2

2 2 L 2 1 Q 1 2 2 2 2 2 ψ D1 2 2 1 L 2 2 222 2 2 Q L 3 2 22 2 2 ψ 2

Figure 2. n =0, 1, 2andnj =2

Let us consider any fundamental domain D for K so that in its interior D0 0 0 are contained points aD ∈ D in the K-orbit of a, bD ∈ D in the K-orbit of b 0 0 and cD ∈ D in the K-orbit of c. Consider the family of restrictions ηj : D → C. Clearly, ηj (aD)=a, ηj (bD)=b and ηj (cD)=c. It follows from Theorem 2.1 in 0 [17]thatηj : D → C is a normal family. If D1, ..., Dn are fundamental domains for K,sothat,foreachk =1, ..., n, 0 the interior Dk of Dk always contains a point in the K-orbit of a, b and c,then 0 → C the previous ensures that ηj : Dk is a normal family. It follows that ηj : ∪n 0 → C k=1Dk is a normal family. As Ω is a countable union of interiors of fundamental domains as above, we ⊂ ⊂ ··· ⊂ ∪∞ may construct a family of open domains Ω1 Ω2 Ωsothat k=1Ωk =Ω and ηj :Ωk → C is a normal family, for each k. If we consider any subsequence of ηj :Ω→ C, then there is a subsequence of ηj :Ω1 → C converging locally uniformly. Now, there is a subsequence of such one whose restriction to Ω2 converges locally uniformly. We now consider such a new subsequence and restrict it to Ω3 and continue inductively such a process. Now we use the diagonal method to obtain a subsequence converging locally uniformly on all Ω. Limit mappings of subsequences. Let us choose any subsequence ηj :Ω→ C that k converges locally uniformly to the conformal map η∞ :Ω→ C. As ηj (a)=a, ηj (b)=b and ηj (c)=c, it follows that η∞(a)=a, η∞(b)=b and η∞(c)=c,inparticular,η∞ is a non-constant conformal mapping. ¡ ∩∞ { } As Kj+1 Kj and j=0Kj = I , we may construct fundamental domains Dj ⊂ ∪∞ 0 for Kj so that Dj Dj+1 and j=1Dj = Ω. Let us denote by Dj the interior of Dj . ≥ 0 Clearly, for each j k, ηj restricted to Dk is an injective conformal mapping.

NUMERICAL SCHOTTKY UNIFORMIZATION 109

0 → C The convergent subsequence of injective conformal mappings ηjk : Dk 0 → C converges uniformly to η∞ : Dk , which is either constant or injective. As we know the last to be non-constant, we obtain that η∞ :Ω→ C is a locally injective conformal map, that is, a local homeomorphism onto its image. ∈ ∩∞ { } ∈ ≥ Let γ K.As j=0Kj = I ,thereissomej0 so that γ/Kj ,forj j0.It follows that η∞ is globally injective on all Ω. By Lemma 2, Ω is also the region of discontinuity of a Schottky group, which is a domain of class OAD (that is, it admits no holomorphic function with finite Dirichlet norm (see [1, pg 241]). It follows from this (see [1, pg 200]) that any one- to-one conformal map on Ω is necessarily the restriction of a Möbius transformation. In particular, η∞ is the restriction of a Möbius transformation. As it fixes three different points, η∞ = I (identity map).

Convergence of all the family. Since any subsequence of (ηj ) has a convergent subsequence, the above asserts that the complete sequence converges locally uniformly to the identity map. Now, as Q = Qj ηj , it follows that the sequence (ψj)converges locally uniformly to Ψ.

5. The generalized Myrberg’s algorithm 5.1. First step. Let us consider a collection of r pairwise disjoint simple arcs, say L1,...., Lr.SetS = C − (L1 ∪···∪Lr) and choose three diﬀerent points on S, say a, b, c ∈ S.LetO be the Riemann orbifold whose underlying Riemann surface is given by the Riemann sphere and the conical points are given by the 2r end points of the above simple arcs Lj , so that both end points for the same arc Lj have the same order nj ≥ 2 (that is, O is a Whittaker orbifold). As consequence of Lemmas 5 and 6, there is a unique Whittaker uniformization (K, Ω,Q:Ω→O), for which L1,..., Lr is a Whittaker fundamental set of arcs and Q is normalized by the rule Q(a)=a, Q(b)=b and Q(c)=c. In this way, there is a set of Whittaker generators of K,sayE1,..., Er so that Ej has order nj, and a corresponding set of fundamental loops, say C1, ..., Cr ⊂ Ω, all of them bounding a common domain D⊂Ω of connectivity 2r,sothatEj (D) ∩D= ∅, Q(Cj)=Lj , for all j =1, .., r, −1 and K = E1, ..., Er. Let us consider the conformal homeomorphism Ψ = Q : S →D.

5.2. Second step. We use the points a, b, c to normalize each opening map, that is, each opening map is assumed to fix these three points. As in [24], we call the arcs L1,..., Lr the first generation slots. 5.2.1. We consider the opening map Φ : C − L → D = C − D∗ and L1 1 L1 L1 the Möbius transformation EL1 of order n1 (a choice of one of the two geometric generators of the corresponding cyclic group). The generator EL1 has both fixed points on the boundary loop C = ∂D and E (D ) ⊂ D∗ . L1 L1 L1 L1 L1

5.2.2. The arcs ΦL1 (L2),...., ΦL1 (Lr)areﬁrst generation slots,and Es (Φ (L )),..., Es (Φ (L )), where s ∈{1, ..., n − 1}, are called the second L1 L1 2 L1 L1 r 1 generation slots.

110 RUBEN´ A. HIDALGO AND MIKA SEPPAL¨ A¨

5.2.3. We now choose any of the new first generation slots to proceed as done with L1. For instance, if we choose the arc L =ΦL1 (L2), then we consider the open- C − ∗ ing map φL defined on the complement of L ontoadiscDL = DL.Wehave again a Möbius transformation of order n2,sayEL,withbothfixedpointsonthe ⊂ ∗ boundary loop ∂DL and EL(DL) DL.ThearcsΦL(ΦL1 (L3)),...., ΦL(ΦL1 (Lr)) are called first generation slots,thearcsΦ (Es (Φ (L ))),...., Φ (Es (Φ (L ))), L L1 L1 2 L L1 L1 r r r EL(ΦL(ΦL1 (L3))),...., EL(ΦL(ΦL1 (Lr))) are called second generation slots and Er (Φ (Es (Φ (L )))),...., Er (Φ (Es (Φ (L )))) are called the third generation L L L1 L1 2 L L L1 L1 r slots. 5.2.4. We continue the process until all first generations slots have been opened and new higher generation slots have been formed. In this way, when opening slots, one forms iteratively new slots which are divided into generations. Order these slots by generation and within a generation. Continue with opening the slots in the order we have chosen above. Iterating this procedure we get a sequence of conformal mappings Φ1,Φ2,... of opening maps. 5.3. Third step. The chosen order for the opening slot process produces in a natural way an admissible sequence Kn so that ψn =ΦnΦn−1 ···Φ2Φ1 : S → C.

The region Ψn(S) is a domain bounded by r pairwise disjoint simple loops, say C1,n,..., Cr,n, contained in some region Ωn ⊂ C. We also have conformal ∗ homeomorphisms of Ωn, (not necessarily Möbius transformations), say E1,n,..., ∗ ∗ Er,n,sothatEj,n has order nj and has exactly two fixed points on Cj,n,and ∗ ∩ ∅ Ej,n(Ψn(S)) Ψn(S)= . ∗ As consequence of Theorem 11, the homeomorphism Ej,n converges to the Möbius transformation Ej ∈ K, necessarily of order nj . Let us consider the unique Möbius transformations E1,n,..., Er,n so that Ej,n has order nj and it has as fixed ∗ points the two fixed points of Ej,n.Set Kn = E1,n, ..., Er,n The above asserts that the sequence of Whittaker groups (Kn) converges algebraically to the Whittaker group K.

6. Application to algebraic curves

If C is an algebraic curve as in (2), then the orbifold fundamental group of OC has a presentation r orb O nj nj π1 ( C )= x1, ..., xr,y1, ..., yr : xj = yj =1(j =1, ..., r), (xjyj )=1 j=1 where we may think of xj (respectively, yj ) as a small loop around aj (respectively, bj ) oriented counterclockwise (that is, the surrounded points is at the left side). As consequence of the Poincar´e-Koebe Uniformization Theorem [10], we may orb O assume π1 ( C ) is a discontinuous group of conformal automorphisms of a simply- C C H2 orb O connected Riemann surface Δ (either , or )sothatΔ/π1 ( C )is(asorb- ifolds) conformally equivalent to OC . orb O Let N be the smallest normal subgroup of π1 ( ) containing the words xj yj (j =1, .., r). It turns out that N is a torsion free subgroup and that K =

NUMERICAL SCHOTTKY UNIFORMIZATION 111

orb O π1 ( C )/N is a Whittaker group of type (n1, ..., nr) with region of discontinu- ∈ −1 ity Ω = Δ/N .IfEj K denotes the quotient class of xj (the same as for yj ), then n1 ∗···∗ nr K = E1 : E1 =1 Er : Er =1. Let G be the kernel of the surjective homomorphism (see also Remark 4)

φ : K → ZL = τ

kj L/nj where φ(Ej)=τ (kj ∈{1, 2, ..., nj − 1} relatively prime with nj). The value L/nj of kj is determined by the rotation number of τ around aj (in reality, around a ﬁxed point of it over aj). As consequence of Lemma 2, G is a Schottky group of index L in K. The surface S is conformally equivalent to Ω/G,whereG¡K is a torsion free C ∼ normal subgroup so that K/G = ZL = τ. Our generalization of Myrberg’s algorithm (see Section 5) permits to provide numerical approximations of K,soofG as desired.

Acknowledgements The authors are very grateful to the referees for their very helpful, deep and accurate comments concerning the previous version of this paper.

References [1] L. Ahlfors and L. Sario. Riemann Surfaces. Princeton Mathematical Series 26. Princeton University Press, Princeton, N.J. (1960) MR0114911 (22:5729) [2] L. Ahlfors. Finitely generated Kleinian groups. Amer. J. of Math. 86 (1964), 413-429. MR0167618 (29:4890) [3] L. Bers. Automorphic forms for Schottky groups. Adv. in Math. 16 (1975), 332-361. MR0377044 (51:13218) [4] L. Bers. On the Ahlfors’ finiteness theorem. Amer. Math. J. 89 (4) (1967), 1078-1082. MR0222282 (36:5334) [5] W. Burnside. Note on the equation y2 = x(x4 − 1). Proc. London Math. Soc. (1) 24 (1893), 17-20. [6] W. Burnside. On a class of Automorphic Functions. Proc. London Math. Soc. 23 (1892), 49-88. [7] P. Buser and R. Silhol. Geodesics, Periods and Equations of Real Hyperelliptic Curves. Duke Math. J. 108 (2001), 211-250. MR1833391 (2002d:30049) [8] V. Chuckrow. On Schottky groups with applications to Kleinian groups. Annals of Math. 88, (1968) 47-61. MR0227403 (37:2987) [9] H. Farkas and I. Kra. Riemann Surfaces. Second edition. Graduate Texts in Mathematics 71. Springer-Verlag, New York (1992). MR1139765 (93a:30047) [10] R. Fricke and F. Klein. Vorlesungenuber ¨ die theorie der automorphen funktionen. Teubner, Lepzig (1926). [11] P. Gianni, M. Seppälä, R. Silhol and B. Trager. Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26 (1998), 789-803. MR1662036 (99m:14055) [12] R.A. Hidalgo. On the Schottky groups with automorphisms. Ann. Acad. Fenn. 19 (1994), 259-289. MR1274083 (95k:30092) [13] R.A. Hidalgo. Lowest uniformizations of closed Riemann orbifolds. Revista Matematica Iberoamericana 26 No.2 (2010), 639-649. MR2677010 (2011g:30093) [14] R.A. Hidalgo and M. Seppälä. Numerical Schottky Uniformizations: Myrberg’s opening process. Lecture Notes in Mathematics 2013 (2011), 195-209. [15] L. Keen. On Hyperelliptic Schottky groups. Ann. Acad. Sci. Fenn. Series A.I. Math. 5 (1) (1980), 165-174. MR595187 (82a:32029) [16] Koebe, P. Uber¨ die Uniformisierung der Algebraischen Kurven II. Math. Ann., 69:1–81, 1910. MR1511577

112 RUBEN´ A. HIDALGO AND MIKA SEPPAL¨ A¨

[17] O. Lehto. Univalent Functions and Teichmüller Spaces.GTM109, Springer-Verlag (1986). MR867407 (88f:30073) [18] B. Maskit. Kleinian groups. Springer-Verlag, Heidelberg-New York-Berlin (1988). MR959135 (90a:30132) [19] B. Maskit. A theorem on planar covering surfaces with applications to 3-manifolds. Ann. of Math. (2) 81 (1965), 341-355. MR0172252 (30:2472) [20] B. Maskit. A characterization of Schottky groups. J. Analyse Math. 19 (1967), 227-230. MR0220929 (36:3981) [21] J.P. Myrberg. Uber¨ die Numerische Ausführung der Uniformisierung. Acta Soc. Scie. Fenn., XLVIII (7) (1920), 1-53. [22] P. Scott. The geometries of 3-manifolds. Bulletin London Math. Soc. 15 (1983), 401-487. MR705527 (84m:57009) [23] A. Selberg. On discontinuous groups in higher-dimensional symmetric spaces. Contributions to function theory, TATA Institute, Bombay (1960), 147-164. MR0130324 (24:A188) [24] M. Seppälä. Myrberg’s numerical uniformization of hyperelliptic curves. Ann. Acad. Scie. Fenn. Math. 29 (2004), 3-20. MR2041696 (2004m:30069) [25] M. Seppälä. Computation of period matrices of real algebraic curves. Discrete Computational Geometry 11 (1994), 65-81. MR1244890 (95a:30035) [26] R. Silhol. Hyperbolic lego and algebraic curves in genus 2 and 3. Contemporary Math. 311 Complex Manifolds and Hyperbolic Geometry, 313-334(2001) MR1940178 (2004b:14046) [27] W.P. Thurston. The geometry and topology of 3-manifolds. Princeton lecture notes (1978- 1981).

Departamento de Matematicas,´ Universidad Tecnica´ Federico Santa Mar´ıa, Val- para´ıso, Chile E-mail address: [email protected] Department of Mathematics, Florida State University, USA and Department of Mathematics and Statistics, University of Helsinki, Finland E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11359

Generalized lantern relations and planar line arrangements

Eriko Hironaka

Abstract. In this paper we show that to each planar line arrangement deﬁned over the real numbers, for which no two lines are parallel, one can write down a corresponding relation on Dehn twists that can be read oﬀ from the combinatorics and relative locations of intersections. This gives an alternate proof of Wajnryb’s generalized lantern relations, and of Endo, Mark and Horn-Morris’ daisy relations.

1. Introduction Braid monodromy is a useful tool for studying the topology of complements of line arrangements as is seen in work of [13], [3], [4]. In this paper, we adapt braid monodromy techniques to generate relations on Dehn twists in the mapping class group MCG(S) of an oriented surface S of finite type. The study of hyperplane arrangements has a rich history in the realms of topology, algebraic geometry, and analysis (see, for example, [14] for a survey). While easy to draw, the deformation theory of real planar line arrangements L holds many mysteries. For example, there are topologically distinct real line arrangements with equivalent combinatorics [1][15]. Moreover, by the Silvester-Gallai theorem [7] there are planar line arrangements defined over complex numbers, whose combinatorics cannot be duplicated by a real line arrangement, for example, the lines joining flexes of a smooth cubic curve. Braid monodromy is a convenient tool for encoding the local and global topology of L. The lantern relation on Dehn twists is of special interest because it with four other simple to state relations generate all relations in the Dehn-Lickorish- Humphries presentation of MCG(S)(see[2], [8], [12], [17]). The lantern relation also plays an important role in J. Harer’s proof that the abelianization of MCG(S) is trivial if S is a closed surface of genus g ≥ 3[9] (cf., [8], Sec. 5.1.2). Let S be an oriented surface of finite type. If S is closed, the mapping class group MCG(S) is the group of isotopy classes of self-homeomorphisms of S.IfS has boundary components, then the definition of MCG(S) has the additional condition that all maps fix the boundary of S pointwise. For a compact annulus A,MCG(A) is isomorphic to Z and is generated by a right or left Dehn twist centered at its core curve. As illustrated in Figure 1, a right Dehn twist takes an arc on A transverse

2010 Mathematics Subject Classiﬁcation. Primary 57M27, 20F36; Secondary 32Q55. This work was partially supported by a grant from the Simons Foundation (#209171to Eriko Hironaka).

c 2012 American Mathematical Society 113

114 ERIKO HIRONAKA

c c

Figure 1. Right Dehn twist on an annulus A. to the core curve to an arc that wraps once around the core curve turning in the right hand direction (a left Dehn twist correspondingly turns in the left direction) as it passes through c. A Dehn twist can also be thought of as rotating one of the boundary components by 360◦ while leaving the other boundary component ﬁxed. Each simple closed curve c on S determines a right Dehn twist on an annulus neighborhood of c, and this Dehn twist extends by the identity to all of S.The isotopy class ∂c of this map is the (right) Dehn twist centered at c and is an element of MCG(S). The original statement and proof of the lantern relation appears in Dehn’s 1938 paper [5] and relates a product of three interior Dehn twists to four boundary twists on a genus zero surface with four boundary components. The relation was rediscovered by D. Johnson [11], and B. Wajnryb gave the following generalized version in [18] (Lemma 17). Theorem . c ⊂ 1.1 (Wajnryb) Let S0,n+1 S be a surface of genus zero with n+1 boundary components d0,d1,...,dn. There is a collection of simple closed curves ≤ ≤ c ai,j , 1 i

(1) for each i, j, ai,j separates di ∪ dj from the rest of the boundary components, and (2) there is a relation on Dehn twists n−2 (1.1) ∂0(∂1 ···∂n) = α1,2 ···α1,nα2,3 ···α2,n ···αn−2,n−1αn−2,nαn−1,n,

where αi,j is the right Dehn twist centered at ai,j ,and∂i is the right Dehn centered at a curve parallel to the boundary components di. We now generalize Theorem 1.1 in terms of line arrangements in R2. Theorem 1.2. Let L be a union of n ≥ 3 distinct lines in the (x, y)-plane over the reals with distinct slopes and no slope parallel to the y-axis. Let I = {p1,...,ps} be the intersection points on L numbered by largest to smallest x-coordinate. For ∈L I∩ c each L ,letμL be the number of points in L.LetS0,n+1 be a surface of genus zero and n+1 boundary components, one denoted dL for each L ∈L,andone extra boundary component d0. Then there are simple closed curve apk , k =1,...,s c on S0,n+1 so that the following holds: (1) each a separates pk dL

pk∈L∈L from the rest of the boundary curves; and

GENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 115

d 0 d0

L 1 a1,3 a a1,2 a d 2,3 L 2 1,3 1 a1,2 d1 2,3 1,2 d3 d3 d2 1,3

a2,3 L 3 d2

Figure 2. Three lines in general position, and curves deﬁning associated lantern relation drawn two ways.

(2) the Dehn twist ∂L centered at dL and αpk centered at apk satisfy − μL 1 ··· (1.2) ∂0 ∂L = αps αp1 . L∈L Remark 1.3. In Equation (1.2), the terms on the left side commute, while the ones on the right typically don’t. Thus, the ordering of p1,...,ps matters, and reﬂects the global (as opposed to local) combinatorics of the line arrangement. The curves apk can be found explicitly (see Section 2.2, Lemma 2.1). Remark . c 1.4 The relations in MCG(S0,n+1) give rise to relations on MCG(S) c → for any surface S admitting an embedding S0,n+1 S. When n = 3, Theorem 1.1 gives the standard Lantern relation

∂0∂1∂2∂3 = α1,2α1,3α2,3. The core curves for these Dehn twists and the corresponding line arrangements are shown in Figure 2. The diagram to the right is the motivation for the name of this relation. Here is a sketch of our proof of Theorem 1.2. First consider a great ball B ⊂ C2 containing all the points of intersection of L.LetCP2 be the projective compact- ification of C2. Then the complement of B in CP2 is a neighborhood of the “line 2 2 2 at infinity” or L∞ = CP \ C .Letρ : C → C be a generic projection. The monodromy of ρ over the boundary γ of a large disk in C depends only on the way L intersects L∞. If no lines in L are parallel to each other, then it is possible to move the lines in L to obtain a new configuration T where all lines meet at a single point without changing any slopes, and hence the topology of CP2 \ B remains the same (Lemma 2.3). Thus, the monodromies over γ defined by L and T are the same. Theorem 1.2 then follows from a description of the monodromy of line arrangements on compactified fibers of a generic projection (Lemma 2.1). The monodromy can be interpreted as point pushing maps, where we keep track of twisting on the boundary components of the compactified fibers using the complex coordinate system of the ambient space C2 (Lemma 2.2). This paper is organized as follows. In Section 2.1 we recall the Moishezon- Teicher braid monodromy representation of a free group associated to a planar line arrangement. We refine the representation so that its image is the the mapping class group of compactified fibers in Section 2.2. In Section 2.3, we prove Theorem 1.2

116 ERIKO HIRONAKA using deformations of line arrangements and give further variations of the lantern relation, including the daisy relation (Theorem 3.1). Acknowledgments: The author is grateful to J. Mortada and D. Margalit for helpful discussions and comments, and to the referee for careful corrections to the original version.

2. Real line arrangements and relations on Dehn twists In this section, we analyze line arrangements L in the complex plane defined by real equations and the monodromy on generic fibers under linear projections C2 \L→C. A key ingredient is B. Moishezon and M. Teicher description of the monodromy as elements of the pure braid group. (See, for example, [13]and[10].) We generalize this braid monodromy by studying the action of the monodromy not only on generic fibers of ρ, but also on their compactifications as genus zero surfaces with boundary. This leads to a proof of Theorem 1.2. The ideas in this section can be generalized to more arbitrary plane curves. An investigation of the topology of plane curve complements using such general projections appears in work of O. Zariski and E. van Kampen [16]. We leave this as a topic for future study.

2.1. Braid monodromy deﬁned by planar line arrangements over the reals. In this section we recall the Moishezon-Teicher braid monodromy associated to a real line arrangement. For convenience we choose Euclidean coordinates (x, y) for C2 so that no line is parallel to the y-axis, and no two intersection points have the same x-coordinate. For i =1,...,n,letLi be the zero set of a linear equation in x and y with real coeﬃcients:

Li = {(x, y); y = mix + ci} mi,ci ∈ R and assume that the lines are ordered so that the slopes satisfy:

m1 >m2 > ···>mn. 2 Let I = I(L)={p1,...,ps}⊂C be the collection of intersections points of L ordered so that the x-coordinates are strictly decreasing. Let ρ : C2 → C be the projection of C2 onto C given by ρ(x, y)=x.Foreach x ∈ C,let −1 Fx = ρ (x) \L.

The y-coordinate allows us to uniformly identify Fx with the complement in C of n points Li(x), where −1 {(x, Li(x))} = ρ (x) ∩ Li.

Thus, we will think of Fx as a continuous family of copies of C minus a ﬁnite set of points, rather than as a subset of C2. Let x0 ∈ R be greater than any point in ρ(I). Then there is a natural map γ :[0, 1] → C \ ρ(I) from arcs based at x0 toabraidonn strands in C parameterized by

{Li(γ(t)) : i =1,...,n}.

GENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 117

f p3 fp 2 f p1

x0 gp gp gp 3 2 1

Figure 3. Simple loop generators for π1(C \ ρ(I)).

Since two homotopic arcs give rise to isotopic braids, and a composition of arcs gives rise to a composition of braids, we have a homomorphism 2 β : π1(C \ ρ(I),x0) → B(S ,n+1) from the fundamental group to the spherical braid group on n + 1 strands. The (braid) monodromy of (C2, L) with respect to the projection ρ and base- point x0 is the homomorphism C \ I → (2.1) σL : π1( ρ( ),x0) MCG(Fx0 ), given by the composition of β and the braid representation 2 → B(S ,n+1) MCG(S0,n+1)=MCG(Fx0 ), from the braid group to the mapping class group on a genus zero surface with n +1 punctures. C \ I We now study the image of simple generators of π1( ρ( ),x0)inMCG(Fx0 ).

C \ I −1 By a simple loop in π1( ρ( ),x0), we mean a arc of the form p = fpgpfp , where p ∈ ρ(I), p > 0,

gp :[0, 1] → C \ ρ(I) 2πit t → p + pe and fp is a arc from x0 to p + p whose image is in the upper half plane except at its endpoints. Since π1(C \ ρ(I),x0) is generated by simple loops, it is enough to understand the monodromy in the image of these elements. In order to describe the monodromy of p we study how Fx is transformed as x follows its arc segments gp and fp First we look at gp.LetLj1 ,Lj2 ,...,Ljk be the lines in L that pass through p. We can assume by a translation of coordinates that p =0,andLjr is deﬁned by an equation of the form

y = mrx ··· where m1 >m2 > >mk.Thenast varies in [0, 1], the intersection of Ljr with −1 ρ (gp(t)) is given by 2πit 2πit Ljr (gp(t)) = (pe ,mrpe ). The other lines in L locally can be thought of as having constant slope, hence −1 their intersections with ρ (gp(t)) retain their order and stay outside a circle loc ⊂ on Fgp(t) enclosing Lj1 (gp(t)),...,Ljk (gp(t)) (see Figure 4). Let ap Fp+ be this circle. The restriction of ρ to C2 \L deﬁnes a trivial bundle over the loc image of fp.Thusap determines a simple closed curve ap on Fx0 separating

Lj1 (x0),...,Ljk (x0) from the rest of the Lj (x0).

118 ERIKO HIRONAKA

Figure 4. Monodromy deﬁned by gp with the real part of L drawn in.

Next we notice that lifting over fp deﬁnes a mapping class on Fx0 .Thisis ∈ R \ I because there is a canonical identiﬁcation of Fx0 and Fx for any x ρ( )given by the natural ordering of L∩ρ−1(x)bythesizeofthey-coordinate from largest to smallest. Thus fp determines a braid on n strands and corresponding mapping ∈ c class βp MCG(Fx0 ). We have shown the following. −1 c Lemma . L 2.1 Let p = fpgpfp . The element σ (p) in MCG(Fx0 ) is the Dehn −1 loc twist αp centered at ap = βp (ap ).

Proof. By the above descriptions of the fibers above the arcs fp and gp,we can decompose σL(p)as −1 ◦ ◦ αp = βp σp βp, loc where σp is a right Dehn twist centered at ap = βp(ap). 2.2. Monodromy on compactified fibers. In this section, we define the C2 \L c c monodromy representation of π1( ,y0)intoMCG(Fx0 ), where Fx0 is a com- pactification of Fx0 as a compact surface with boundary. C2 L ∪n As before choose coordinates for , and let = i=1Li be a planar line arrangement defined over the reals with distinct slopes. Assume all points of intersection I have distinct x-coordinates. Let >0 be such that the radius disks N(p) around the points p ∈ ρ(I)aredisjoint.Letδ>0 be such that the δ radius tubular neighborhoods N (L ) around L are disjoint in the complement of δ i i −1 ρ (N(p)). p∈ρ(I)

GENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 119

Let D be a disk in C containing all points of ρ(I) in its interior, and having x0 on its boundary. Let N∞ be a disk centered at the origin of C so that C×N∞ contains L∩ρ−1(D). For each x ∈ C \ ρ(I), let c −1 ∩ C × \∪n ⊂ \ Fx = ρ (x) ( N∞ i=1Nδ(Li)) Fx Nδ(Li). For each x ∈ D and i =1,...,n,let

di(x)=∂Nδ(Li) ∩ Fx. We are now ready to define the monodromy on the compactified fibers c → c σL : π1(Fx0 ,y0) MCG(Fx0 ). Let η be the inclusion homomorphism c → η :MCG(Fx0 ) MCG(Fx0 ), c ⊂ that is, the homomorphism induced by inclusion Fx0 Fx0 . Then we would like to have a commutative diagram

c σL π (C \ ρ(I),y ) /MCG(F c) 1 PP0 0 PPP PσPL PPP η PP( MCG(F0). The kernel of η is generated by Dehn twists centered at the boundary components c 8 c of Fx0 (Theorem 3.18, [ ]). Thus, in order to describe σL, we need to understand what twists occur near boundary components in the monodromy associated to the arcs gp and fp deﬁned in Section 2.1. Consider the simplest case when L ⊂ C2 is a single line deﬁned by y = L(x)=mx. 2 Let Nδ(L) be the tubular neighborhood around L in C

Nδ(L)={(x, L(x)+y):|y| <δ}.

Then Nδ(L) ∩ Fg(t) is a disk centered at L(g(t)) of radius δ. The boundary ∂Nδ(L) is a trivial bundle over C \ δ(ρ(I)) with trivialization deﬁned by the framing of C by real and purely imaginary coordinates. ∈ I Now assume that there are several lines Lj1 ,...,Ljk meeting above p ρ( ).

Let L be a line through p with slope equal to the average of those of Lj1 ,...,Ljk , ∩ and let δ>0 be such that Nδ(L) Fgp(t) contains dj1 (gp(t)),...,djk (gp(t)), but no other boundary components of F c , for all t.Letd (t) be the boundary gp(t) p component of Fgp(0) given by ∩ dp(t)=∂N(L) Fgp(t).

Let dp = dp(0) and dji = dji (gp(0)).

Then looking at Figure 4, we see that the points Lj1 (gp(t)),...,Ljk (gp(t)) ro- tate as a group 360◦ in the counterclockwise direction as t ranges in [0, 1]. The corresponding mapping class on the bounded portion of Fgp(0) enclosed by dp is the composition of a clockwise full rotation of dp and a counterclockwise rotation around dj1 ,...,djk . It can also be thought of as moving the inner boundary components dj1 (gp(0)) in a clockwise direction while leaving all orientations of boundary components ﬁxed with respect to the complex framing of C.

120 ERIKO HIRONAKA

Figure 5. The mapping class ∂dp .

Figure 6. The monodromy deﬁned by gp.

Figure 5 illustrates the Dehn twist ∂dp centered at a simple closed curve parallel c to dp and Figure 6 shows the monodromy σL(gp) in the case when L is a union of 4 lines meeting at a single point p. In both ﬁgures, the middle picture illustrates the ﬁber Fgp(0.5) half way around the circle traversed by gp.Fromtheabove discussion, we have c −1 σL(gp)=(∂d1 ∂d2 ∂d3 ∂d4 ) ∂dp . More generally we have the following lemma. Lemma . 2.2 Let Lj1 ,...,Ljk be the lines meeting above p,andlet

gp :[0, 1] → C \L t → p + e2πit. Then the monodromy on F c defined by g is given by gp(0) p c −1 σL(g )=(∂ ···∂ ) ∂ . p dj1 djk dp 2.3. Deformations of line arrangements. To finish our proof of Theo- rem1.2 we analyze the effect of deforming a line arrangement. Let n L = Li i=1 be a finite union of real lines in the Euclidean plane, R2 with no two lines parallel. Let T be the complexified real line arrangement with all n lines intersecting at a 2 single point p0.Letρ : C → C be a generic projection, and let D ⊂ C be a disk of radius r centered at the origin containing ρ(I)andρ(p0) in its interior. Let γ :[0, 1] → C t → re2πit.

GENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 121

x x xN 2 1 x x x1 N 2

Figure 7. Two representatives of γ in π1(C \ ρ(I)).

c c Lemma 2.3. The monodromies σL(γ) and σT (γ) are the same.

Proof. Let −1 D∞ = N∞ × C ∩ ρ (∂D).

Then D∞ \T and D∞ \L are isomorphic as ﬁber bundles over γ and hence the monodromies over γ deﬁned by L and T are the same up to isotopy. c c Proof of Theorem 1.2. By Lemma 2.3, σL(γ)=σT (γ). Figure 7 gives an illustration of two equivalent representations of the homotopy type of γ. By Lemma 2.1 and Lemma 2.2, we have c ··· −1 σT (γ)=(∂d1 ∂dn ) ∂dp .

Let p1,...,ps be the elements of I numbered by decreasing x-coordinate. Then for each i =1,...,s,wehave

c −1 −1 σL(f g f )=(∂ ···∂ ) α pi pi pi dj1 djk pi where αpi is the pullback of dpi along the arc fpi .Thus,

c μ1 μs −1 σL(γ)=(∂ ···∂ ) α ···α , d1 ds ps p1 where μi is the number of elements in I∩Li. To show that Theorem 1.1 follows from Theorem 1.2, we need to show that the ordering given in Equation (1.1) can be obtained by a union of lines L satisfying the conditions. To do this, we start with a union of lines T intersecting in a single point. Let L1,...Ln be the lines in T ordered from largest to smallest slope. Translate L1 in the positive x direction without changing its slope so that the intersections of the translated line L1 with L2,...,Ln have decreasing x-coordinate. Continue for each line from highest to lowest slope, making sure with each time that the shifting L creates new intersections lying to the left of all previously created ones. More generally, we can deform the lines through a single point T to one in general position L so that the only condition on the resulting ordering on the pairs of lines is the following. A pair (i, j) must preceed (i, j +1)foreach1≤ i

Theorem 2.4. Let {p1,...,ps} be an ordering of the pairs (i, j), 1 ≤ i

(i, i +1), (i, i +2),...,(i, n) is strictly decreasing. Then there a lantern relation of the form ··· n−2 ··· ∂0(∂1 ∂n) = αp1 αps .

122 ERIKO HIRONAKA

Figure 8. Line arrangement, and associated arrangement of curves (n=6).

3. Applications Although it is known that all relations on the Dehn-Lickorish-Humphries generators can be obtained from the braid, chain, lantern and hyperelliptic relations, there are some other nice symmetric relations that come out of line arrangements that are not trivially derived from the four generating ones. We conclude this paper with a sampling.

3.1. Daisy relation. Consider the line arrangements given in Figure 8. As pointed out to me by D. Margalit, this relation was recently also discovered by H. Endo, T. Mark, and J. Van Horn-Morris using rational blowdowns of 4-manifolds [6]. We follow their nomenclature and call this the daisy relation. c Let S0,n+1 denote the compact surface of genus 0 with n + 1 boundary components. Consider the configuration of simple closed curves shown in Figure 8. c Let d0,...,dn be the boundary components of S0,n+1.Letd1 be the distinguished boundary component at the center of the arrangement, and let d0,d2,...,dn be the boundary components arranged in a circle (ordered in the clockwise direction around d1). Let a1,k be a simple closed loop encircling d1 and dk,where k =0, 2, 3,...,n.Let∂i be the Dehn twist centered at di, and let α1,k be the Dehn twist centered at a1,k. Theorem . ≥ c 3.1 (Daisy relation) For n 3, the Dehn twists on S0,n+1 satisfy the relation n−2 ··· ··· ∂0∂1 ∂2 ∂n = α1,0α1,n α1,2 where ∂i is the Dehn twist centered at the boundary component di,andα1,j is the Dehn twist centered at curves a1,j . When n = 3, Theorem 3.1 specializes to the usual lantern relation. Proof. We associate the boundary component di with Li for i =1,...,n, and d0 with the “line at infinity”. Theorem 1.2 applied to the line arrangement in Figure 8 gives: ··· −1 ∂0(∂1 ∂n) = Rpn ...Rp1 where p1,...,pn are the intersection points of the line arrangement L ordered by largest to smallest x-coordinate. For this configuration, pk gives rise to −1 Rpk =(∂1∂k+1) α1,k+1,

GENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 123

d1 L 1

L 2 d0 d2

L 3 a 1,0 a1,2 a L 4 1,4 a1,3 d 4 d3

Figure 9. Alternate drawing of the daisy conﬁguration (n=4).

Figure 10. Configuration of lines giving rise to the doubled daisy relation. for k =1,...,n− 1. Noting that the loop that separates d2 ∪···∪dn from d0 ∪ d1 canbewrittenasa1,0,wehave ··· −1 Rpn =(∂2 ∂n−1) α1,0 yielding the desired formula. Remark 3.2. Let 2 β : B(S ,n+1)→ MCG(S0,n+1) be the braid representation from the spherical braid group to the mapping class group. Recall the relation R in B(S2,n+1)givenby 2 −1 2 ··· −1 −1 ··· −1 2 ··· ··· 2 ··· (σ1)(σ1 σ2σ1) (σ1 σ2 σn−1σnσn−1 σ1)=σ1 σn−1σnσn−1 σ1. =1 This induces a relation R in MCG(S0,n+1). The daisy relation can be considered as the lift of R under the inclusion homomorphism η. 3.2. Doubled daisy relation. As a final example, we consider a configuration of n ≥ 5 lines, with n−2 meeting in a single point. There are several ways this can be drawn. We give one example in Figure 10. Other line arrangements satisfying these conditions will give similar relations, but the drawings of the associated curves will be more complicated. c As before, let d0,...,dn be the boundary components of S0,n+1. The boundary component di is associated to the line Li for i =1,...,n,andd0 is the boundary

124 ERIKO HIRONAKA

d d 4 d3 d2 0

Figure 11. The doubled daisy relation for n =5.

Figure 12. Drawing of the general doubled daisy conﬁguration.

component associated to the “line at inﬁnity”. Let ai,j betheloopinFigure12 encircling di ∪ dj and no other boundary component. Let c be the loop encircling d2,...,dn−1 in Figure 12 (or, when n =5,d2,d3, and d4 in Figure 11).

Theorem 3.3 (Doubled daisy relation). Let ∂i be the right Dehn twist centered at di, αi,j the right Dehn twist centered at ai,j ,andβ the right Dehn twist centered at c.Then n−2 ··· n−2 ··· ··· ∂0∂1 ∂2 ∂n−1∂n = αn−1,nαn−2,n α2,n βα1,nα1,n−1 α1,2 Proof. Theorem 2 applied to the line arrangement in Figure 10 gives the equation ··· −1 ··· ··· ∂0(∂1 ∂n) = Rsn−2 Rs1 RqRpn−2 Rp1 , where −1 Rpk =(∂1∂k+1) α1,k+1 −1 Rq =(∂2 ···∂n−1) β −1 Rsk =(∂n∂k+1) αk+1,n.

(As one sees from Figure 10 and Figure 12, the order of Rpn−1 and Rq may be interchanged.) Putting these together yields the desired formula.

GENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 125

References 1. B. Artal, J. Ruber, J. Cogolludo, and M. Marco, Topology and combinatorics of real line arrangements,Comp.Math.141 (2005), no. 6, 1578–1588. MR2188450 (2006k:32055) 2. J. Birman, Mapping class groups of surfaces, Braids (Santa Cruz, CA, 1986), vol. 78, Amer. Math. Soc., Providence, RI, 1988. MR975076 (90g:57013) 3. D. Cohen and A. Suciu, Braid monodromy of plane algebraic curves and hyperplane arrangements, Comm. Math. Helv. 72 (1997), 285–315. MR1470093 (98f:52012) 4. R. Cordovil, The fundamental group of the complement of the complexification of a real arrangement of hyperplanes, Adv. App. Math. 21 (1998), 481–498. MR1641238 (99g:52015) 5. M. Dehn, Die gruppe der abbildungsklassen,ActaMath.69 (1938), 135–206. MR1555438 6. H. Endo, T. E. Mark, and J. Van Horn-Morris, Monodromy substitutions and rational blowdowns, J. Topol. 4 (2011), no. 1, 227–253. MR2783383 (2012b:57051) 7. P. Erdös and R. Steinberg, Three point collinearity, American Mathematical Monthly 51 (1944), no. 3, 169–171. MR1525919 8. B. Farb and D. Margalit, A primer on mapping class groups, Princeton University Press, 2011. MR2850125 9. J. Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), no. 2, 221–239. MR700769 (84g:57006) 10. E. Hironaka, Abelian coverings of the complex projective plane branched along configurations of real lines,Mem.oftheA.M.S.105 (1993). MR1164128 (94b:14020) 11. D. Johnson, Homeomophisms of a surface which act trivially on homology, Proc. of the Amer. Math. Soc. 75 (1979), 119–125. MR529227 (80h:57008) 12. M. Matsumoto, A simple presentation of mapping class groups in terms of Artin groups, Sugaku Expositions 15 (2002), no. 2, 223–236. MR1944137 (2004g:20051) 13. B. Moishezon and M. Teicher, Braid group technique in complex geometry. I. Line arrangements in CP2, Braids (Santa Cruz, CA, 1986), Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 425–555. MR975093 (90f:32014) 14. P. Orlik and H. Terao, Arrangement of hyperplanes, Grundlehren der math. Wissenschaften, vol. 300, Springer-Verlag, Berlin, 1992. MR1217488 (94e:52014) 15. G. Rybnikov, On the fundamental group of the complement of a complex hyperplane arrangement., DIMACS: Technical Report (1994), 33–50. 16. E. van Kampen, On the fundamental group of an algebraic curve,Am.Jour.Math.55 (1933), 255–260. 17. B. Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983), 157–174. MR719117 (85g:57007) 18. , Mapping class group of a handlebody, Fund. Math. 158 (1998), 195–228. MR1663329 (2000a:20075)

Department of Mathematics, Florida State University, Tallahassee, Florida 32306- 4510 E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11373

Eﬀective p-adic cohomology for cyclic cubic threefolds

Kiran S. Kedlaya

This paper is an updated form of notes from a series of six lectures given at a summer school on p-adic cohomology held in Mainz in the fall of 2008. (They may be viewed as a sequel to the author’s notes from the Arizona Winter School in 2007 [51].) The goal of the notes is to describe how to use p-adic cohomology to make effective, provably correct numerical computations of zeta functions. More specifically, we discuss three techniques in detail: • use of the Hodge filtration to infer the zeta function from point counts; • the “direct cohomological method” of computing the Frobenius action on the p-adic cohomology of a single variety; • the “deformation method” of computing the Frobenius structure on the p-adic cohomologies of a one-parameter family of varieties, using the associated Picard-Fuchs differential equation. We demonstrate the effective nature of these methods by describing how to make them explicit for cyclic cubic threefolds, i.e., smooth cubic threefolds in P4 admitting an automorphism of order 3. This example has the features of being rich enough to allow us to illustrate some useful features of p-adic cohomology (e.g., behavior with respect to automorphisms, and effect of the Hodge filtration) while simple enough that the final computations are still tractable. A number of references will be made to computations that can be made using the Sage open-source computer algebra system, including a numerical example over the field F7 to which we return frequently. We have prepared a worksheet containing all of these computations in the form of a Sage notebook available at the author’s web site [54]; however, one key calculation requires the additional nonfree system Magma [67] to be installed. (It is also worth noting that our Sage code depends implicitly upon the commutative algebra package Singular [89], which Sage incorporates.)TimingsquotedarebasedonexecutionsonanAMD Opteron 246 (64-bit, 2 GHz) with 2 GB of RAM. The structure of the six lectures is as follows. (Note that subsections marked “Optional” were not intended for presentation in the lectures.) In lecture 1, we recall some generalities about zeta functions of varieties over finite fields, specialize to the case of cyclic cubic threefolds, then demonstrate with the Fermat cubic

2010 Mathematics Subject Classiﬁcation. Primary 14G10; Secondary 14F30. The author was supported by NSF CAREER grant DMS-0545904, a Sloan Research Fellow- ship, and the NEC Fund of the MIT Research Support Committee.

c 2012 American Mathematical Society 127

128 KIRAN S. KEDLAYA and with a more generic example over F7. In lecture 2, we recall the formalism of algebraic de Rham cohomology, then make it explicit for cyclic cubic threefolds. In lecture 3, we recall the formalism of p-adic cohomology, including the divisibilities imposed on the zeta function by the Hodge ﬁltration; we then apply this knowledge to our generic example of a cyclic cubic threefold, and fully recover the zeta function. In lecture 4, we describe how to directly compute the Frobenius action on the p-adic cohomology of a variety, and illustrate using our generic example; however, we do not include a computational demonstration because the method we had in mind at the time of preparation of these notes appears to be infeasible. (It subsequently became clear that this diﬃculty is not insurmountable; see Remark 4.4.8.) In lecture 5, we introduce relative de Rham cohomology and Picard-Fuchs-Manin (Gauss-Manin) connections, and compute an example for a pencil of cyclic cubic threefolds including our generic example. In lecture 6, we describe Frobenius structures on Picard-Fuchs-Manin connections, compute the Frobenius structure for the connection from the previous lecture, and recover the zeta function of our generic example. The appendix contains many references and remarks omitted from the main text in order to streamline the exposition.

Acknowledgments. Thanks to Duco van Straten, Ralf Gerkmann, and Kira Samol for organizing the summer school in Mainz, supported by SFB/TR 45 “Peri- ods, Moduli Spaces, and Arithmetic of Algebraic Varieties”. Thanks to Jim Carlson for the suggestion to consider cyclic cubic threefolds, to Alan Lauder for helpful discussions about Frobenius structures on connections, and to Jan Tuitman for pointing out an error in our original analysis of t-adic precision (now resolved in [56]).

1. Zeta functions: generalities In this lecture, we recall the notion of the zeta function of an algebraic variety, and the formalism of Weil cohomology theories which can be used to interpret the Weil conjectures on zeta functions. We illustrate by computing the zeta function of the Fermat cubic threefold over F7; this example will be needed later as an initial condition for solving a Picard-Fuchs-Manin connection.

1.1. Zeta functions of algebraic varieties. Definition 1.1.1. Let X be a variety (reduced separated scheme of finite type) over the finite field Fq.Thezeta function of X is the formal power series ! " ∞ T n ζ (T )=exp #X(F n ) ; X n q n=1 we can also write ζX (T ) as an Euler product [κx:Fq] −1 ζX (T )= (1 − T ) x∈X over closed points x of X (where κx denotes the residue field of x), so ζX (T ) ∈ ZT . Remark 1.1.2. One motivation for computing zeta functions of varieties over finite fields is that they can be used to compute L-functions of varieties over number fields, which carry enormous amounts of global arithmetic information. For

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 129 instance, for E an elliptic curve over Q,forp a prime of good reduction, we have

Lp(T ) ζEF (T )= p (1 − T )(1 − pT ) 2 for Lp(T ) a polynomial of the form 1 − apT + pT . (Note that ap can be computed as p +1− #E(Fp).) For an appropriate definition of Lp(T )forp not of good reduction, the L-function of an elliptic curve over Q is defined as the product −s L(E,s)= Lp(p ). p This product converges absolutely for Real(s) > 3/2, but is now known to extend to an analytic function on all of C. The conjecture of Birch and Swinnerton-Dyer predicts that the order of vanishing of L(E,s)ats = 1 equals the rank of the group E(Q) of rational points of E. The methods developed in this paper can be used in particular to compute zeta functions for cyclic cubic threefolds. In subsequent work, we plan to use these techniques to gather some data concerning L-functions of cyclic cubic threefolds over Q. 1.2. The Weil conjectures. The following theorem encompasses what were formerly (and still commonly) called the Weil conjectures. For historical details, see the references in the appendix. Theorem 1.2.1. Let X be a variety (separated scheme of finite type) over the finite field Fq. Then the zeta function of X is the power series representation of a rational function in T . Moreover, if X is smooth and proper over Fq, then there is auniquewaytowrite 2dim(X) (−1)i+1 (1.2.1.1) ζX (T )= Pi(T ) i=0 for some polynomials Pi(T ) ∈ Z[T ] with Pi(0) = 1, satisfying the following conditions. (i) We have

i −i deg(Pi)/2 − deg(Pi) Pi(1/(q T )) = ±q T Pi(T ),

with the sign being + whenever i is odd. In other words, the roots of Pi are invariant under the map r → q−i/r,andifi is odd then the multiplicities of ±q−i/2 are even. −i/2 (ii) The roots of Pi in C all have complex absolute value q . (This is commonly called the Riemann hypothesis for zeta functions of varieties over finite fields.) ∼ (iii) If X = XFq for some smooth proper scheme X over the local ring R = oK,p for some number field K and some prime ideal p of oK with residue field Fq, then for any embedding K→ C, i an deg(Pi)=dimC H ((X ×R C) , C).

In other words, deg(Pi) equals the i-th Betti number of X ×R C. Remark 1.2.2. Using p-adic cohomology, one can reﬁne assertion (iii) of The- orem 1.2.1 to take into account the Hodge numbers of X in addition to the Betti numbers. See Theorem 3.3.1.

130 KIRAN S. KEDLAYA

When computing zeta functions, it will be helpful to account for the Riemann hypothesis using the following lemma from [50] (applied to the reverse of one of the Pi).

Lemma 1.2.3. Given positive integers q, d, j, and complex numbers a1,...,aj−1, d j/2 there exists a certain explicit disc of radius j q which contains every aj for which we can choose aj+1,...,ad ∈ C so that the polynomial d j R(T )=1+ aj T j=1 has all roots on the circle |T | = q−1/2.

By contrast, bounding ad−j directly gives the far inferior bound d j/2 |a − |≤ q . d j j

Proof. Let sj denote the sum of the (−j)-th powers of the roots of R.From the Newton-Girard identities, j−1 sj + jaj = − sj−hah; h=1

j/2 given a1,...,aj−1, we may explicitly compute s1,...,sj−1.Since|sj |≤dq ,this d j/2 limits aj to an explicit disc of radius j q . Remark 1.2.4. The bound in Lemma 1.2.3 is typically not very tight except when j is very small. See Remark 4.3.3 for an example in the context of these lectures, and [50] for additional examples. 1.3. Weil cohomology. We now recall Weil’s proposed cohomological interpretation of Theorem 1.2.1. Our discussion is quite incomplete; see the references in the appendix for further details.

Definition 1.3.1. Fix a finite field Fq and a field F of characteristic zero. A Weil cohomology over F consists of a collection of contravariant functors Hi(·)from smooth proper varieties X over Fq to finite dimensional F -vector spaces, satisfying a number of additional conditions which we will not list completely (see [57]fora full account). Instead, we will simply enumerate the ones we need as we use them. i i For one, H (X) is canonically isomorphic to H (XFqn ) for any n. For another, if we let Fq : X → X denote the q-power Frobenius, and put i Pi(T ) = det(1 − TFq,H (X)) (i =0,...,2 dim(X)), then we must have that Pi(T ) ∈ Z[T ] and (1.2.1.1) holds. This last claim is equivalent to the Lefschetz trace formula: for any positive integer n, 2dim(X) F − i n i (1.3.1.1) #X( qn )= ( 1) Trace(Fq ,H (X)). i=0 (This equivalence requires the coefficient field to have characteristic zero.)

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 131

We will make extensive use of a slightly stronger form of (1.3.1.1): for any automorphism ι of X, 2dim(X) i i (1.3.1.2) #{x ∈ X(Fq):(Fq ◦ ι)(x)=x} = (−1) Trace(Fq ◦ ι, H (X)). i=0 Remark 1.3.2. The existence of a Weil cohomology, plus the Lefschetz trace formula (1.3.1.1), together imply the rationality of ζX (T ). To deduce property (i) in Theorem 1.2.1, one needs Poincaré duality for Weil cohomology. For property (iii), one needs a comparison theorem between the given Weil cohomology and singular cohomology over C. Property (ii) lies somewhat deeper; we will not discuss its proof here. Remark 1.3.3. The Lefschetz trace formula (1.3.1.2) can be extended to more general endomorphisms, and even to correspondences, but the counting function on the left side must be replaced by a more complicated sum of local terms. In the case of Fq ◦ ι, the graph of the morphism has transverse intersection with the diagonal × inside X Fq X, so the fixed points are isolated and occur with multiplicity 1. One other case in which one can describe the trace formula is for an automorphism of order prime to the characteristic of Fq; in that case, the left side of the Lefschetz formula becomes the Euler characteristic of the fixed locus. Remark 1.3.4. The first Weil cohomology to be constructed was étale cohomology, in which the coefficient field may be taken to be the -adic numbers Q for any prime distinct from the characteristic of K. See appendix for references. 1.4. Cyclic cubic threefolds. We now specialize the discussion to the particular class of varieties we will be using as examples in this paper. Definition 1.4.1. Let K be a field of characteristic not equal to 3. A cyclic P4 cubic threefold over K is a hypersurface of degree 3 in K invariant under the action of a cyclic group of order 3. Throughout these notes, when discussing cyclic cubic P4 threefolds, we will take homogeneous coordinates w, x, y, z, a on K and restrict to cyclic cubic threefolds defined by polynomials of the form S = a3 − Q with Q ∈ K[w, x, y, z] homogeneous of degree 3. (This is the most general form for K algebraically closed.) Lemma 1.4.2. ThecycliccubicthreefolddefinedbyS = a3 − Q is smooth if P3 and only if the cubic surface in K defined by Q is smooth.

Proof. Let Sw denote the partial derivative of the polynomial S with respect to the variable w, and so forth. Then 3 2 (S, Sw,Sx,Sy,Sz,Sa)=(a − Q, Qw,Qx,Qy,Qz, 3a ), so the saturation of this ideal contains a and hence Q. Consequently, this ideal contains a power of (w, x, y, z, a) if and only if (Q, Qw,Qx,Qy,Qz) contains a power of (w, x, y, z). In other words, (S, Sw,Sx,Sy,Sz,Sa) deﬁnes the empty subscheme of Proj K[w, x, y, z, a] if and only if (Q, Qw,Qx,Qy,Qz) deﬁnes the empty subscheme of Proj K[w, x, y, z]; this is the desired result.

Observation 1.4.3. Let X be a cyclic cubic threefold over Fq.BytheLefschetz hyperplane section property of a Weil cohomology, for i =0, 1, 2, 4, 5, 6, we have

132 KIRAN S. KEDLAYA a canonical isomorphism Hi(X) ∼ Hi(P3 ). Thus the zeta function of X has the = Fq form P (T ) ζ (T )= X (1 − T )(1 − qT)(1 − q2T )(1 − q3T ) 3 for P (T ) = det(1 − TFq,H (X)). We will show using algebraic de Rham cohomology (see Observation 2.3.1) that the middle Betti number of any lift of X is 10, so dim H3(X)=deg(P ) = 10. It will then follow that P (1/(q3T )) = q−15T −10P (T ), and the complex roots of P lie on the circle |T | = q−3/2.

Definition 1.4.4. Given a choice of a primitive cube root ζ3 ∈ K,wewrite [ζ3] for the automorphism

[ζ3]([w : x : y : z : a]) = [w : x : y : z : ζ3a] on any cyclic cubic threefold X over K.IncaseK = Fq with q ≡ 1(mod3), 3 [ζ3] splits H (X) into two eigenspaces of dimension 5, on which [ζ3]actsbymul- tiplication by the two primitive cube roots of 1 in the coefficient field. (This will be apparent for rigid cohomology from the explicit description we will give; for an arbitrary Weil cohomology, this can be deduced from the Lefschetz trace formula 2 for the automorphisms [ζ3]and[ζ3] , as described in Remark 1.3.3.) Consequently, P (T )factorsoverZ[ζ3] into two factors of degree 5. In case K = Fq with q ≡ 2(mod3),[ζ3] is not defined over Fq,soitdoesnot 2 commute with Fq; rather, we have Fq ◦[ζ3]=[ζ3] ◦Fq. In fact, we may see explicitly 3 3 that #X(Fq)=#P (Fq): for each w, x, y, z, the equation a = Q(w, x, y, z)has 3 exactly one solution a ∈ Fq. Hence Trace(Fq,H (X)) = 0, and similarly for any 2 odd power of Fq. This forces P (T ) to be a polynomial of degree 5 in T ,which we can recover by computing the zeta function of XF . We will thus concentrate q2 mainly on the case q ≡ 1 (mod 3) hereafter. Remark 1.4.5. The dichotomy we have just encountered is analogous to the situation of an elliptic curve with complex multiplication. In that case, whether the curve has ordinary or supersingular reduction is determined by whether the prime of reduction is split or inert in the CM field. 1.5. A special example: the Fermat cubic threefold. As an explicit illustration of the properties of zeta functions, we compute the action of Frobenius on the Weil cohomology of a very special cubic threefold. Definition 1.5.1. Let K be a field of characteristic not equal to 3. The P4 Fermat cubic threefold over K is the threefold X in K defined by the polynomial w3 + x3 + y3 + z3 + a3 = 0; we will identify it with the cyclic cubic threefold defined by S = a3 − Q for Q = w3 + x3 + y3 + z3. If K contains a primitive cube root ζ3, the analysis of the Fermat cubic threefold 5 is aided greatly by the action of the group G = μ3 acting by c0 c4 c0 ··· c4 (ζ3 ,...,ζ3 )[w : x : y : z : a]=[ζ3 w : : ζ3 a]. This action on X factors through the quotient by the diagonal subgroup generated by (ζ3,...ζ3). However, we prefer to use G instead of the quotient so we can have also an action on homogeneous polynomials.

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 133

Procedure 1.5.2. Consider the Fermat cubic threefold X over Fq with q ≡ 1 (mod 3); then the action of G is deﬁned over Fq, so it commutes with Fq.Wecan 3 then compute the trace of Fq on each of the eigenspaces of H (X)forG using the Lefschetz trace formula (1.3.1.2), as follows. (q−1)/3 1/3 Choose a cubic nonresidue r in Fq with ζ3 = r . Fix also a cube root r F c0 c4 ∈ of r in q.Forc =(ζ3 ,...,ζ3 ) G with c4 = 0, put

−c0 3 −c1 3 −c2 3 −c3 3 P˜c = r w + r x + r y + r z and let X˜c denote the corresponding cyclic cubic threefold. The variety X˜c is a twist of X; that is, it is isomorphic to X over Fq. Speciﬁcally, we may identify the Fq-points of X with those of X˜c via the map [w : x : y : z : a] → [rc0/3w : rc1/3x : rc2/3y : rc3/3z : a].

Under this identification, the fixed points of Fq ◦ c on X are identified with the fixed points of Fq on X˜c. Thus (1.3.1.2) may be rewritten in this case as 6 i i #X˜c(Fq)= (−1) Trace(Fq ◦ c, H (X)) i=0 2 3 3 =1+q + q + q − Trace(Fq ◦ c, H (X)). 3 We can thus compute Trace(Fq ◦ c, H (X)) by counting the points of #X˜c(Fq). For q small, we may as well do this by enumerating the points themselves; for some procedures that make more sense when q is large, see Procedure 1.7.1 and Remark 1.7.3. We may describe the character group G of G as (Z/3Z)5, where the character (d0,...,d4):G → μ3 acts as

c0 c4 c0d0+···+c4d4 (d0,...,d4)(ζ3 ,...,ζ3 )=ζ3 .

Given a primitive cube root of unity ζ3,F ∈ F ,wemayembedμ3 into F and separate H3(X) into eigenspaces for the characters of G.Inparticular,forthe 5 eigenspace corresponding to the character d =(d0,...,d4) ∈ (Z/3Z) , we compute the trace on that eigenspace as 2 1 − − − − ζ c0d0 c1d1 c2d2 c3d3 Trace(F ◦ c, H3(X)). 34 3 q c0,c1,c2,c3=0 Example 1.5.3. For q = 7, we may carry out Procedure 1.5.2 by explicitly counting the F7-points of all of the X˜c (see worksheet). We ﬁx the cube root ζ3 =2 in F7. For the eigenspaces corresponding to the characters (1.5.3.1) (2, 1, 1, 1, 1), (1, 2, 1, 1, 1), (1, 1, 2, 1, 1), (1, 1, 1, 2, 1), (2, 2, 2, 2, 1), (1.5.3.2) (1, 1, 1, 1, 2), (1, 2, 2, 2, 2), (2, 1, 2, 2, 2), (2, 2, 1, 2, 2), (2, 2, 2, 1, 2), we obtain the traces

(1.5.3.3) 21ζ3,F +7, 21ζ3,F +7, 21ζ3,F +7, 21ζ3,F +7, −21ζ3,F − 14, 2 2 2 2 − 2 − (1.5.3.4) 21ζ3,F +7, 21ζ3,F +7, 21ζ3,F +7, 21ζ3,F +7, 21ζ3,F 14, respectively. It follows that each of these eigenspaces is one-dimensional, there are no other eigenspaces, and the polynomial P (T ) in the zeta function of X equals the product of 1 − αT for α running over the values in (1.5.3.3) and (1.5.3.4).

134 KIRAN S. KEDLAYA

Remark 1.5.4. One has the same eigenspace decomposition, with the same characters, for any q ≡ 1 (mod 3). For a general Weil cohomology, this can be proved using the Lefschetz trace formula for the elements of G (Remark 1.3.3; compare Definition 1.4.4). For rigid cohomology, this will follow from an explicit description using algebraic de Rham cohomology (Example 2.3.2) and the comparison theorem with rigid cohomology (Theorem 3.2.1). Remark 1.5.5. The general formalism of Weil cohomologies does not provide a specific way to match up the primitive cube roots of unity in Fq and F . We will see later that the formalism of p-adic cohomology does provide such a matching. 1.6. A generic example. We now introduce a less special example, to which we will return throughout the lectures. Example 1.6.1. Consider the polynomial Q = w3 + x3 + y3 + z3 +(w + x)(w +2y)(w +3z)+3xy(w + x + z) over F7. One computes (see worksheet) that the Jacobian ideal (Qw,Qx,Qy,Qz) of Q is zero-dimensional, so Q is nonsingular. Consequently, we have a cyclic cubic 3 threefold X over F7 with defining equation S = a − Q. We fix the choice ζ3 =2inF7, and let ζ3,F be a primitive cube root of 1 in the 3 coefficient field F .LetH1,H2 be the eigenspaces of [ζ3]onH (X) with eigenvalues 2 ∈ F (7−1)/3 ζ3,F ,ζ3,F , respectively. Let b 7 be a cubic nonresidue with b = 2, and fix 1/3 acuberootb of b in Fq. As in Procedure 1.5.2, for k =0, 1, 2, we identify the −k 3 Fq-rational points of the cubic threefold Xq,k defined by b a − Q with the fixed k points of Fq ◦ [ζ3] ,viathemap [w : x : y : z : a] → [w : x : y : z : b−k/3a]. Using the extended Lefschetz trace formula (1.3.1.2), we find that for j =1, 2, 2 1 − Trace(F ,H )=− ζ jk#X (F ). q j 3 3,F q,k q k=0 By enumerating points (see worksheet), we obtain the following table after about 15 minutes of computation. (Note that we infer the counts for k = 2 from the other two columns, using the fact that each row must sum to 3(q3 + q2 + q +1).)

#Xq,k(Fq) k =0 k =1 k =2 q =7 407 365 428 q =72 120933 118728 120639 q =73 40464740 40484291 40465769 We thus obtain the series approximations 2 det(1 − TFq,H1)=1+(3ζ3,F + 2)(7T )+(8ζ3,F + 5)(7T ) 3 4 +(7ζ3,F − 14)(7T ) + O(T ) − 2 2 2 det(1 TFq,H2)=1+(3ζ3,F + 2)(7T )+(8ζ3,F + 5)(7T ) 2 − 3 4 +(7ζ3,F 14)(7T ) + O(T ). Since each of these is a polynomial of degree 5, we do not have enough data from the point counts alone to determine ζX (T ).Thiswouldremaintrueevenifwe computed a fourth row of the table; we estimate that this would have taken us

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 135 about one week of computation. (We did not attempt to combine this data with the Riemann hypothesis bound using Lemma 1.2.3; see appendix for discussion.) 1.7. Optional: Counting points on diagonal threefolds. For completeness, we describe some more intelligent procedures for counting points on diagonal cubic threefolds. We start with a procedure that is still simple but improves greatly upon counting points directly for q of moderate size.

Procedure 1.7.1. Recall that we wish to count the Fq-points of the twisted Fermat cubic threefold X˜c corresponding to the polynomial − − − − P˜ = r c0 w3 + r c1 x3 + r c2 y3 + r c3 z3, × ≡ ∈ Z ∈ F for q 1(mod3).Forj, j ,letaj,j be the number of x q such that rjx3 +1 equals rj times a nonzero cubic residue; this only depends on j, j modulo 3. The aj,j can be computed using cubic Jacobi sums (see Remark 1.7.3 for the case q = p); for now, we instead compute a0,1,a0,2 by iterating over all x ∈ Fq, then use the identities

aj,j = aj,j a = a− − #j,j j,j j q − 4 j ≡ 0(mod3) aj,j = q − 1 j ≡ 0(mod3) j to infer the other aj,j . For i ∈{0, 1, 2, 3, 4} and j ∈{0, 1, 2} ∪ {∗}, put # # $ r−j(F×)3 j =0, 1, 2 i+1 −c0 3 −ci 3 q Ci,j =# (u0,...,ui) ∈ F : r u + ···+ r u ∈ . q 0 i {0} j = ∗

For i =0, 1, 2, 3, 4 in succession, we compute the Ci,j for all j as follows. For i =0, we have q − 1 C ∗ =1,C= (j =0, 1, 2). 0, 0,j 3 Given the C − for some i>0, we compute i 1,j # (q − 1)Ci−1,∗ j ≡ ci (mod 3) Ci,j = Ci−1,j + Ci−1,kac −k,c −j + i i 0 j = c (mod 3) k=0,1,2 i

Ci,∗ = Ci−1,∗ +3Ci−1,ci . Then we have 1 X˜(F )= (C − 1). q q − 1 4,0 Observation 1.7.2. If q ≡ 2 (mod 3), there is no need to count anything over Fq because all diagonal cubics have as many points as projective space itself. However, one may wish to carry out Procedure 1.7.1 over Fq2 . In this case, the base calculation of aj,j is made somewhat easier by the fact that a0,1 = a0,2. Hence it 3 3 suﬃces to calculate a0,0, but this is also easy: since the elliptic curve x +1=y over Fq has zeta function 1+qT 2 (q ≡ 2(mod3)), (1 − T )(1 − qT)

136 KIRAN S. KEDLAYA we have q2 +2q − 8 q2 − q − 4 a = ,a= a = . 0,0 3 0,1 0,2 3

We next describe a computation of the aj,j based on cubic Jacobi sums in the case q = p ≡ 1(mod3).

Remark 1.7.3. For two Dirichlet characters χ1,χ2 on Fp, deﬁne the Jacobi sum J(χ1,χ2)= χ1(u)χ2(v)

u,v∈Fp:u+v=1 × ∈ F 2 j 3 We may interpret 3aj,j as the number of pairs (x, y) ( p ) for which r x + j 3 r y =1.Letχ be the cubic Dirichlet character on Fp sending r to ζ3.Then { ∈ F2 j 3 j 3 } # (x, y) p : r x + r y =1 2 2 −ij i −i j i = ζ3 χ (u) ζ3 χ (v) u+v=1 i=0 i=0 −ij−i j i i = ζ3 J(χ ,χ ) i,i − −j−2j − −2j−j −j−j −2j−2j 2 2 = q ζ3 ζ3 + ζ3 J(χ, χ)+ζ3 J(χ ,χ ), where the last line follows from the one before by standard identities [42, §8.3, Theorem 1]. By [3, Theorem 3.1.3], we have 1 √ J(χ, χ)= (α + iβ 3) 2 where α, β are uniquely determined by the requirements α2 +3β2 =4p α ≡ 1(mod3) β ≡ 0(mod3) 3β ≡ (2r(p−1)/3 +1)α (mod p). These α and β can be found in time polylogarithmic in p, e.g., by performing the (p−1)/3 Euclidean algorithm on p andr ˜ − ζ3 in Z[ζ3] for anyr ˜ ∈ Z lifting r. Remark 1.7.4. In the case q = p, an explicit (but complicated) formula to compute the Ci,j directly can be found in [3, Theorem 10.6.1]. 2. Algebraic de Rham cohomology We next describe the formalism of algebraic de Rham cohomology, then specialize to the case of cyclic cubic threefolds. This will be used for our explicit descriptions of p-adic cohomology in the next lecture. 2.1. Cohomology of smooth varieties. We first recall the definition of algebraic de Rham cohomology for smooth varieties. Definition 2.1.1. Let X be a smooth variety over a field K of characteristic 0. Let ΩX/K be the sheaf of Kähler differentials; since X is smooth, by the Jacobian i criterion ΩX/K is coherent and locally free of rank dim(X/K). Let ΩX/K be the i- O 0 th exterior power of ΩX/K over the structure sheaf X/K ,soinparticularΩX/K =

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 137

O 1 O → X/K and ΩX/K =ΩX/K . There is a universal derivation d : X/K ΩX/K , i → i+1 ◦ using which we obtain maps d :ΩX/K ΩX/K satisfying d d =0.Wethusobtain the de Rham complex of sheaves → 0 → 1 →··· 0 ΩX/K ΩX/K . i The algebraic de Rham cohomology HdR(X)ofX is deﬁned to be the hypercohomol- Hi · ogy (ΩX/K ) of this complex. If X is aﬃne, this coincides with the cohomology of i the complex of global sections (so in particular HdR(X) vanishes for i>dim(X)); · otherwise the coherent cohomology of each ΩX/K intervenes, so we only have the i weaker vanishing result that HdR(X)=0fori>2 dim(X). By recalling how to compute hypercohomology, we identify an important extra structure on de Rham cohomology.

Definition 2.1.2. Let {Ul} be a finite cover of X by affine open subschemes. Let Ci,j be the j-th term of the Cechˇ complex (with differentials dˇ) associated to i { } i,j the sheaf ΩX/K and the cover Ul .WemayviewC as a double complex with differentials d and dˇ; the total complex with differential on Ci,j given by d +(−1)idˇ Hi · i computes the hypercohomology (ΩX/K )=HdR(X). i s,i−s More precisely, HdR(X) consists of classes supported on C for s =0,...,i. j i We may define a descending filtration Fil HdR(X) by taking classes supported s,i−s i only on C for s = j,...,i; this defines the Hodge filtration on HdR(X), which turns out to be independent of the choice of the affine covering. For instance, i i Fil HdR(X) consists of classes represented by holomorphic i-forms on X.More j i i generally, Fil HdR(X)istheimageinHdR(X) of the hypercohomology of the truncated de Rham complex → j →···→ dim(X) → 0 ΩX/K ΩX/K 0 h in which ΩX/K is still placed in degree h. Theorem 2.1.3 (Grothendieck). Given an embedding K→ C,weobtain i ⊗ C canonical isomorphisms from HdR(X) K to the following: • the singular cohomology of X with coefficients in C; • the smooth de Rham cohomology of X with coefficients in C; • the holomorphic de Rham cohomology (Dolbeaut cohomology) of X. Moreover, the Hodge filtration on algebraic de Rham cohomology coincides with Hodge’s filtration on smooth de Rham cohomology (defined using harmonic forms). Remark 2.1.4. Hodge actually defined a decomposition, not just a filtration, on smooth de Rham cohomology. However, only the filtration admits an algebraic description. 2.2. The Griffiths-Dwork construction. In general, computing the algebraic de Rham cohomology of a nonaffine variety can be awkward, due to the need to consider hypercohomology. In the case of a smooth hypersurface in projective space, one can get around this awkwardness by passing to a related affine variety. Definition 2.2.1. Again, let K be a field of characteristic 0. Let S be a homogeneous polynomial of degree d in K[u0,...,un] which is nonsingular (i.e., the ideal generated by S and its partial derivatives contains a power of (u0,...,un)).

138 KIRAN S. KEDLAYA

Pn Then S defines a smooth hypersurface X in the projective space K .PutU = Pn \ K X,sothatU is affine with coordinate ring equal to the degree 0 part of the −1 localization K[u0,...,un,S ]. Theorem 2.2.2. There is a canonical map Hn−1(X) → Hn(U);ifn is even, then this map is an isomorphism, otherwise it is surjective with one-dimensional kernel spanned by the Lefschetz class c(O(1))(n−1)/2,wherec denotes the first Chern class. (In other words, Hn(U) computes the primitive part of Hn−1(X).) Proof. This follows from the excision property for algebraic de Rham cohomology. Definition 2.2.3. Put n i % Ω= (−1) ui du0 ∧···∧dui ∧···∧dun, i=0 where the hat denotes omission. It is straightforward to check that Hn(U)maybe identified with the quotient of the group of n-forms AΩ/Si,wherei is an arbitrary positive integer and A ∈ K[u0,...,un] is homogeneous of degree id − n − 1, by the subgroup generated by (∂ A)Ω A(∂ S)Ω (2.2.3.1) j − i j Si Si+1 for all nonnegative integers i,allj ∈{0,...,n}, and all homogeneous polynomials ∂ A ∈ K[u0,...,un]ofdegreeid − n.(Here∂j is shorthand for .) ∂uj Besides giving an explicit description of the cohomology of X, this construction also makes the Hodge filtration readily apparent. − − Theorem 2.2.4 (Griffiths). Define Filn 1 i Hn(U) as the image in Hn(U) of · the set of forms AΩ/Si+1 with A homogeneous of degree id−n−1.ThenFil Hn(U) corresponds to the Hodge filtration on the primitive part of Hn−1(X). Remark 2.2.5. More generally, there is a similar recipe for computing the algebraic de Rham cohomology of a smooth complete intersection inside any toric variety. 2.3. Cyclic cubic threefolds. We now use the Griffiths-Dwork recipe to study the de Rham cohomology of a cyclic cubic threefold. Observation 2.3.1. Suppose that X is a cyclic cubic threefold as in Defini- tion 1.4.1. Using Griffiths’s theorem, we recover the Hodge numbers (2.3.1.1) h0,3 = h3,0 =0,h1,2 = h2,1 =5. 3 3 In particular, dimK H (X) = 10. We also see that the action of [ζ3] splits HdR(X) into two subspaces H1 ⊕ H2,whereH1 transforms like a and has dimK (H1 ∩ 2 3 2 ∩ 2 3 Fil HdR(X)) = 4, while H2 transforms like a and has dimK (H2 Fil HdR(X)) = 1. More explicitly, if b is a generator of the degree 4 subspace of the Jacobian ring

JX = K[w, x, y, z]/(Qw,Qx,Qy,Qz), then a basis for H1 is given by wΩ xΩ yΩ zΩ bΩ , , , , , S2 S2 S2 S2 S3

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 139

2 with the ﬁrst four basis elements spanning Fil H1. Similarly, if b1,b2,b3,b4 form a basis of the degree 3 subspace of JX , then a basis for H2 is given by aΩ ab Ω ab Ω ab Ω ab Ω , 1 , 2 , 3 , 4 , S2 S3 S3 S3 S3

2 with the ﬁrst basis element spanning Fil H2.

In this light, let us consider our special and generic examples.

Example 2.3.2. For the Fermat cubic, we may make particularly convenient choices of b, b1,b2,b3,b4 in Observation 2.3.1: we take

b = wxyz, b1 = xyz, b2 = wyz, b3 = wxz, b4 = wxy.

Using the chosen bases, H1 and H2 split into eigenspaces for the G-action with characters

H1 :(2, 1, 1, 1, 1), (1, 2, 1, 1, 1), (1, 1, 2, 1, 1), (1, 1, 1, 2, 1), (2, 2, 2, 2, 1)

H2 :(1, 1, 1, 1, 2), (1, 2, 2, 2, 2), (2, 1, 2, 2, 2), (2, 2, 1, 2, 2), (2, 2, 2, 1, 2), as predicted by Example 1.5.3.

Example 2.3.3. In Example 1.6.1, one checks (see worksheet) that b = wxyz and b = wxyz + w4 have nonzero images in the Jacobian ring, so give rise to good bases of H1. Similarly, one checks (see worksheet) that b1 = xyz, b2 = wyz, b3 = wxz, b4 = wxy are linearly independent in the Jacobian ring, so give rise to a good basis of H2.

2.4. Optional: Intermediate Jacobians. We recall a construction of Clemens and Griﬃths [15].

Definition 2.4.1. For X any smooth cubic threefold in P4 (not necessarily cyclic), there exists a canonical abelian variety A and a canonical isomorphism ∼ H3(X) = H1(A)(1) respecting all extra structures, e.g., the Hodge ﬁltration if K is of characteristic zero, or the action of Frobenius if K is a p-adic ﬁeld and X has good reduction (see next section). We call A the intermediate Jacobian of X.

Remark 2.4.2. We amplify Remark 1.1.2 slightly: our intended application of the calculation of p-adic cohomology of cyclic cubic threefolds is to compute the L-function of the intermediate Jacobian of a cyclic cubic threefold over Q.Note that the intermediate Jacobian inherits the action of ζ3 on X.

3. de Rham cohomology and p-adic cohomology We now give a brief description of one particular Weil cohomology theory, Berthelot’s theory of p-adic rigid cohomology, then explain how it can be computed in many cases using algebraic de Rham cohomology. This comparison leads to a relationship between the Hodge ﬁltration of a variety and its zeta function; we will use this to ﬁnish the computation of the zeta function of our generic example of a cyclic cubic threefold, as initiated in Example 1.6.1.

140 KIRAN S. KEDLAYA

3.1. Rigid cohomology.

Definition 3.1.1. For q apoweroftheprimep,wewriteQq for the unramified extension of the p-adic field Qp having residue field Fq.WewriteZq for the integral closure of Zp in Qq. Definition . F i 3.1.2 For X a variety over the finite field q,letHrig(X)denote the i-th rigid cohomology of X. This is a Weil cohomology which we will not construct explicitly in general; instead, we will describe some special cases in detail, and refer for the rest to the book of le Stum [66] (and to additional references discussed in the appendix). The construction of rigid cohomology is contravariantly functorial, so in particular the p-power Frobenius morphism Fp : X → X induces i an endomorphism of Hrig(X). This endomorphism on cohomology is σp-semilinear Q for σp the Witt vector Frobenius on q; raising to the power logp(q) gives a q-power Frobenius morphism Fq which on cohomology is Qq-linear. Remark 3.1.3. In these notes, we will mostly consider the case q = p in examples. However, in some applications (notably, in the use of hyperelliptic curves in cryptography) one wishes to take q to be a large power of p. In these cases, it is much more efficient to compute with the p-power Frobenius first, then extrapolate results for the q-power Frobenius, than to work with the q-power Frobenius directly. 3.2. Comparison theorems. In the computations described in these lectures, we access rigid cohomology via the following comparison theorem. Theorem 3.2.1 (Berthelot, Baldassarri-Chiarellotto). Let (X, Z) be a smooth proper pair over Zq (i.e., X is smooth proper over Zq and Z is a relative normal crossings divisor). Then there is a canonical isomorphism i \ ∼ i \ HdR(XQq ZQq ) = Hrig(XFq ZFq ). In order to control p-adic precision in computations, we need also an integral comparison theorem. Theorem 3.2.2 (Berthelot, Shiho). Let (X, Z) be a smooth proper pair over Zq. Then there is a canonical isomorphism i ∼ i HdR(X, Z) = Hcrys(XFq ,ZFq ), where the left side denotes the hypercohomology of the logarithmic de Rham complex, while the right side denotes logarithmic crystalline cohomology. Again, the right side in this isomorphism carries an action of Frobenius, so i → i \ the image of the map HdR(X, Z) HdR(XQq ZQq ) is a lattice stable under the Frobenius action.

3.3. p-adic divisibility and the Hodge ﬁltration. When computing zeta functions, it is often helpful to account for the following theorem of Mazur, which relates the Hodge ﬁltration to p-adic divisibility of the Frobenius matrix.

Theorem 3.3.1. Let X be a smooth proper scheme over Zq. Assume that p>i. j i Then for j =0,...,i, the image of Fil HdR(X) under the action of the p-power i j Frobenius on Hcrys(XFq ) is divisible by p .

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 141

Corollary 3.3.2. Let X be a smooth proper scheme over Zq. Assume that p>i.Letpe1 ≤ ··· ≤ ped denote the elementary divisors of the matrix of the i q-power Frobenius acting on some basis of Hcrys(XFq ). Then for j =1,...,d, ej is 0,i · at least the j-th partial sum of the sequence consisting of h copies of 0 logp(q), 1,i−1 · h copies of 1 logp(q), and so on. Moreover, equality holds for j = d.

Corollary 3.3.3. Let X be a smooth proper scheme over Zq. Assume that p>i. Then the Newton polygon of the characteristic polynomial of the q-power i Frobenius on Hrig(XFq ) lies on or above the Hodge polygon, with the same endpoints. j,i−j (The Hodge polygon is deﬁned to have slope j logp(q) with multiplicity h .) Remark 3.3.4. Beware that the analogue of Theorem 3.3.1 for the q-power Frobeniusisfalseforq = p. However, Corollary 3.3.2 is nonetheless correct as written: the relationship between the Hodge polygon and the elementary divisors of the q-power Frobenius matrix can be deduced from the p-power case, but this does not say anything about the action of Frobenius relative to the Hodge ﬁltration. We will later use the following lemma to take into account the Hodge divisibility in the Frobenius matrix.

Lemma 3.3.5. Let Φ be a d × d matrix over Zq whose reduction modulo q has m i rank e. Then for any matrix Δ ∈ q Zq, the coefficients of T in det(1 − T Φ) and det(1 − T (Φ + Δ)) differ by a multiple of qmax{m,m+i−e−1}. Proof. See [53, Theorem 4.4.2] or [1, Proposition 1.6.3]. 3.4. p-adic cohomology of cyclic cubic threefolds. We make the previous discussion explicit for cyclic cubic threefolds over finite fields, including our special and generic examples.

Observation 3.4.1. Suppose q ≡ 1(mod3)andthatFq has characteristic p ≥ 5. Let X be a cyclic cubic threefold over Fq defined by the polynomial Q.By Theorem 3.2.1, the rigid cohomology of X is isomorphic to the de Rham cohomology of the cyclic cubic threefold defined by any cubic polynomial Q˜ ∈ Zq[w, x, y, z] lifting Q. By Theorem 3.2.2, the matrix Φ of action of Fq on our chosen basis has entries in Zq. (This requires p ≥ 5 to ensure that the basis we wrote down is indeed a basis of the integral de Rham cohomology module.) Moreover, since h0,3 = h3,0 =0 (Observation 2.3.1), Φ is divisible by q. Since the cyclic automorphism lifts, we see that the spaces H1 and H2 of Ob- servation 2.3.1 are stable under Fq. We may thus use Theorem 3.3.1 to deduce divisibilities in det(1 − TFq,Hi)fori =1, 2, provided that we correctly match up the cube roots of unity in Fq and Qq. The correct matching is to match a cube root r of 1 in Fq with its Teichmüller liftr ˜ in Qq; this has the effect of distinguishing Z a one of the two prime ideals p in [ζ3]abovep.Puta =logp q and q = p . With this in mind, write −1 5 det(1 − q TFq,H1)=1+a1T + ···+ a5T −1 5 det(1 − q TFq,H2)=1+b1T + ···+ b5T , so that aj,bj ∈ Z[ζ3] are conjugates for j =1,...,5. Taking into account the 2 3 intersection of Fil Hrig(X) with H1 and H2,weseethataj is divisible by the ideal j−1 q for j =2, 3, 4, 5, while b5 is divisible by q (so a5 is divisible by q).

142 KIRAN S. KEDLAYA

Example 3.4.2. In the case of the Fermat cubic threefold over F7,wemay check the consistency of Theorem 3.3.1 with the computation of Example 1.5.3. In (1.5.3.1), the first four entries correspond to the eigenspaces in H1 belonging 2 3 to Fil Hrig(X); correspondingly, the first four eigenvalues in (1.5.3.3) are divisible by 7(ζ3 − 2) (see worksheet). Similarly, in (1.5.3.2), the fifth entry corresponds 2 3 tothesingleeigenspaceofH2 belonging to Fil Hrig(X); correspondingly, the fifth eigenvalue in (1.5.3.4) is divisible by 7(ζ3 − 2) (see worksheet). Example 3.4.3. In the case of our generic example (Example 1.6.1), we can use Observation 3.4.1 to completely determine the zeta function. What we know so far from the computation in Example 1.6.1 is that −1 2 3 4 5 det(1 − 7 TFq,H1)=1+(3ζ3 +2)T +(8ζ3 +5)T +(7ζ3 − 14)T + a4T + a5T − −1 2 2 2 2 − 3 4 5 det(1 7 TFq,H2)=1+(3ζ3 +2)T +(8ζ3 +5)T +(7ζ3 14)T + a4T + a5T for some a4,a5 ∈ Z[ζ3]. From Observation 3.4.1, we get the additional information 3 3 that a4 is divisible by (ζ3 − 2) while a5 is divisible by 7(ζ3 − 2) . Using the symmetry of the zeta function, we also have P (T/7) = 1+T +9T 2 +2T 3+?T 4+?T 5+?T 6 +98T 7 +3087T 8 +2401T 9 +16807T 10. Thisgivesustheequations

16807 = a5a5

2401 = a4a5 + a5a4 2 − − 3087 = a4a4 +(7ζ3 14)a5 +(7ζ3 14)a5. 3 5 Since 7(ζ3 − 2) already has norm 16807 = 7 , the ﬁrst equation only has the solutions k 3 a5 =(−ζ3) 7(ζ3 − 2) (k =0,...,5). The second and third equations can be viewed as computing the trace and norm of 4 a4a5/7 ∈ Z[ζ3]; namely, 4 Trace(a4a5/7 )=1 4 Norm(a4a5/7 )=−2, 11, 22, 20, 7, −4(k =0,...,5) (see worksheet for the second computation). We thus have # $ 1 1 a a /74 ∈ ± i x − : x = −2, 11, 22, 20, 7, −4 , 4 5 2 4 but only the value x = 7 leads to an element of Z[ζ3]. We thus must take k =4, yielding a5 = −133ζ3 − 126 and a4 ∈{16ζ3 − 39, −35ζ3 +21}. Only the ﬁrst choice is consistent with the equation − 2 − 2 98=(8ζ3 +5)a5 +(7ζ3 14)a4 +(7ζ3 14)a4 +(8ζ3 +5)a5 (see worksheet) so we compute −1 2 3 det(1 − 7 TFq,H1)=1+(3ζ3 +2)T +(8ζ3 +5)T +(7ζ3 − 14)T 4 5 +(16ζ3 − 39)T +(−133ζ3 − 126)T − −1 2 2 2 2 − 3 det(1 7 TFq,H2)=1+(3ζ3 +2)T +(8ζ3 +5)T +(7ζ3 14)T 2 − 4 − 2 − 5 +(16ζ3 39)T +( 133ζ3 126)T

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 143 and P (T/7) = 1 + T +9T 2 +2T 3 − 31T 4 − 45T 5 − 217T 6 +98T 7 + 3087T 8 + 2401T 9 + 16807T 10 (see worksheet). One checks that P (T/7) indeed has all complex roots of norm 7−1/2 (see worksheet).

4. The direct method for cyclic cubic threefolds In this lecture, we describe one application of the direct method for using p-adic cohomology to compute zeta functions, in the case of cyclic cubic threefolds. This will be only a theoretical discussion, however; we will see that the direct method is rather impractical for cyclic cubic threefolds, at least in the form given here. (Recent work of David Harvey suggests that the direct method may ultimately be practical in cases like this; see Remark 4.4.8 and the appendix for discussion.) 4.1. Frobenius actions on aﬃne varieties. The direct method is based on an explicit description of the Frobenius action on the rigid cohomology of an aﬃne variety, via the interpretation of rigid cohomology in terms of Monsky-Washnitzer cohomology.

Definition 4.1.1. Let (X, Z) be a smooth proper pair over Zq such that U = X \ Z is affine. Let A =Γ(U, OU ) be the coordinate ring of U.LetA be the p-adic completion of A.LetA† be the subring of A defined by the following condition: we have x ∈ A† if and only if there exists some a>0 such that for each positive integer n, the reduction of x modulo pn has poles of order at most an along each component of Z. (The ring A† is also known as the weak p-adic completion of A.) Theorem . i 4.1.2 (Berthelot) There is a canonical isomorphism between Hrig(UFq ) † ⊗ Q and the cohomology of the de Rham complex of A Zq q. Moreover, if (X ,Z ) is another smooth proper pair, and we define (A)† similarly, then any ring homomor- † † i i phism f : A → (A ) induces the functoriality morphism H (UF ) → H (U ) rig q rig Fq corresponding to the map X → XF given by reducing f mod p. (Note that the Fq q morphism A† → (A)† need not be induced by a map X → X; even if such a map exists, that map need not carry Z into Z.) 4.2. The direct method. We now describe how to execute the direct method for computing the zeta function of a cyclic cubic threefold. This is a summary of the approach described in more detail (and in more generality) in [1]. Procedure 4.2.1. Suppose q isapowerofaprimep ≥ 5. Let X be the cyclic cubic threefold over Fq associated to the nonsingular polynomial Q ∈ Fq[w, x, y, z]. Let o be the ring of integers in some number field, such that there exists an ideal p of o unramified above p with residue field Fq; we identify the p-adic completion of o with Zq. Choose a homogeneous cubic polynomial Q˜ ∈ o[w, x, y, z] lifting Q,and put S˜ = a3 − Q˜ ∈ o[w, x, y, z, a]. Let X˜ be the cyclic cubic threefold over the local ring op associated to Q˜. ∼ ˜ To compute the numerator P (T ) of the zeta function of X = XFq ,weusethe comparisons 3 ˜ ∼ 3 ˜ ∼ 4 ˜ ∼ 4 ˜ Hrig(XFq ) = HdR(XQq ) = HdR(UQq ) = Hrig(UFq )

144 KIRAN S. KEDLAYA for U˜ = P4 \ X˜. Note however that the Frobenius action on H4 (U) is not quite Zq rig 4 the same as the one on Hrig(XFq ); rather, it is twisted by an extra factor of q. −1 4 Consequently, we compute the action of q Fq on Hrig(U) rather than that of Fq. 3 ⊕ We split the integral de Rham cohomology HdR(X) as a direct sum H1 H2 of eigenspaces for the action of [ζ3]. We obtain integral bases of both H1 and H2 by applying the recipes from Observation 2.3.1 modulo p and lifting to elements of the same degree. (This succeeds in giving integral bases because p ≥ 5.) We apply Theorem 4.1.2 to the map induced by the algebraic map F on P4 q Zq acting on the variables by ∗ →∗q (∗ = w, x, y, z, a). The induced action on S˜−1 carries it to (4.2.1.1)! " −1 ∞ F (S˜) − S˜q −1 S˜−q 1+ q = (a3q −Q˜(wq,xq,yq,zq)−(a3−Q˜)q)iS˜−q(i+1). ˜q i S i=0 Note that this map does not carry X˜ into itself, but Theorem 4.1.2 requires no such hypothesis. All that matters is that the powers of p in the numerator of the summand accrue at a linear rate compared to the powers of S˜ in the denominator. To compute the Frobenius matrix, apply the map Fq formally to each basis vector, using the formula −1 3 q−1 q Fq(Ω) = q (wxyza) Ω. The result is an infinite series, so we cannot compute it exactly; we must neglect those terms divisible by a sufficiently large power of p. This has the effect of eliminating terms with sufficiently many factors of S˜ in the denominator, so we obtain an algebraic differential; we use the relations (2.2.3.1) to rewrite the resulting algebraic differential as an exact differential plus a Q-linear combination of basis vectors. For instance, this can be done by first eliminating the poles of highest order, then the next highest order, and so on. 3 The end result is a p-adic approximation of the matrix of Frobenius on Hrig(XFq ); we must make some side analysis to determine exactly how accurate this matrix is. This gives a p-adic approximation of the characteristic polynomial of this matrix, again with some known precision; if this precision is sufficient, there will be a unique monic polynomial with coefficients in Z and complex roots of absolute value q3/2 agreeing with this approximation. The reverse of this polynomial must then equal P (T ). This completes the description aside from the analysis of the initial and final precision needed for the computation. We address these issues later in this lecture. Remark 4.2.2. When q = p, one normally computes the p-power Frobenius first and then recovers the q-power Frobenius. The most important thing to remember is that the p-power Frobenius is not linear on scalars; it acts via the Witt vector Frobenius map. See [1] for more details. 4.3. Final precision. Of the two precision questions in Procedure 4.2.1, the easier one to answer is how much p-adic precision is needed in an approximation of the Frobenius matrix in order to uniquely determine its characteristic polynomial; we answer this using the Riemann hypothesis condition.

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 145

Observation 4.3.1. Retain notation as in Procedure 4.2.1. Let Φ denote the 3 matrix of action of Frobenius on an integral basis of Hrig(X). We have noted earlier (Observation 3.4.1) that Φ is divisible by q, so we work with q−1Φinstead. We are trying to determine the degree 10 polynomial P (T/q)=det(1−q−1T Φ). Thanks to the symmetry P (1/(q3T )) = q−15T −10P (T ), it is enough to determine the coefficients of T j in P (T/q)forj =1, 2, 3, 4, 5. Lemma 1.2.3 implies that once we determine the coefficients of T k for k qj/2 (j =1, 2, 3, 4, 5), j then we can uniquely reconstruct P (T/q). For q>16, this occurs as soon as m ≥ 4 (see worksheet); for q = 7, we instead must take m =5. Observation 4.3.2. In case q ≡ 1 (mod 3), we can do better by computing the matrix Φ1 via which Fq acts on the chosen basis of H1, as follows. Let q be the ideal defined in Observation 3.4.1; in particular, q has norm q,andζ3 reduces modulo q to the chosen cube root of 1 in Fq. m −1 m−1 Suppose we have computed Φ1 modulo q , or equivalently q Φ1 modulo q . j −1 By Lemma 3.3.5, the coefficient of T in det(1 − q T Φ1) is determined modulo qm−1, qm−1, qm, qm+1, qm+2 (j =1, 2, 3, 4, 5). −1 m−2 On the other hand, the entries of q Φ1 have relative precision at least q ;thatis, each is known to be a particular power of p timesaunitinZq which is known modulo m−2 −1 −1 2 −1 q . It follows that the same is true of the entries of q(q Φ1) = q Φ1 .Since m−2 this matrix has entries in Zq, it is known modulo q . Hence by Lemma 1.2.3, j − 2 −1 the coefficient of T in det(1 q T Φ1 ) is determined modulo qm−2, qm−2, qm−2, qm−2, qm−1 (j =1, 2, 3, 4, 5). However, these coefficients are the complex conjugates of the coefficients of det(1 − −1 q T Φ1). Hence the latter are determined modulo qm−2q,qm−2q,qm−2q2,qm−2q3,qm−1q3 (j =1, 2, 3, 4, 5). The minimum complex norm of a nonzero element of one of these ideals is the square root of the norm of the ideal. Hence if we have the five inequalities qm−3/2 > 10q1/2 qm−3/2 > 5q 10 qm−1 > q3/2 3 5 qm−1/2 > q2 2 qm+1/2 > 2q5/2 −1 then we can reconstruct det(1 − q T Φ1) and hence all of the zeta function. For q sufficiently large, these five inequalities hold for m =3;forq =7,they hold for m = 4 (see worksheet). These are each one less than the bounds obtained

146 KIRAN S. KEDLAYA in Observation 4.3.1; this will lead to significant runtime improvements in our calculations. Remark 4.3.3. As noted in Remark 1.2.4, one can sometimes compute zeta functions using less p-adic precision than one might initially predict, by accounting for the Riemann hypothesis condition. We can see this explicitly for the zeta function computed in Example 3.4.3, using the Sage package associated to the paper [50]. For example, we find that the polynomial P (T/7) is already determined uniquely when m = 4 (i.e, by its reduction modulo 73; see worksheet), whereas Observation 4.3.1 only predicts this for m = 5. For another example, if we take m =3,thenP (T/7) is determined within a list of 7 possibilities, but six of these have irreducible factors over Q(ζ3) of degree greater than 5 (see worksheet). So again P (T/7) is uniquely determined. 4.4. Initial precision. It remains to specify how much initial precision is needed in the calculation of the Frobenius action on forms in Procedure 4.2.1, in order to obtain a specific precision on the resulting Frobenius matrix. This analysis of precision loss is one of the trickiest aspects of the direct method. Remark 4.4.1. The analysis of precision loss serves two functions. On one hand, it is needed in order to make provably correct calculations. On the other hand, even if one is merely interested in experimental results which are probably correct, one would like to generate these efficiently; analysis of precision loss suggests how to balance speed against precision in order to avoid generating garbage data. Inthecaseofcycliccubicthreefolds,wefirstrecasttheprecisionlossproblem as follows. Problem 4.4.2. Given a form AΩ/S˜i for A a polynomial with coefficients in Zp, bound the denominators appearing when this form is written as an exact differential plus a Qq-linear combination of basis forms. Given a good enough solution of Problem 4.4.2, we can bound the precision of the error term created by omitting terms with S˜j for j ≥ i in the denominator. We can then determine where this truncation may be made to achieve the desired final precision. Example 4.4.3. In the case of cyclic cubic threefolds, simply counting divisions by p gives a bound on the denominator in Problem 4.4.2 which is linear in i.This is not good enough; by a somewhat complicated argument using an analysis of integral logarithmic de Rham cohomology [1, Proposition 3.4.6], one obtains the following bound which is logarithmic in i.

i Proposition 4.4.4. Any form AΩ/S˜ ,withA ∈ Zp[w, x, y, z, a], is cohomol- −c ogous to a linear combination of integral basis vectors with coeﬃcients in p Zp for 4 { − } c = logp max 1,i j . j=1 Example 4.4.5. Suppose that we wish to compute the matrix Φ modulo pm. We can write each basis diﬀerential as AΩ/S˜3 for some polynomial A; by (4.2.1.1),

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 147 its image under Frobenius is (4.4.5.1) ∞ −3 p3(wxyza)p−1F (A) (a3p − Q˜(wp,xp,yp,zp) − (a3 − Q˜)p)iS˜−p(i+3). p i i=0 We wish to compute a quantity N such that if if we consider the terms of (4.4.5.1) for which i ≥ N, then their reductions to the basis vectors have coefficients in m 3+i p Zp.Thei-th term in the sum is divisible by p , so it would suffice to have − ≥ − (4.4.5.2) 3 + i m 4 logp(p(i +3) 1) (i = N,N +1,...). For p = 7, we know by Observation 4.3.2 that it suffices to take m = 4 to recover the zeta function. In this case, (4.4.5.2) holds for N = 9 but not for any smaller value (see worksheet for a check up to i = 75). Remark 4.4.6. By further accounting for the Frobenius action [1, Proposi- tion 3.4.9], one gets a bound which is asymptotically 3 logp(i). While this is sus- pected to be asymptotically optimal, it seems to be suboptimal for small values. Improving the bound may lead to significant runtime improvements in practice, by reducing the degrees of the polynomial approximations needed in the truncations of Frobenius. In the particular case of Example 1.6.1, taking p =7andm =3(asinRe- mark 4.3.3), we may apply [1, Algorithm 3.4.10] (using the associated Magma code from [1]) to see that we can ignore all terms in the expansion divisible by p9.In our notation, this means we may take N =6. However, even this level of precision is difficult to achieve in practice; we must work with polynomials in five variables with coefficients in Z/7nZ for n at least 9, of total degree 3·p·(N −1) = 105. We will thus not carry out any demonstration of the direct method here. (See the associated Magma code of [1] for a demonstration for surfaces, where the situation is somewhat less dire. See also Remark 4.4.8 below.) Remark 4.4.7. The analogous analysis of precision loss in Kedlaya’s algorithm is [47, Lemmas 2 and 3]; however, note the erratum which corrects the latter. The erratum also points out that the analysis in [47], while not phrased in terms of integral de Rham cohomology, can indeed be interpreted this way. Remark 4.4.8. After the original version of these notes was prepared, David Harvey proposed an alternate reduction algorithm for de Rham cohomology in this setting, in which one structures the reduction in order to use only sparse polynomials. This may render the direct cohomological method much more practical than we had previously anticipated. See the appendix for further discussion.

5. Picard-Fuchs-Manin connections In this lecture, we discuss the relative version of algebraic de Rham cohomology. This gives rise to certain special diﬀerential systems classically called Picard-Fuchs systems, and often nowadays called Gauss-Manin connections. We will use these in the next lecture to execute the deformation method for computing zeta functions.

5.1. Connections on vector bundles. Before describing Picard-Fuchs-Manin connections, we recall the general notion of a connection on a vector bundle over a subset of P1.

148 KIRAN S. KEDLAYA

Definition 5.1.1. Let K be a field of characteristic zero. Let B be a nonempty P1 E E open subscheme of K .Let be a vector bundle over B.Aconnection on is a bundle map ∇ : E→E⊗ΩB/K which is additive and satisfies the Leibniz rule: for V ⊆ B open, s ∈ Γ(V,O), and v ∈ Γ(V,E), ∇(sv)=s∇(v)+v ⊗ ds. Observation 5.1.2. In order to compute with connections, we will describe them in terms of matrices as follows. Keep notation as in Definition 5.1.1, but assume now that ∞ ∈/ B and that v1,...,vn is a basis of sections of E. Define the n × n matrix N over Γ(B,O) by the equation n ∇(vj)= Nijvi ⊗ dt (j =1,...,n). i=1 By additivity and the Leibniz rule, we can recover ∇ from N. The simplest way to express that statement is to use the basis to identify sections of E with column vectors of functions; then ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ f1 d(f1) f1 dt ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∇ . . . ⎝ . ⎠ = ⎝ . ⎠ + N ⎝ . ⎠ . fn d(fn) fn dt In other words, ∇ = d + Ndt. Observation 5.1.3. The effect of changing basis in Observation 5.1.2 is as follows. Let w1,...,wn be a second basis of E. Define the change of basis matrix U from v1,...,vn to w1,...,wn to be the n × n matrix satisfying n (5.1.3.1) wj = Uij vi (j =1,...,n). i=1

Then the matrix representing the connection in terms of w1,...,wn is d U −1NU + U −1 (U). dt We will be interested in a special class of connections. Definition 5.1.4. With notation as in Observation 5.1.2, and z ∈ Kalg,we say the basis v1,...,vn is regular at z (or Fuchsian at z) if the matrix (t − z)N is holomorphic in a neighborhood of z.WesaythatE is regular at z if it admits a regular basis on some neighborhood of z. Another way to say this is that E can be extended across z so that the connection has only logarithmic singularities at z. This definition is invariant under automorphisms of P1.Itthusmakessenseto extend it to z = ∞ by using any coordinate change moving ∞ to a finite point (since the resulting definition will not depend on the choice of the coordinate change). For instance, we may use the substitution t → t−1; it then follows that a basis is regular at ∞ if and only if each entry of the matrix N has a zero at t = ∞. In concrete terms, for each entry of N, the degree of the numerator must be strictly less than the degree of the denominator. Definition 5.1.5. With notation as in Observation 5.1.2, and z ∈ Kalg, suppose that the basis v1,...,vn is regular at z.Theresidue matrix at z of this basis is the matrix obtained from (t−z)N by reducing modulo t−z;ifz ∈ B,thismatrix is zero. The exponents at z of the basis are the eigenvalues of the residue matrix.

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 149

Note that one can change from one regular basis to another without preserving the exponents; for instance, changing basis to (t−z)v1,...,(t−z)vn replaces N by −1 N +(t−z) In, which increases each exponent by 1. However, it can be shown that as a multisubset of the quotient group Kalg/Z, the set of exponents of a regular basis is independent of the choice of the regular basis; we call this the set of exponents of E at z. If z = ∞, we may define the residue matrix to be the matrix obtained from −tN by reducing modulo t−1, and proceeding similarly. The following lemma demonstrates the use of shearing transformations. Lemma 5.1.6. With notation as in Definition 5.1.5, suppose that the exponents alg of the basis v1,...,vn at some z ∈ K are integers in the range {−a,...,b}. Then there exists an invertible n × n matrix U over K(t) such that (t − z)bU and a −1 (t−z) U are regular at z,andthebasisw1,...,wn of E (over some neighborhood of z) defined by (5.1.3.1) is regular at z with all exponents equal to 0. Proof. This reduces to the fact that one can shift the largest exponent down by 1 using a change of basis matrix U such that (t − z)U and U −1 are regular at z. We may first use a change of basis defined over K (which acts on N by simple conjugation, since its derivative vanishes) to ensure that the reduction of (t − z)N modulo (t − z) is a block matrix with each block corresponding to the generalized eigenspace of a different eigenvalue. We then change basis by the block diagonal matrix U which is (t−z)−1 times the identity on the block with the largest exponent, and the identity on the other blocks. Remark 5.1.7. Associated to a connection is a representation of the topological fundamental group π1(B,x) called the monodromy representation. It is defined as follows. Construct a basis of local horizontal sections at the base point x of the fundamental group. For any loop in B, analytically continue these horizontal sections along the loop. The image of the monodromy representation on this loop is the linear transformation on the fibre Ex taking the restriction to x of each basis section to the restriction of x of its analytic continuation. In general, it is somewhat hard to identify the eigenvalues of a monodromy transformation. However, if E is regular at z with exponents λ1,...,λn,then the eigenvalues of the monodromy transformation corresponding to a loop going counterclockwise once around z (and enclosing no other points of P1 \ B)are − − e 2πiλ1 ,...,e 2πiλn . 5.2. Relative de Rham cohomology. Definition 5.2.1. Let f : X → B be a smooth morphism over a field K of P1 characteristic zero, for B a nonempty open subscheme of K .Therelative de Rham q cohomology of X/B is the collection of sheaves HdR(X/B) whose sections over an open affine V ⊂ B are the hypercohomology of the relative de Rham complex · ΩX/V . The fact that this gives a sheaf follows from the preservation of coherent cohomology under flat base change. For f proper, this construction also commutes with arbitrary base change; this follows from Grothendieck’s comparison theorem (Theorem 2.1.3). This fails if f is not proper; consider Spec K[x, y, z]/((x + y)(x − y)z − 1) → Spec K[x], in which the Betti numbers of the fibre x = 0 differ from the generic values.

150 KIRAN S. KEDLAYA

Since mixed partial derivatives commute and the computation of relative de Rham cohomology only involves “vertical” differentiation (along fibres), the result should carry an action of “horizontal” differentiation (along the base). This is in fact the case; this is captured by a construction of Katz and Oda. Definition . · 5.2.2 Equip the de Rham complex ΩX/K with the decreasing filtration i ·−i ∗ i . ⊗O → F = image[ΩX/K X π (ΩB/K) ΩX/K ], then form the corresponding spectral sequence. The E1 term of the result has p,q p q ⊗O E1 =ΩB/K B HdR(X/B); the algebraic Picard-Fuchs-Manin (Gauss-Manin) connection is the differential d1 : 0,q → 1,q E1 E1 . Remark 5.2.3. In practice, we will compute only in the case where X is affine. In this case, the definition of d1 amounts to the following: lift a relative cohomology class to an absolute differential form (no longer a cocycle), differentiate, and project the result back into relative cohomology. Definition 5.2.4. Suppose that K is a subfield of C. Then the fibration f : X → B is locally trivial in the category of real differentiable manifolds. On a contractible open subset of B, we may canonically identify the complex homology q classes of the fibres; this gives a real differentiable connection on HdR(X/B), called the topological Picard-Fuchs-Manin connection. It turns out that this is holomorphic(see[31]), and that it agrees with the algebraic Picard-Fuchs-Manin connection (see [46]). Theorem 5.2.5. With notation as in Definition 5.2.2, the algebraic Picard- P1 Fuchs-Manin connection is regular at every geometric point of K ,withallexpo- nents in Q/Z. 5.3. Pencils of cyclic cubic threefolds. We now explain how to compute the Picard-Fuchs-Manin connection for certain families of cyclic cubic threefolds.

3 3 Procedure 5.3.1. Let K be a ﬁeld of characteristic zero. Take Q0 = w +x + y3 + z3, and let Q ∈ K[w, x, y, z] be a second homogeneous polynomial of degree 3 alg such that Q1 = Q0 + Q is nonsingular. Put Qt = Q0 + tQ.Fort ∈ K ,let

Jt = K[w, x, y, z]/(Qt,w,Qt,x,Qt,y,Qt,z) be the Jacobian ring of Qt.(HereQt,w denotes the partial derivative with respect to w of the polynomial Qt, and similarly.) → P1 We wish to consider the pencil π : X K of cyclic cubic threefolds defined 3 − → P1 P3 \ by S = a Qt, as well as the complementary family τ : U K for U = P1 X. K By Lemma 1.4.2, a fibre Xt is smooth if and only if the cubic surface defined by Qt is smooth; in particular, the fibres X0,X1 are smooth. ⊂ P1 ∈ Let B K be the open subscheme over which π is smooth; since 0, 1 B, B E 3 4 is nonempty. Put = HdR(XB/B), which we will also interpret as HdR(UB/B). Using the order 3 automorphism a → ζ3a, we split E = E1 ⊕E2 with Ei transforming like ai. The Picard-Fuchs-Manin connection on E splits into separate connections for E1 and E2, which we now describe individually.

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 151

Let us start with E1.Chooseb ∈ K[w, x, y, z] such that b generates the degree 4 subspace of J0,J1.LetB1 be the open subscheme of B consisting of those t for which b spans the degree 4 subspace of Jt; by construction, 0, 1 ∈ B1. Now differentiate each of the basis elements wΩ xΩ yΩ zΩ bΩ , , , , S2 S2 S2 S2 S3 with respect to t, obtaining 2wQΩ 2xQΩ 2yQΩ 2zQΩ 3bQΩ , , , , , S3 S3 S3 S3 S4 then reduce each of these back into the desired form using the relations A Ω (3i2 + tQ )AΩ (5.3.1.1) i ≡−j i (i ∈{w, x, y, z}). Sj Sj+1 This amounts to a large linear algebra calculation over K(t), and for best results it may be preferable to implement it that way. However, we found it easiest to implement this using Gröbner basis methods to express a form as a linear combination of terms amenable to (5.3.1.1). In any case, the entries of the resulting matrix N1 will belong to the coordinate ring of B1. LetusnowconsiderE2.Chooseb1,b2,b3,b4 ∈ K[w, x, y, z]whichspanthe degree 3 subspace of J0,J1.LetB2 be the open subscheme of B of those t for which b1,b2,b3,b4 span the degree 3 subspace of Jt; again by construction, 0, 1 ∈ B2. Again, differentiate each of the basis elements aΩ ab Ω ab Ω ab Ω ab Ω , 1 , 2 , 3 , 4 S2 S3 S3 S3 S3 with respect to t, obtaining 2aQΩ 3ab QΩ 3ab QΩ 3ab QΩ 3ab QΩ , 1 , 2 , 3 , 4 , S3 S4 S4 S4 S4 then reduce each of these back into the desired form using the relations aA Ω a(3i2 + tQ )AΩ (5.3.1.2) i ≡−j i (i ∈{w, x, y, z}). Sj Sj+1

This time, the entries of the resulting matrix N2 will belong to the coordinate ring of B2.

Example 5.3.2. We calculate the matrix N1 for K = Q, 3 3 3 3 Q1 = w + x + y + z +(w + x)(w +2y)(w +3z)+3xy(w + x + z) (as in Example 1.6.1), and b = wxyz (see worksheet). This computation was carried out using Gröbner basis methods over the coefficient field Q(t), as implemented in Magma (see Remark 5.3.4 for the reason why); it required about twenty seconds to complete. We then analyze the singular points of the connection as follows. The matrix N1 has entries in Q(t), and the least common denominator Δ ∈ Z[t] of the entries factors as Δ = Δ1Δ2Δ3 where Δ1 = t +3,Δ2 is a polynomial of degree 23, and Δ3 is a polynomial of degree 26 (see worksheet). In particular, Δ is squarefree, so our chosen basis is regular at all finite points.

152 KIRAN S. KEDLAYA

We next compute the exponents at each of these singular points. For i =1, 2, 3, we compute the characteristic polynomial of N Δ/Δ(t)inQ[t]/(Δ ); we get 1 i 1 x3 (x +1) x + (i =1) 2 x4 (x − 1) (i =2) 7 x4 x + (i =3) 6

(see worksheet). In particular, the points of Δ1, Δ3 have a nonintegral exponent and so must be true singularities of the connection, whereas we cannot tell about Δ2. We will see below (Example 5.3.3) that in fact the singularities at Δ2 can be eliminated by a change of basis. Finally, we analyze the situation at infinity. The given basis is not regular here, because the last row contains entries which are regular but nonvanishing at t = ∞ (see worksheet). However, if we change basis using the matrix ⎛ ⎞ 10000 ⎜ ⎟ ⎜01000⎟ ⎜ ⎟ U = ⎜00100⎟ , ⎝00010⎠ 0000t then we get a regular basis (see worksheet). Computing the characteristic polynomial of the residue matrix yields 3 4 5 3 x − x − x − 2 3 3 (see worksheet). Example 5.3.3. We calculate the connection matrix again as in Example 5.3.2, 4 but this time with b = wxyz + w (see worksheet). Let N˜1 be the new connection matrix, and let Δ˜ be the least common denominator of the entries of N˜1.Thenwe compute that gcd(Δ, Δ)˜ = Δ1Δ3 (see worksheet); we deduce that the singularities of N1 at Δ2 can be removed by changing basis (i.e., they are so-called apparent singularities). Remark 5.3.4. ThereasonthatweusedMagma instead of Sage for this calculation is that we use Gröbner bases for polynomials over the field Q(t), which are well supported in Magma. By contrast, Sage does not support such polynomials directly; one can directly call Singular to work with such polynomials, but this does not work well in the version of Sage that we tried, as even basic operations take an unacceptably long time to complete. (By contrast, working over Fq(t)causesno such problems.) 5.4. Optional: Exponents in a pencil of cyclic cubic threefolds. We include some discussion of the possible exponents of a Picard-Fuchs-Manin connection associated to a pencil of cyclic cubic threefolds. Observation 5.4.1. By Theorem 5.2.5, the exponents of the Picard-Fuchs- Manin connection associated to a family of cyclic cubic threefolds at any (necessarily regular) singular point are rational numbers. We may bound the lowest common denominator of these numbers as follows. The exponents of the full connection are

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 153 the union of the exponents of E1 and E2. The corresponding sets of local monodromy eigenvalues are interchanged by any automorphism in Gal(Q/Q) which does not ﬁx ζ3. Hence, any ζn with n not divisible by 3, if it occurs at all, occurs together with all of its conjugates in each of E1 and E2. This can only happen if φ(n) ≤ 5, i.e., if n ∈{1, 2, 4, 5, 8, 10}.Ifn is divisible by 3, then ζn and its conjugates split between E1 and E2; we must still have φ(n) ≤ 10, so n ∈{3, 6, 9, 12, 15, 18, 24, 30}. Definition 5.4.2. A Lefschetz pencil is a pencil of hypersurfaces in which each singular ﬁbre contains a single rational double point and no other singularities. Unfortunately, a pencil of cyclic cubic threefolds can never be a Lefschetz pencil, because the generic degeneration is a rational triple point. However, we can ask for the underlying pencil of cubic surfaces to be a Lefschetz pencil; in that case, a Hodge-theoretic argument shows that the denominators of the exponents (in the Picard-Fuchs-Manin connection for the family of cyclic cubic threefolds) in fact always divide 6. (Thanks to Jim Carlson for pointing this out.)

6. The deformation method for cyclic cubic threefolds In this lecture, we describe the Frobenius actions on Picard-Fuchs-Manin connections obtained by relating relative de Rham cohomology to relative rigid cohomology. We then execute the deformation method for computing the zeta function for our generic example of a cyclic cubic threefold. This amounts to solving the differential equation imposed on the Frobenius structure by its compatibility with the connection, using the Frobenius matrix of the Fermat cubic as an initial condition.

6.1. Frobenius structures. Definition 6.1.1. Let q be a prime power. Let σ : P1 → P1 denote the Qq Qq q map induced by the σq-semilinear map carrying t to t ,forσq the Witt vector i ∈ Q ∗ q-Frobenius. That is, if x = i cit with ci q, then the pullback σ (x)equals qi i σq(ci)t . We will normally use the case q = p,inwhichcaseσq is the identity map and σ is just the substitution t → tp. Definition 6.1.2. Let V be a rigid (or Berkovich) analytic subspace of P1 Qq such that σ−1(V ) ⊆ V .LetE be a vector bundle with connection on V .AFrobenius ∗ ∼ structure on E is an isomorphism F : σ E = E of vector bundles with connection on σ−1(V ). We typically view F as a σ-semilinear map on E;thatis,forf a section of O and s a section of E, F (fs)=σ(f)F (s). Most Frobenius structures arise from the following construction. Theorem 6.1.3 (Berthelot). Let B be an open formal subscheme of the completion of P1 along its special fibre. Let Π:X → B be a smooth proper morphism Zq of formal schemes over Spf Zq.LetX, B be the Raynaud generic fibres of X, B,in the category of rigid analytic spaces. Let V be the subspace of B consisting of points i with reduction in BFq . Then the restriction of HdR(X/B) to V admits a Frobenius ∈ ∈ an structure with the property that for any t BFq ,for[t] B the Teichmüller lift i of t,therestrictionofF to Π[t] gives the Frobenius action on Hrig(Xt). Observation 6.1.4. In order to compute with a Frobenius structure, we need to make explicit how it acts in terms of differential systems. Let us do this now.

154 KIRAN S. KEDLAYA

E d Suppose v1,...,vn is a basis of ,andthatN is the matrix of action of dt as in Observation 5.1.2. Deﬁne the n × n matrix Φ by setting 5 F (vj)= Φijvi (j =1,...,5). i=1

The matrix Φ will have entries in the p-adic completion of Qq(t) for the Gauss norm (that is, the norm of a polynomial is the maximum norm of any of its coefficients). More precisely, modulo any power of p, the entries of Φ will be congruent to rational functions with no poles in V . The action of Φ on column vectors is given by ⎛ ⎞ ⎛ ⎞ f1 σ(f1) ⎜ . ⎟ ⎜ . ⎟ F ⎝ . ⎠ =Φ⎝ . ⎠ . fn σ(fn) Hence the effect of changing basis by a matrix U is to replace Φ by U −1Φσ(U). The fact that Φ is an isomorphism of vector bundles with connection, not just an isomorphism of vector bundles, is expressed by the compatibility equation d (6.1.4.1) NΦ+ (Φ) = qtq−1Φσ(N). dt Given N, this expresses Φ as the solution of a differential system; that observation is the basis of the deformation method. Example 6.1.5. In the case of cyclic cubic threefolds, the Frobenius structure and the cyclic automorphisms interact via the commutation relation

q F ◦ [ζ3]=[ζ3] ◦ F. (compare Definition 1.4.4). If q ≡ 1 (mod 3), this means that the Frobenius structure acts separately on E1 and E2. That is, when written in terms of a basis as in Procedure 5.3.1, the matrix Φ splits as a block diagonal matrix in which the diagonal blocks Φ1, Φ2 describe the Frobenius structures on the chosen bases of E1, E2. If q ≡ 2 (mod 3), then E1 and E2 are interchanged rather than preserved by the Frobenius structure. Thus Φ is again a block matrix, but now it is the off-diagonal blocks which are nonzero. 6.2. Solving for the Frobenius structure. As noted above, the compatibility equation (6.1.4.1) imposes a differential equation on the entries of the matrix describing a Frobenius structure on a connection. To solve this equation, it is convenient to first solve the connection itself; we may do this using power series expansions around a point. Lemma . ∞ i × Q 6.2.1 Let N = i=0 Nit be an n n matrix over q t . Then there × ∞ i Q is a unique n n matrix U = i=0 Uit over q t with U0 = In satisfying d (6.2.1.1) NU + (U)=0. dt

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 155

Proof. (Compare [53, Proposition 7.3.6].) Extracting the coeﬃcient of ti−1 on the left side of (6.2.1.1) gives the equation i−1 iUi = − Ni−j Uj , j=0 which determines Ui in terms of U0,...,Ui−1. Definition 6.2.2. With notation as in Lemma 6.2.1, we call U the fundamental solution matrix of N. Remark 6.2.3. One can give a quadratically convergent algorithm to compute U, in the manner of the Newton-Raphson method of approximating roots of polynomials. Start with U = I , then repeatedly replace U with &n d U I − (U −1NU + U −1 (U)) dt . n dt

i After i iterations, U will agree with the fundamental solution matrix modulo t2 . However, if one is working with p-adic approximate numbers rather than exact rationals, one must be careful about p-adic numerical precision; see Remark 6.4.6 below. We can now compute the Frobenius matrix given the initial condition of its value at t = 0, as follows. Lemma . ∞ i ∞ i × 6.2.4 Let N = i=0 Nit and Φ= i=0 Φit be n n matrices over Qqt satisfying (6.1.4.1).LetU be the fundamental solution matrix for N.Then −1 (6.2.4.1) Φ = UΦ0σ(U) . Proof. The compatibility equation (6.1.4.1) is preserved by the change of basis → −1 −1 d → −1 d −1 N U NU+U dt (U), Φ U Φσ(U). This implies dt (U Φσ(U)) = 0; since −1 U ≡ In (mod t), we must have U Φσ(U)=Φ0. This proves the claim. 6.3. The deformation method. We can now describe the deformation method in the case of cyclic cubic threefolds. Procedure 6.3.1. Retain notation as in Procedure 4.2.1, but assume that q ≡ 1 (mod 3). Use Procedure 5.3.1 to compute the Picard-Fuchs-Manin connection associated to the pencil of cyclic cubic threefolds in which the fibre at t = 0 is the Fermat cubic, while the fibre at t = 1 is the cyclic cubic threefold associated to the ˜ d E polynomial Q.LetN1 denote the matrix of action of dt on the chosen basis of 1. Let Φ1 denote the matrix of action of the Frobenius structure constructed using Theorem 6.1.3 on the chosen basis of E1. At t = 0, each basis vector is an eigenvector for the group action on the Fermat cubic given in Procedure 1.5.2. Hence we may read off the matrix Φ1(0)asthedi- agonal matrix with eigenvalues computed as in Procedure 1.5.2, once we remember our choice of the identification of Z[ζ3] with a subring of Zq (see Observation 3.4.1). We now compute a t-adic approximation to the fundamental solution matrix U of N1, to a precision to be specified later (Subsection 6.5). In Qqt,wemaythus −1 write Φ1 = UΦ1(0)σ(U) by Lemma 6.2.4. By Theorem 3.3.1 applied at the generic point, Φ1 has entries in Zqt. Modulo m q , we may identify the reduction of Φ1 as the series expansion of a rational function

156 KIRAN S. KEDLAYA with all poles congruent modulo p to poles of N1. (This requires having a bound on the number of these poles, so we can be sure to have carried enough t-adic precision. See Subsection 6.5.) With this done, we can evaluate this rational function at t = 1 to obtain the 3 m Frobenius matrix on Hrig(X) modulo q .Form as in Observation 4.3.2, this suffices to determine the zeta function of X. Let us now carry out this computation for our chosen example of a cyclic cubic threefold over F7 (Example 1.6.1). For this computation, we take m = 3; while this is not quite enough to be sure aprioriof uniquely determining the zeta function (Observation 4.3.2 only guarantees this for m = 4), we know from our previous computation that it suffices in this case (Remark 4.3.3). (Note that we also need to know that we have an integral basis of each of E1, E2; this follows from the fact that the basis conditions in Observation 2.3.1 are satisfied over F7.) Example 6.3.2. We first compute an approximation modulo t500 to the fundamental solution matrix U for N1 (see worksheet), using a quadratically convergent algorithm (Remark 6.2.3). It is somewhat time-consuming to compute this series with exact rational coefficients; since we will end up reducing modulo a small power of 7 later, we work with 7-adic coefficients with maximum relative precision 150. Even so, this requires about 15 minutes; however, it should be possible to speed this up substantially. See Remark 6.4.6. Note that the minimum 7-adic valuation of any coefficient appearing in our approximation of U is only −3 (see worksheet). By contrast, the proof of Lemma 6.2.1 only guarantees that the entries of U modulo t500 have coefficients with 7-adic valuation at least ' ( ' ( ' ( 500 500 500 − − − = −82. 7 72 73 This discrepancy is explained qualitatively by the fact that the existence of a Frobe- nius structure forces the entries of U to converge for t in the whole open unit disc. It is explained more quantitatively by certain explicit convergence bounds for p-adic differential equations; see Theorem 6.4.3.

Example 6.3.3. We next compute the matrix Φ1 of action of the Frobenius structure on the chosen basis of E1, using the formula (6.2.4.1). In this equation, U is as computed in the previous example, while Φ1(0) is the Frobenius matrix for the Fermat cubic threefold. By Proposition 1.5.2, the latter matrix is diagonal with diagonal entries

21ζ3 +7, 21ζ3 +7, 21ζ3 +7, 21ζ3 +7, −21ζ3 − 14 as computed in Example 1.5.3. Here we identify ζ3 with the Teichm¨uller lift of 2 in Q7. −1 After computing Φ1, we check (see worksheet) that 7 Φ1 has entries in Z7t, and all of the columns except the rightmost one have entries in 7Z7t. −1 2 Example 6.3.4. We next reduce 7 Φ1 modulo 7 and multiply by the degree 13 7 218 polynomial Δ1 Δ2Δ3. In the resulting matrix, each entry is congruent modulo t500 to a polynomial of degree at most 211 (see worksheet). This suggests that we have carried enough t-adic precision to identify these series as rational functions with divisors no less than

−13(Δ1) − (Δ2) − 7(Δ3)+7(∞).

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 157

We will prove that this is the case in Subsection 6.5. Example 6.3.5. Finally, we evaluate these polynomials at t = 1, then divide 13 7 −1 2 by (Δ1 Δ2Δ3)(1) to get a 7-adic matrix which is congruent to 7 Φ1 modulo 7 . Let A be a lift of this matrix to Z. Then the coeﬃcient of T j in det(1 − TA) agrees with the expected answer

2 3 4 5 1+(3ζ3 +2)T +(8ζ3 +5)T +(7ζ3 − 14)T +(16ζ3 − 39)T +(−133ζ3 − 126)T from Example 3.4.3 modulo p3, p2, p3, p4, p5 (j =1, 2, 3, 4, 5)

(see worksheet). Here as before, p is the prime ideal (ζ3 − 2) of Z[ζ3], which has norm 7. This is consistent with Observation 4.3.2, which predicts this agreement modulo p2, p2, p3, p4, p5 (j =1, 2, 3, 4, 5) Meanwhile, the coefficient of T j in det(1−7TA−1) agrees with the expected answer 2 2 2 2 − 3 2 − 4 − 2 − 5 1+(3ζ3 +2)T +(8ζ3 +5)T +(7ζ3 14)T +(16ζ3 39)T +( 133ζ3 126)T from Example 3.4.3 modulo p, p, p, p, p2 (j =1, 2, 3, 4, 5) (see worksheet). This is also consistent with Observation 4.3.2. 6.4. p-adic precision. One can significantly reduce the p-adic precision required for computing Frobenius structures by using effective bounds for convergence of solutions of fundamental solution matrices.

Notation 6.4.1. Let Qqt0 be the subring of Qqt consisting of series with bounded coefficient; that is, Q Z ⊗ Q q t 0 = q t Zq q Let |·|denote the supremum norm. Even without accounting for Frobenius structures, one obtains an extremely strong effective convergence bound for convergent solutions of bounded nonsingular differential equations on the unit disc. Theorem 6.4.2 (Dwork-Robba). For i ≥ 0, define i { } f(i)= logp max 1,j . j=i−n+2 ∞ i × Q ∞ i Let N = i=0 Nit be an n n matrix over q t 0.LetU = i=0 Uit be the fundamental solution matrix of N. Suppose that the entries of U and U −1 are convergent on the open unit disc (but not necessarily bounded). Then

f(i) n−1 |Ui|≤p max{1, |N| } (i ≥ 0). However, it is better in general to account for Frobenius structures when possible, as follows.

158 KIRAN S. KEDLAYA

Theorem 6.4.3. For i ≥ 0, deﬁne − g(i)=f(iq logq i ), → q for f(i) as in Theorem 6.4.3.Letσ denote the σq-semilinear substitution t t Q ∞ i × Q ∞ i on q t .LetN = i=0 Nit be an n n matrix over q t 0.LetA = i=0 Ait be a matrix over Qqt0 with A0 invertible, and suppose that d NA+ (A)=qtq−1Aσ(N). dt Then: (a) the fundamental solution matrix U of N satisﬁes

−1 U Aσ(U)=A0; (b) we have − − | |≤| |n 1 g(i) | 1|| | logq i Ui N p ( A0 A ) ; in particular, U converges on the open unit disc.

Example 6.4.4. In Example 6.3.2, the matrix N1 has supremum norm bounded by 1. Theorem 6.4.2 predicts that the fundamental solution matrix modulo t500 − − has coefficients of valuation at least 4 log7(500) = 12. On the other hand, we have a Frobenius structure given by a matrix A with |A| =1and|A−1| =7, so Theorem 6.4.3 implies that the fundamental solution matrix modulo t500 has coefficients of valuation at least −3. The latter agrees with the computed value from Example 6.3.2. Remark . | −1|| | 6.4.5 The quantity A0 A in Theorem 6.4.3 is determined by the difference between the greatest and least Hodge slopes of A. In case the Newton polygon of A lies strictly above the Hodge polygon, one can refine the bounds by replacing the Frobenius structure by a power of itself, whose Hodge slopes are closer to the Newton slopes (as in [44, Corollary 1.4.4]). We will not take advantage of this refinement here. Remark 6.4.6. In the situation of Theorem 6.4.2, one would like to able to compute U in a manner which is p-adically numerically stable, i.e., which does not require as much intermediate p-adic precision as is needed in the case when the entries of U really do have fast-growing denominators. The best one can hope for, in case |N| = 1, is to compute U modulo (pm,tN )givenN modulo (ph+m,tN ), where h is the number of factors of p appearing in the denominators of U modulo tN . The algorithm of Remark 6.2.3 is quite far from this; one can do slightly better by taking p into account in a limited fashion. For instance, one can proceed as in Remark 6.2.3 but first eliminating only terms ti with i not divisible by p, then with i not divisible by p2, and so on. A much better algorithm would be to directly imitate a proof of Theorem 6.4.3 (see references in the appendix), but this is somewhat more complicated to implement. In the context of Picard-Fuchs-Manin connections, one can usually maintain p-adic numerical stability by writing the differential equation as a linear recursion of finite length (with matrix coefficients). That way, one can control the p-adic precision loss rather directly; for instance, see [61, Theorem 5.1] or [40, Theorem 2].

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 159

6.5. t-adic precision. To complete the description of the deformation method, we must explain how to bound the degree of a rational function given by reducing a Frobenius matrix modulo a power of p, so that we can provably recover this rational function by computing some specific number of coefficients of its Taylor expansion around t = 0. We start with a qualitative result. Definition 6.5.1. Set notation as in Theorem 6.1.3. A strict neighborhood of V is a rigid analytic subspace W of P1 containing V , consisting of a closed disc Qq of radius strictly greater than 1 around the origin, minus finitely many open discs of discs of radius strictly less than 1 centered at points in the closed unit disc. In more geometric terms, W is a neighborhood of V within P1 containing V in its Qq relative interior. Theorem 6.5.2 (Berthelot). Set notation as in Theorem 6.1.3. Then the Frobe- nius structure F extends over some strict neighborhood of V . Remark 6.5.3. This implies that the reduction of the matrix Φ modulo pn is a rational function of total degree bounded by some linear function of n. However, we do not obtain an effective bound either on the slope or the constant term of this linear function. For this, we need the quantitative Theorem 6.5.10 below. Remark 6.5.4. In the language of p-adic cohomology, Theorem 6.5.2 asserts that the relative rigid cohomology in this setting forms an overconvergent F -isocrystal on the smooth locus of the family. For more discussion of this concept, see the appendix. To obtain a quantitative refinement of Theorem 6.5.10, one could apply known precision bounds for the direct method (Subsection 4.4) to the generic fibre. How- ever, since these bounds are known experimentally to be suboptimal, this will result is a suboptimal refinement. One can do much better by making a careful analysis of Frobenius structures on connections over a p-adic disc, as follows. We first observe that we can convert the Frobenius structure from one Frobenius lift to another, using Taylor series. Theorem 6.5.5. With notation as in Theorems 6.1.3 and 6.5.2,letσ : P1 → Qq P1 be any σ -semilinear map carrying t to something congruent to tq mod p.Then Qq q i HdR(X/B) also admits a Frobenius structure F on a strict neighborhood of V with respect to σ, defined by ∞ 1 di (6.5.5.1) F (v)= (σ(t) − σ(t))iF (v) . i! dti i=0

This computes the Frobenius matrix on a ﬁbre Xt by specialization to the unique lift of t carried to its σq-image by σ . Proof. The series converges on a strict neighborhood because the presence of the Frobenius structure forces the generic radius of convergence of the connection to equal 1 [53, Proposition 17.2.3]. Given that, the Leibniz rule implies ﬁrst that on the trivial connection module (i.e., functions on P1 ), Qq F (t)=t +(σ(t) − σ(t))F (t)=σ(t). (This observation is a good way to remember the formula (6.5.5.1).) The Leibniz rule then implies that on any connection module, F is semilinear for σ.Formore details, see references in the appendix.

160 KIRAN S. KEDLAYA

Example 6.5.6. In the situation of Example 6.3.2, we compute the Frobenius structure with respect to the map σ given by σ(t)=(t +3)7 − 3, 3 modulo 7 .LetΦ1 be the matrix of action on our chosen basis. Given the series −1 3 representation of the matrix 7 Φ1 modulo 7 , we compute by (6.5.5.1) 7−1Φ ≡ 7−1Φ +((t +3)7 − 3 − t7)7−1Φ σ(N ) 1 1 1 1 1 d + ((t +3)7 − 3 − t7)27−1Φ σ N 2 + (N ) (mod 73). 2 1 1 dt 1

To recover the characteristic polynomial of the ﬁbre of XF7 above t =1,wemust specialize Φ1 to the unique point in that residue disc which is ﬁxed by the map → 7 − 2 − ≡ 3 t (t+3) 3. This point is none other than ζ3 3 15 (mod 7 ) (see worksheet). Clearing denominators and then evaluating at this point, we obtain another characteristic polynomial with the same accuracy as in Example 6.3.5 (see worksheet). The point of converting the Frobenius structure is to be able to take advantage of the following fact [53, Proposition 17.5.1]. Lemma . ∞ i × 6.5.7 Let N = i=−1 Nit be an n n matrix such that the entries of Q tN are power series over q convergent on the open unit disc, and N−1 is a nilpotent ∞ i × matrix. Let Φ= i=−∞ Φit be an n n matrix whose entries are Laurent series over Qq convergent on some open annulus with outer radius 1. Suppose that N,Φ satisfy (6.1.4.1).ThenΦi =0for i<0,soΦ converges on the whole open unit disc. Corollary 6.5.8. Retain notation as in Lemma 6.5.7, except now assume only that the eigenvalues λ1,...,λn of N−1 are rational numbers with denominators coprime to p.ThenΦi =0whenever

i2. Fix a positive integer m.LetB be an open dense subscheme of P1 whose complement Z consists of points with distinct Qq reductions modulo p, one of which is the point ∞.LetE be a vector bundle with

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 161 connection on B which is everywhere regular, with all exponents in Q ∩ Zp.Let E d v1,...,vn be a basis of ,andletN be the matrix of action of dt on this basis. For z ∈ Z, deﬁne the quantities f(z) and g(z) as follows. Choose a matrix U Q over q(t) such that the basis wj = i Uijvi is regular at z.Letλz,1,...,λz,n be the exponents of this basis at z.Put −1 f(z)=−qvt(U ) − vt(U) − q min{λj } +max{λj }, j j where vt denotes the (t−z)-adic valuation, and vt(U)=mini,j {vt(Uij)}.Putg(z)= 0 if the residue matrix of w1,...,wn vanishes at z,orifz ∈{0, ∞};otherwise,let g(z) be the least nonnegative integer i for which i −(i − 1)/(p − 1)≥m − 1. Let V be the rigid analytic subspace of P1 given as the complement of the Qq residue discs containing points of Z. Suppose that E admits a Frobenius structure F on a strict neighborhood of V .LetΦ be the matrix of action of F on v1,...,vn. Assume that both N and Φ have nonnegative Gauss valuation. Then Φ is congruent modulo pm to a rational function with divisor bounded below by −(f(z)+qg(z))(z). z∈Z Proof. It suﬃces to check that the contribution of each z ∈ Z to the divisor is at least −(f(z)+qg(z))(z). To see this, we see that using a Frobenius lift carrying (t − z)to(t − z)q,wegetapoleoforderatmostf(z)atz by Corollary 6.5.8. We then apply Theorem 6.5.5 to convert back to the original Frobenius (this step being unnecessary if z =0, ∞), noting that the p-adic valuation of the term (σ(t)− σ(t))i/i!isatleasti −(i − 1)/(p − 1).

Example 6.5.11. In Example 6.3.4, we use Theorem 6.5.10 to bound the pole −1 2 divisor of 7 Φ1 modulo 7 ; this is valid because the poles are distinct mod 7. In concrete terms, the roots of Δ1Δ3 in an algebraic closure of Q7 lie in the integral closure of Z7 and are distinct modulo 7 (see worksheet). For z arootofΔ1, Δ2, Δ3 or the value ∞, using the computation of exponents in Example 5.3.2, we compute respectively f(z)=7, 1, 8, −7 g(z)=1, 0, 1, 0. The values of g are clear, but it is worth explaining where the values of f came from. For z = −3 the unique root of Δ1, we computed f(z) = 7 in Example 6.5.9. For z arootofΔ2,wehavef(z)=1− 7 · 0 = 1 because the exponents are 0, 1. For z arootofΔ3,wehave ' ( ' ( 7 49 f(z)= 0 − 7 · − = =8. 6 6 For z = ∞, we get a contribution of 1 from the change of basis matrix U to a regular basis, and a contribution of ' ( ' ( 5 4 23 − 7 · = − = −8 3 3 3 from the exponents.

162 KIRAN S. KEDLAYA

We thus get a lower bound of

−14(Δ1) − (Δ2) − 15(Δ3)+7(∞) for the pole divisor. In particular, this divisor has degree −14 · 1 − 1 · 23 − 15 · 26 + 8 · 1=−419, so we need the Taylor series expansions around t =0within O(t420) to guarantee that we have correctly identiﬁed the rational functions. Since we computed to order O(t500) in Example 6.3.4, the computation is validated. Remark 6.5.12. Note the discrepancy between the lower bound

−14(Δ1) − (Δ2) − 15(Δ3)+7(∞) given in Example 6.5.11 and the computed divisor

−13(Δ1) − (Δ2) − 7(Δ3)+7(∞). This correctly suggests that there is a lot more work to be done in the area of ana- lyzing the pole orders of Frobenius structures of Picard-Fuchs-Manin connections. We make a few remarks in the optional addendum to this lecture, but otherwise the subject is very much open.

2 Remark 6.5.13. By making the substitution t1 = t − 3 and changing basis on the Frobenius matrix in Example 6.5.9, we can get a matrix which is holomorphic at t1 = 0. The reduction modulo t1 has eigenvalues congruent to 21, 161, 35, 14, 324 (mod 73)

(see worksheet). The last of these is the reduction of ζ3 modulo p. The other four are supposed to appear in the zeta function of the singular cubic threefold deﬁned 3 by a = Q−3 over F7. (To prove this relationship requires either an appeal to Dwork’s deformation theory for singular hypersurfaces, or a comparison theorem between de Rham cohomology and Hyodo-Kato cohomology which does not seem to have been written down yet.) 6.6. Optional: Further analysis of t-adic precision. In some cases, the following reﬁnement of Corollary 6.5.8 may be useful. Lemma 6.6.1. With notation as in Corollary 6.5.8,foreachα ∈ Q,put

Sα = {λ1,...,λn}∩(α + Z). If i

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 163

For X concentrated in a single off-diagonal block corresponding to the congruence classes α + Z, β + Z, the operation X →−XN−1 + N−1X + jX has eigenvalues in the set ±{α − β + Z}, which does not contain zero. We can thus choose X so j j that changing basis using W (In + t X) puts N into the correct form modulo t . We may thus proceed by induction to deduce the claim. If we apply Lemma 6.5.7 to the result of changing basis by a suitably t-adically close approximation of V , we may now deduce the desired result. Remark 6.6.2. If one has a family of pencils of varieties, one gets a family of Picard-Fuchs-Manin connections admitting Frobenius structures. If one can use Theorem 6.5.10 to bound the pole orders of the Frobenius modulo pn for a generic member of this family, the same bound will apply to each special member, even if it does not have all of its poles in distinct residue classes mod p. In practice, this may significantly improve the range of applicability of the deformation method. Remark 6.6.3. In light of Remark 6.6.2, it would be useful to have a completely general analogue of Theorem 6.5.10 that makes no hypothesis on the poles of the connection being distinct mod p. Some more experimentation may be necessary in order to correctly formulate an appropriate conjecture. Remark 6.6.4. One can improve the bound in Theorem 6.5.10 so that g(z)−m is only logarithmic in m, rather than linear in m, by using effective convergence bounds for p-adic differential equations, as in Theorem 6.4.3. (This would allow allow for the use of p = 2, which is impossible with a bound of the form given in Theorem 6.5.10.) See [56].

Appendix A. Notes and further reading In this appendix, we provide references omitted in the main text, in a sequence of subsections keyed to the six lectures. We ﬁnish with suggestions for further reading.

A.1. Zeta functions: generalities. There is a useful, if brief, introduction to zeta functions and the Weil conjectures in Hartshorne’s algebraic geometry text- book [35, Appendix C]. See also the survey by Osserman [78]. The analytic continuation of the L-function of an elliptic curve over Q was proved by Breuil, Conrad, Diamond, and Taylor [8] following the method introduced by Wiles [95] and Taylor-Wiles [91]. Lemma 1.2.3 is taken from [50], where it is used to give an algorithm (implemented in Sage) for searching for Weil polynomials subject to congruence conditions. However, this algorithm is only designed to handle polynomials over Z;we are not aware of any algorithms designed to handle situations where a Weil polynomial is known to have a factorization over a larger field, as happens for cyclic cubic threefolds. The strongest notion of a Weil cohomology theory includes Poincaré duality, cycle class maps, the Künneth decomposition theorem, a Lefschetz hyperplane theorem, plus additional compatibilities. See [57] for more details. For a development of étale cohomology, see the books of Freitag and Kiehl [27], Milne [73], and Tamme [90]. The course notes of Milne [74] may also be helpful. The computation of the zeta function of diagonal hypersurfaces is originally due to Weil. It was one of the two main justifications for his original assertion

164 KIRAN S. KEDLAYA of the Weil conjectures, the other being his proof of the conjectures for curves (generalizing Hasse’s theorem bounding the number of points on an elliptic curve over a finite field). For further discussion of Jacobi sums, including the case q = p, the definitive reference is [3]. A.2. Algebraic de Rham cohomology. The standard development of algebraic de Rham cohomology is that of Hartshorne [34]. However, it might be helpful to become acquainted with the complex-analytic situation first, by reading about it in Griffiths and Harris [31]. For Grothendieck’s comparison theorem (Theorem 2.1.3), see [32]. To compute the algebraic de Rham cohomology of a smooth complete intersection inside a toric variety, one has a generalization of the Griffiths-Dwork method; this calculation has become fashionable of late because it can be used to generate putative instances of mirror symmetry. See Cox and Katz [17]. A.3. de Rham cohomology and p-adic cohomology. A useful overview of p-adic cohomologies is the survey of Illusie [41]. The subject has developed considerably since that survey was written; a more recent but more advanced survey is [52]. BeforethebookofleStum[66] appeared, there was no proper foundational treatise on rigid cohomology; instead, one was forced to infer much of the theory from the articles of Berthelot. Fortunately, these are quite readable, and even now may prove helpful; we suggest in particular the introductory article [4]for the general construction, and the later article [5] for details on the comparisons between rigid cohomology, Monsky-Washnitzer cohomology, and crystalline cohomology. Theorem 3.2.1 is a logarithmic version of Berthelot’s original comparison theorem, given by Baldassarri and Chiarellotto [2]. The integral version (Theo- rem 3.2.2) is due in the logarithmic case to Shiho [84, 85]. The fact that p-adic cohomology is a Weil cohomology includes a great many assertions, some of which were only proved quite recently. For example, finite di- mensionality was established by Berthelot in [5], while Poincaré duality and the Künneth formula were established by Berthelot [6]. The Riemann hypothesis component of the Weil conjectures in p-adic cohomology was originally proved by Katz and Messing [45] by reducing to Deligne’s -adic version [19]; see [12] for a similar argument. Purely p-adic proofs were later given by Faltings [26] and Kedlaya [49]. The construction of cycle classes is due to Petrequin [79]; this is needed for the full Lefschetz trace formula (Remark 1.3.3). Mazur’s theorem comparing the Hodge filtration with divisibility in the Frobe- nius matrix (originally a conjecture of Katz) was originally announced in [69]and proved in [70]. Another treatment is given by Berthelot and Ogus in [7]. See also the discussion in [41]. A.4. The direct method for cyclic cubic threefolds. The Frobenius action on affine varieties comes from the comparison between rigid and Monsky- Washnitzer cohomology. The original development of the latter occurs in the three papers [77, 75, 76]; compare also [94]. One may also use this comparison to deduce the cases of the Lefschetz trace formula in rigid cohomology that we need, as the proof for the Frobenius map given in [76] extends to cover the composition of Frobenius with an automorphism. (This only applies to affine varieties; to deduce

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 165 the general case, one must first apply Poincaré duality to switch to cohomology with compact supports, then use the excision property there.) The original use of p-adic methods for computing zeta functions arose in the context of finding suitable curves for elliptic curve cryptography, namely those for which the group of rational points has order equal to a prime number times a very small cofactor. Methods introduced in this setting include the canonical lift method of Satoh [82] and the AGM iteration of Mestre [72]. These can in principle be extended to higher genus (see for example [65] for a higher genus version of Mestre’s method), but the dependence on the genus is quite poor; most practical interest has concentrated on genera 2 and 3, which also have some relevance for cryptography. (See [16] for a survey of elliptic and hyperelliptic curve cryptography circa 2005.) Thefirstattempttousep-adic methods to give more general algorithms for computing zeta functions was given by Lauder and Wan [64]. They gave a general algorithm based on Dwork’s proof of the rationality of the zeta function. This can be interpreted as an application of p-adic cohomology, but where one computes not in the cohomology but in the chain complex, in which the the terms are infinite-dimensional vector spaces which must be truncated appropriately. The first algorithm involving a calculation on p-adic cohomology itself was Kedlaya’s algorithm for hyperelliptic curves in odd characteristic [47]; see also the exposition by Edixhoven [25], and note the correction to the precision bound given in the errata to [47]). An analogous algorithm for characteristic 2 was given by Denef and Ver- cauteren [21]. The method has been generalized to rather general families of curves (nondegenerate curves in toric surfaces) by Castryck, Denef, and Vercauteren [10]. (See [48] for a survey of this subject circa 2004.) Less work has been carried out in higher dimensions, partly because the case of curves carried some external interest from cryptography, and partly because in higher dimensions the deformation method has better asymptotic complexity. The approach we describe here was given by Abbott, Kedlaya, and Roe in [1], but that paper only gives experimental results for surfaces. A closer analogue of Kedlaya’s original algorithm, for cyclic covers of projective spaces, has been implemented by de Jong [18] but currently lacks rigorous error bounds. (We expect that one can adapt the analysis of [1] to de Jong’s situation, but to our best knowledge no one has attempted to do so.) It might be feasible to use de Jong’s method to compute zeta functions of cyclic cubic threefolds, but we did not investigate this thoroughly; we used the approach from [1] instead so that we could reuse the setup to derive the Picard-Fuchs-Manin connection. It is also worth mentioning the work of Harvey [36], who found a restructuring of Kedlaya’s algorithm for hyperelliptic curve that reduces the complexity in the characteristic p of the finite field from linear to square-root. This has had the effect of making p-adic cohomology applicable in far larger characteristics than had been previously expected; this was demonstrated experimentally for hyperelliptic curves of genus 2 and 3 by Kedlaya and Sutherland [55]. (Interestingly, Harvey’s motivation was not computing zeta functions, but rather computing cyclotomic p- adic canonical heights of elliptic curves over Q using the method of Mazur, Stein and Tate [71].) Even more recently, Harvey has described a higher-dimensional analogue of his work for hyperelliptic curves, which might make the direct method feasible for

166 KIRAN S. KEDLAYA such examples as cyclic cubic threefolds. One key diﬀerence from [1]isthatthe reduction algorithm is translated from a problem of commutative algebra into a reasonably compact problem of linear algebra. This has the eﬀect of avoiding the use of dense multivariate polynomials, leading to improved asymptotic behavior especially as the characteristic grows. As of this writing, Harvey’s work is still in preparation, but see [37].

A.5. Picard-Fuchs-Manin connections. We are not sure where the name “Gauss-Manin connection” comes from. In [32, footnote 13], Grothendieck proposes the existence of a “canonical connection” on relative algebraic de Rham cohomology, inspired by Manin’s use of such a construction in his proof of the analogue of the Mordell conjecture over complex function fields [68]. (Grothendieck suggests that such a connection could be used to define a Leray spectral sequence; Defi- nition 5.2.2 shows that the reverse is actually what happens!) This explains the inclusion of Manin’s name; the reference to Gauss appears to invoke the theory of hypergeometric differential equations while skipping over the intervening history of Picard-Fuchs differential equations. For the holomorphicity of the topological Picard-Fuchs-Manin connection, see [31]. For the fact that it agrees with the algebraic connection, see [46]. Theorem 5.2.5 is a theorem originally due to Griffiths, but many proofs are possible. See [30, Theorem 3.1] for an overview. For more on the use of Lefschetz pencils in algebraic geometry, see Katz’s exposés in SGA 7 [20,Exposés XVII, XVIII].

A.6. The deformation method for cyclic cubic threefolds. The existence of a Frobenius structure on a Picard-Fuchs-Manin connection was originally observed in a number of examples by Dwork, notably including the Legendre family of elliptic curves [22]. See van der Put [94] for a modern treatment of this example. Theorem 6.1.3 is a corollary of Theorem 6.5.2, for more on which see below. The original idea of using the Frobenius structure on a Picard-Fuchs-Manin connection to compute zeta functions is due to Lauder [59], who described an algorithm for smooth projective hypersurfaces using Dwork cohomology. Lauder later gave an alternate development using relative Monsky-Washnitzer cohomology [60]; a similar development was given by Gerkmann [28], and this is what we have followed in these notes. The deformation method has also been used by Hubrechts [38] to give more efficient point counting algorithms for elliptic curves than is possible using the direct method. Additional work has been done for hyperelliptic curves by Hubrechts [39, 40], and for Ca,b curves by Castryck, Hubrechts, and Vercauteren [11]. (It is worth studying Hubrechts’s work for its significant improvements over what we have described here, in the space and memory requirements used for carrying out the deformation method.) It is an interesting open question whether there is an analogue of Harvey’s method (reducing the dependence on p to square-root) for the deformation method. The recent work of Hubrechts on memory-efficient use of the deformation method [40] provides a clue, as it uses the same baby step-giant step trick (due to Chud- novsky and Chudnovsky) as in Harvey’s method; however, there is an additional step needed of repackaging the algorithm so that one never explicitly writes down a power series involving O(p)terms.

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Theorem 6.4.2 is due to Dwork and Robba [24]; see also [23, Theorem IV.3.1]. Theorem 6.4.3 is an effective version due to Kedlaya [53, Theorem 18.3.3] of a bound due to Chiarellotto and Tsuzuki [13, Proposition 6.10]. Theorem 6.5.2 is due to Berthelot [4,Théorème 5]. A conjecture of Berthelot (in the constant coefficient case) and Shiho (in general) asserts that more generally, for any smooth proper morphism between varieties over a field of characterisic p>0, the relative rigid cohomology should exist as an overconvergent F -isocrystal. This is known in certain cases by work of Tsuzuki [93] and Shiho [86, 87, 88]; it is likely to be proved soon using Caro’s construction of a category of p-adic coefficients (overholonomic arithmetic D-modules), as completed recently by Caro and Tsuzuki [9]. In previous published work on the deformation method, e.g. [28], the t-adic precision is controlled by the method suggested above Theorem 6.5.5, i.e., by using p-adic precision loss bounds in the direct method as applied over the generic fibre of the base. To the best of our knowledge, the method described here has not been used previously, though Alan Lauder informs us that he is using it currently. (See also [63] for another appearance of this technique.) Theorem 6.5.5 is implicit in the work of Berthelot [4]; it follows from the overconvergence of the Taylor series map, which is built into Berthelot’s definition of an overconvergent F -isocrystal. The argument has been made explicit several times in the literature, e.g., [92, Theorem 3.4.10] and [53, Proposition 17.3.1].

A.7. Additional suggestions. These notes are loosely inspired by the author’s notes for the 2007 Arizona Winter School [51]. We have attempted here to focus more on computational aspects of the deformation method; consequently, comparing the two documents may be profitable. For varieties of dimension greater than 1, Lauder has also introduced a “fibration method” for computing zeta functions [61]. This shares the advantage held by the deformation method of involving only one-dimensional varieties at any given step, but does not require the auxiliary construction of the Frobenius structure on an entire Picard-Fuchs-Manin connection. Lauder has obtained good experimental results in the case of elliptic surfaces; these appear in [62]. In principle, all three of the approaches to effective p-adic cohomology (direct, deformation, fibration) should be applicable to appropriate classes of mildly singular varieties, but a fair bit of care must be applied. Some analysis along these lines, including some positive numerical results, has been made by Kloosterman [58]. Dealing with singular fibres properly requires effective convergence bounds for logarithmic connections with nilpotent residues (improving upon work of Christol- Dwork); these can be found in [53, Chapter 18]. It also requires checking some compatibilities between Hyodo-Kato Frobenius actions and Picard-Fuchs-Manin connections; to our knowledge, these have not been checked in general.

References [1] T.G. Abbott, K.S. Kedlaya, and D. Roe, Bounding Picard numbers of surfaces using p- adic cohomology, in Arithmetic, Geometry and Coding Theory (AGCT 2005),Séminaires et Congrès 21, SocietéMathématique de France, 2009, 125–159. MR2483951 (2010e:14014) [2] F. Baldassarri and B. Chiarellotto, Algebraic versus rigid cohomology with logarithmic coefficients, in Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math. 15, Academic Press, San Diego, 1994, 11–50. MR1307391 (96f:14024)

168 KIRAN S. KEDLAYA

[3] B.C. Berndt, R.J. Evans, and K.S. Williams, Gauss and Jacobi Sums, Canadian Math. Soc. Monographs 21, Wiley-Interscience, 1998. MR1625181 (99d:11092) [4] P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique p, Introductions aux cohomologies p-adiques (Luminy, 1984), Mém. Soc. Math. France 23 (1986), 7–32. MR865810 (88a:14020) [5] P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. 128 (1997), 329–377. MR1440308 (98j:14023) [6] P. Berthelot, DualitédePoincaréetformuledeKünneth en cohomologie rigide, C.R. Acad. Sci. Paris Sér. I Math. 325 (1997), 493–498. MR1692313 (2000c:14023) [7] P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Princeton Univ. Press, Prince- ton, 1978. MR0491705 (58:10908) [8] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939. MR1839918 (2002d:11058) [9] D. Caro and N. Tsuzuki, Overholonomicity of overconvergent F -isocrystals on smooth varieties, arXiv:0803.2015v1 (2008). [10] W. Castryck, J. Denef, and F. Vercauteren, Computing zeta functions of nondegenerate curves, Int. Math. Res. Papers 2006, article ID 72017 (57 pages). MR2268492 (2007h:14026) [11] W. Castryck, H. Hubrechts, and F. Vercauteren, Computing zeta functions in families of Ca,b curves using deformation, in Algorithmic Number Theory (ANTS VIII), Lecture Notes in Computer Science 5011, Springer, New York, 2008, 296–311. MR2467856 (2010d:11148) [12] B. Chiarellotto and B. le Stum, Sur la pureté de la cohomologie cristalline, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 961–963. MR1649945 (99f:14024) [13] B. Chiarellotto and N. Tsuzuki, Logarithmic growth and Frobenius filtrations for solutions of p-adic differential equations, J. Inst. Math. Jussieu 8 (2009), 465–505. MR2516304 (2010k:12008) [14] G. Christol and B. Dwork, Effective p-adic bounds at regular singular points, Duke Math. J. 62 (1991), 689–720. MR1104814 (93c:12009) [15] C.H. Clemens and P.A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972), 281–356. MR0302652 (46:1796) [16] H. Cohen and G. Frey (eds.), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, 2005. MR2162716 (2007f:14020) [17] D.A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Math. Surveys and Mono- graphs 68, Amer. Math. Soc., 1999. MR1677117 (2000d:14048) [18] A.J. de Jong, Frobenius project; see http://math.columbia.edu/~dejong/. [19] P. Deligne, La conjecture de Weil. I, Publ. Math. IHES´ 43 (1974), 273–307. MR0340258 (49:5013) [20] P. Deligne and N. Katz (eds.), Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II): Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Math. 340, Springer-Verlag, Berlin, 1973. MR0354657 (50:7135) [21] J. Denef and F. Vercauteren, An extension of Kedlaya’s algorithm to hyperelliptic curves in characteristic 2, J. Cryptology 19 (2006), 1–25. MR2210897 (2007d:11069) [22] B. Dwork, Lectures on p-adic Differential Equations, Grundlehren 253, Springer-Verlag, 1982. MR678093 (84g:12031) [23] B. Dwork, G. Gerotto, and F. Sullivan, An Introduction to G-Functions, Annals of Math. Studies 133, Princeton University Press, Princeton, 1994. MR1274045 (96c:12009) [24] B. Dwork and P. Robba, Effective p-adic bounds for solutions of homogeneous linear differential equations, Trans. Amer. Math. Soc. 259 (1980), 559–577. MR567097 (81k:12022) [25] B. Edixhoven, Point counting after Kedlaya, course notes at http://www.math.leidenuniv. nl/~edix/oww/mathofcrypt/. [26] G. Faltings, F -isocrystals on open varieties: results and conjectures, in The Grothendieck Festschrift, Vol. II, Progr. Math., 87, Birkhüser, Boston, 1990, 219–248. MR1106900 (92f:14015) [27] E. Freitag and R. Kiehl, Etale´ Cohomology and the Weil Conjectures (translated from the German by B.S. Waterhouse and W.C. Waterhouse), Ergebnisse der Math. 13, Springer- Verlag, Berlin, 1988. MR926276 (89f:14017) [28] R. Gerkmann, Relative rigid cohomology and deformation of hypersurfaces, Int. Math. Res. Papers 2007, article ID rpm003 (67 pages). MR2334009 (2008f:14036)

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 169

[29] P.A. Griffiths, On the periods of certain rational integrals I, II, Ann. of Math. 90 (1969), 460–495; ibid. 90, 496–541. MR0260733 (41:5357) [30] P.A. Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems, Bull. Amer. Math. Soc. 76 (1970), 228–296. MR0258824 (41:3470) [31] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978. MR507725 (80b:14001) [32] A. Grothendieck, On the De Rham cohomology of algebraic varieties, Publ. Math. IHES´ 29 (1966), 95–103. MR0199194 (33:7343) [33] G.-M. Greuel and Gerhard Pfister, A Singular Introduction to Commutative Algebra, Springer-Verlag, Berlin, 2002. MR1930604 (2003k:13001) [34] R. Hartshorne, On the De Rham cohomology of algebraic varieties, Publ. Math. IHES´ 45 (1975), 5–99. MR0432647 (55:5633) [35] R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, New York, 1977. MR0463157 (57:3116) [36] D. Harvey, Kedlaya’s algorithm in larger characteristic, Int. Math. Res. Notices 2007, article ID rnm095 (29 pages). [37] D. Harvey, Counting points on projective hypersurfaces, lecture notes available at http:// www.cims.nyu.edu/~harvey/. [38] H. Hubrechts, Quasi-quadratic elliptic curve point counting using rigid cohomology, confer- ence MEGA 2007 (2007). [39] H. Hubrechts, Point counting in families of hyperelliptic curves in characteristic 2, LMS J. Comput. Math. 10 (2007), 207–234. MR2320829 (2008f:11066) [40] H. Hubrechts, Point counting in families of hyperelliptic curves, Found. Comput. Math. 8 (2008), 137–169. MR2403533 (2009e:11122) [41] L. Illusie, Crystalline cohomology, in Motives, Proc. Sympos. Pure Math., vol. 55, part 1, Amer. Math. Soc. Providence, RI, 1994, 43–70. MR1265522 (95a:14021) [42] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, second edition, Graduate Texts in Math. 84, Springer-Verlag, 1990. MR1070716 (92e:11001) [43] K. Kato, Logarithmic structures of Fontaine-Illusie, in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, 1989, 191– 224. MR1463703 (99b:14020) [44] N.M. Katz, Slope filtration of F -crystals, Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. I, Astérisque 63 (1979), 113–163. MR563463 (81i:14014) [45] N.M. Katz and W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields. Invent. Math. 23 (1974), 73–77. MR0332791 (48:11117) [46] N.M. Katz and T. Oda, On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ. 8 (1968), 199–213. MR0237510 (38:5792) [47] K.S. Kedlaya, Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology, J. Ramanujan Math. Soc. 16 (2001), 323–338; errata, ibid. 18 (2003), 417–418. MR1877805 (2002m:14019) [48] K.S. Kedlaya, Computing zeta functions via p-adic cohomology, in D.A. Buell (ed.), Al- gorithmic Number Theory (ANTS 2004), Lecture Notes in Comp. Sci. 3076, 2004, 1–17. MR2137340 (2006a:14033) [49] K.S. Kedlaya, Fourier transforms and p-adic “Weil II”, Compos. Math. 142 (2006), 1426– 1450. MR2278753 (2008b:14024) [50] K.S. Kedlaya, Search techniques for root-unitary polynomials, in K.E. Lauter and K. Ri- bet (eds.), Computational Arithmetic Geometry, Contemp. Math. 463, Amer. Math. Soc., 2008, 71–82. Associated Sage package available at http://math.mit.edu/~kedlaya/papers/. MR2459990 (2009h:26022) [51] K.S. Kedlaya, p-adic cohomology: from theory to practice, in D. Savitt and D.S. Thakur (eds.), p-adic Geometry, University Lecture Series 45, Amer. Math. Soc., 2008, 175–203. MR2482348 (2010h:14031) [52] K.S. Kedlaya, p-adic cohomology, in D. Abramovich et al. (eds.), Algebraic Geometry: Seat- tle 2005, Proc. Symp. Pure Math. 80, Amer. Math. Soc., 2009, 667–684. MR2483951 (2010e:14014) [53] K.S. Kedlaya, p-adic Differential Equations, Cambridge Studies in Advanced Math. 125, Cambridge Univ. Press, 2010. MR2663480 (2011m:12016)

170 KIRAN S. KEDLAYA

[54] K.S. Kedlaya, Effective p-adic cohomology for cyclic cubic threefolds, supporting Sage worksheet available at http://math.mit.edu/~kedlaya/papers/. MR2483951 (2010e:14014) [55] K.S. Kedlaya and A.V. Sutherland, Computing L-series of hyperelliptic curves, in Algorithmic Number Theory (ANTS VIII), Lecture Notes in Computer Science 5011, Springer, New York, 2008, 312–326. MR2467855 (2010d:11070) [56] K.S. Kedlaya and J. Tuitman, Effective convergence bounds for Frobenius structures, arXiv:1111.0136v2 (2012). To appear in Rend. Sem. Mat. Padova. [57] S. Kleiman, Algebraic cycles and the Weil conjectures, in Dix exposés sur la cohomologie des schémas, North-Holland, 1968, 359–386. MR0292838 (45:1920) [58] R. Kloosterman, Point counting on singular varieties, in Algorithmic Number Theory (ANTS VIII), Lecture Notes in Computer Science 5011, Springer, New York, 2008, 327–341; updated version at http://www.iag.uni-hannover.de/~kloosterman/publ.php. MR2467857 (2010d:14028) [59] A.G.B. Lauder, Counting solutions to equations in many variables over finite fields, Found. Comput. Math. 4 (2004), 221–267. MR2078663 (2005f:14048) [60] A.G.B. Lauder, Rigid cohomology and p-adic point counting, J. Théor. Nombres Bordeaux 17 (2005), 169–180. MR2152218 (2006a:14034) [61] A.G.B. Lauder, A recursive method for computing zeta functions of varieties. LMS J. Comput. Math. 9 (2006), 222–269. MR2261044 (2007g:14022) [62] A.G.B. Lauder, Ranks of elliptic curves over function fields, LMS J. Comput. Math. 11 (2008), 172–212. MR2429996 (2009f:11064) [63] A.G.B. Lauder, Degenerations and limit Frobenius structures in rigid cohomology, London Math.Soc.J.Comp.Math.14 (2011), 1–33. MR2777002 (2012d:11138) [64] A.G.B. Lauder and D. Wan, Counting rational points on varieties over finite fields of small characteristic, in J.P. Buhler and P. Stevenhagen (eds.), Algorithmic Number Theory: lattices, number fields, curves and cryptography, MSRI Publications 44, Cambridge Univ. Press, Cambridge, 2008. MR2467558 (2009j:14029) [65] R. Lercier and D. Lubicz, A quasi quadratic time algorithm for hyperelliptic curve point counting, Ramanujan J. 12 (2006), 399–423. MR2293798 (2008b:11069) [66] B. le Stum, Rigid Cohomology, Cambridge Univ. Press, 2007. MR2358812 (2009c:14029) [67] Magma version 2.17-5(2011), information available at http://magma.maths.usyd.edu.au/. [68] Yu. Manin, Rational points of algebraic curves over function fields (Russian), Izv. Akad. Nauk. SSSR Ser. Math. 27 (1963), 1395–1440; English translation, AMS Transl. 50 (1966), 189–234. MR0157971 (28:1199) [69] B. Mazur, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc. 78 (1972), 653–667. MR0330169 (48:8507) [70] B. Mazur, Frobenius and the Hodge filtration (estimates), Ann. of Math. 98 (1973), 58–95. MR0321932 (48:297) [71] B. Mazur, W. Stein, and J. Tate, Computation of p-adic heights and log convergence, Doc. Math. Extra Vol. (2006), 577–614. MR2290599 (2007i:11089) [72] J.-F. Mestre, Algorithmes pour compter des points en petite caractéristique en genre 1 et 2, unpublished preprint. available at http://www.math.univ-rennes1.fr/crypto/2001-02/ mestre.ps. [73] J.S. Milne, Etale´ Cohomology, Princeton Math. Series 33, Princeton Univ. Press, Princeton, 1980. MR559531 (81j:14002) [74] J.S. Milne, Etale´ cohomology, course notes available at http://www.jmilne.org/math/. [75] P. Monsky, Formal cohomology. II: The cohomology sequence of a pair, Annals of Math. 88 (1968), 218–238. MR0244272 (39:5587) [76] P. Monsky, Formal cohomology. III: Fixed point theorems, Annals of Math. 93 (1971), 315– 343. MR0321931 (48:296) [77] P. Monsky and G. Washnitzer, Formal cohomology. I, Annals of Math. 88 (1968), 181–217. MR0248141 (40:1395) [78] B. Osserman, The Weil conjectures, in W.T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton Univ. Press, 2008, 729–732; also available online at http://www. math.ucdavis.edu/~osserman. MR2467561 (2009i:00002) [79] D. Petrequin, Classes de Chern et classes de cycles en cohomologie rigide, Bull. Soc. Math. France 131 (2003), 59–121. MR1975806 (2004b:14030)

EFFECTIVE p-ADIC COHOMOLOGY FOR CYCLIC CUBIC THREEFOLDS 171

[80] J. Pila, Frobenius maps of abelian varieties and finding roots of unity in finite fields, Math. Comp. 55 (1990), 745–763. MR1035941 (91a:11071) [81] Sage version 4.6.1 (2011), available at http://sagemath.org/. [82] T. Satoh, The canonical lift of an ordinary elliptic curve over a finite field and its point counting, J. Ramanujan Math. Soc. 15 (2000), 247–270. MR1801221 (2001j:11049) [83] R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp. 44 (1985), 483–494. MR777280 (86e:11122) [84] A. Shiho, Crystalline fundamental groups, I: Isocrystals on log crystalline site and log convergent site, J. Math. Sci. Univ. Tokyo 7 (2000), 509–656. MR1800845 (2002e:14031) [85] A. Shiho, Crystalline fundamental groups, II: Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), 1–163. MR1889223 (2003c:14020) [86] A. Shiho, Relative log convergent cohomology and relative rigid cohomology, I, arXiv:0707.1742v2 (2008). [87] A. Shiho, Relative log convergent cohomology and relative rigid cohomology, II, arXiv:0707.1743v2 (2008). [88] A. Shiho, Relative log convergent cohomology and relative rigid cohomology, III, arXiv:0805.3229v1 (2008). [89] Singular version 3.1.1 (2010), available at http://www.singular.uni-kl.de/. [90] G. Tamme, Introduction to Etale´ Cohomology (translated from the German by M. Kolster), Springer-Verlag, Berlin, 1994. MR1317816 (95k:14033) [91] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553–572. MR1333036 (96d:11072) [92] N. Tsuzuki, Slope filtration of quasi-unipotent overconvergent F -isocrystals, Ann. Inst. Fourier (Grenoble) 48 (1998), 379–412. MR1625537 (99e:14023) [93] N. Tsuzuki, On base change theorem and coherence in rigid cohomology, Doc. Math. Extra Vol. (2003), 891–918. MR2046617 (2004m:14031) [94] M. van der Put, The cohomology of Monsky and Washnitzer, Introductions aux cohomologies p-adiques (Luminy, 1984), Mém. Soc. Math. France 23 (1986), 33–59. MR865811 (88a:14022) [95] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443– 551. MR1333035 (96d:11071)

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 E-mail address: [email protected] URL: http://math.mit.edu/~kedlaya

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11366

Generating sets of aﬃne groups of low genus

K. Magaard, S. Shpectorov, and G. Wang

Abstract. We describe a new algorithm for computing braid orbits on Nielsen classes. As an application we classify all families of aﬃne genus zero systems; that is all families of coverings of the Riemann sphere by itself such that the monodromy group is a primitive aﬃne permutation group.

1. Introduction Let G be a finite group. By a G-curve we mean a compact, connected Riemann surface X of genus g such that G ≤ Aut(X). By a G-cover we mean the natural projection π of a G-curve X to its orbifold X/G. In our situation X/G is a Riemann surface of genus g0 and π is a branched cover. We are interested in Hurwitz spaces Hin which are moduli spaces of G-covers. By r (G, g0)wemeantheHurwitzspace of equivalence classes of G-covers which are branched over r points, such that g(X/G)=g0. We are mostly interested in the case g0 = 0 in which case we will Hin simply write r (G). Hurwitz spaces are used to study the moduli space Mg of curves of genus g. For example Hurwitz himself showed the connectedness of Mg by first showing that every curve admits a simple cover onto P 1C and then showing that the Hurwitz space of simple covers is connected. The study of Hurwitz spaces is also closely related to the inverse problem of Galois theory. The precise connection was given by Fried and Völklein in [7]. Theorem 1.1 (Fried-Völklein). The following are true: Hin Q (1) r (G) is an affine algebraic set which is defined over . (2) If G is a group with Z(G)=1, then there exists a Galois extension of Q(x), regular over Q, with Galois group isomorphic to G and with r branch Hin Q Q points if and only if r (G) has a -rational point. (This also holds if is replaced throughout by any field of characteristic 0). Hin Q¯ Q Q The space r (G) admits an action of Gal( / ). Thus a -rational point of Hin(G) must lie in an irreducible component which is defined over Q. r ∼ Hurwitz spaces are covering spaces. In our situation, where X/G = P 1C,the Hin 1C base space of r (G) is the configuration space of P with r marked points. That is the space

2010 Mathematics Subject Classiﬁcation. Primary 14H10, 14H30; Secondary 14H45, 14H37. Key words and phrases. Aﬃne groups, Hurwitz spaces, covers of curves.

c 2012 American Mathematical Society 173

174 K. MAGAARD, S. SHPECTOROV, AND G. WANG

1 Dr := {S ⊂ P C : |S| = r}/PGL2(C)

1 r−3 =(P C \{0, 1, ∞}) \ Δr−3. Where

1 r−3 Δr−3 := {(x1,...,xr−3) ∈ (P C \{0, 1, ∞}) : ∃i, j with xi = xj }.

The fundamental group of Dr is the Hurwitz braid group on r-strings and is a Hin quotient of the Artin braid group. Thus the connected components of r (G)are the orbits of the fundamental group of Dr on the ﬁbres. We deﬁne Tr(G)tobe r those elements (τ1,...,τr) ∈ G such that

G = (τ1,...,τr),

r τi =1, i=1 and r |G|( (|τi|−1)/|τi|)=2(|G| + g − 1). i=1 Hin ∈F T The fibres of r (G) are parametrized by elements of τ r(G):= r(G)/G , r where the action of G on G , and hence on Tr(G), is via diagonal conjugation. The action of the fundamental group of Dr on Tr(G)isthewellknownactionofthe Artin braid group which we will define in the next section. This action commutes with the action of G via diagonal conjugation and hence induces a well defined action on Fr(G). From the definition of the action it is clear that the action of the G Artin braid group preserves the set of conjugacy classes Ci := τi of elements of τi ∈ τ. For an r-tuple of conjugacy classes C1,...,Cr we define the subset

N i(C1,...,Cr):={τ ∈Fr(G):∃σ ∈ Sr such that τi ∈ Ciσ for all i}, called the Nielsen class of C1,...,Cr. The braid group action on Fr(G)pre- Hin serves Nielsen classes which implies that connected components of r (G)are in parametrized by braid orbits on Nielsen classes. The subset H(G, C1, ..., Cr) ⊂ H in (G)r of G-curves X with g(X/G) = 0 is a union of components parametrized by N i(C1,...,Cr). By slight abuse of notation it is also called a Hurwitz space. Generally it is very diﬃcult to determine the number of braid orbits on Nielsen classes and hence not too much is known in general. There is the celebrated result of Clebsch, alluded to above, where he shows that if G = Sn and all elements of in τ are transpositions, then the corresponding Hurwitz space H(G, C1, ..., Cr) is connected. His result was recently generalized by Liu and Osserman [17]whoshow that if G = Sn and Ci is represented by gi where each gi is a single cycle of length |gi|,thenH(G, C1,...,Cr) is connected. On the other hand, Fried [3] showed that if G = An, g>0andallCi are represented by 3-cycles, then H(G, g0,C1,...,Cr), the space of G curves with g0 = g(X/G) and ramiﬁcation in classes C1,...,Cr, has one component if g0 =0and two components if g0 > 0. In the latter case the components are separated by the lifting invariant.

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 175

Table 1. Aﬃne Primitive Genus Zero Systems: Number of Components

Degree #Group # ramiﬁcation #comp’s #comp’s #comp’s #comp’s #comp’s #comp’s isom. types types r=3 r=4 r=5 r=6 r=7 total 8 2 50 29 17 9 3 1 59 16 14 86 441 69 18 2 - 530 32 1 24 169 3 - - - 172 64 14 34 69 - - - - 69 9 4 26 14 10 2 - - 26 27 6 20 32 2 - - - 34 81 4 5 6 - - - - 6 25 7 19 16 3 - - - 19 125 1 2 8 - - - - 8 49 1 2 6 - - - - 6 121 1 2 10 - - - - 10 Totals 55 270 800 104 29 5 1 939

Finally we mention the theorem of Conway-Parker, see [7], which shows that if the Schur multiplier of G is generated by commutators and the ramification involves all conjugacy classes of G sufficiently often, then the corresponding Hurwitz space is connected, hence defined over Q. Nevertheless deciding whether or not H(G, g0,C1,...,Cr) is connected is still an open problem, both theoretically and algorithmically. The algorithmic diffi- culties are due to the fact that the length of the Nielsen classes involved grows quickly. The package BRAID developed by Magaard, Shpectorov and Völklein [19] computes braid orbits algorithmically. This package is being upgraded by James, Magaard, Shpectorov [14] to generalize to the situation of orbits of the mapping class group on the fibres of the Hurwitz space H(G, g0,C1, ..., Cr)ofG-curves X with g(X/G)=g0. In this paper we introduce an algorithm which is designed to deal with long Nielsen classes. Our idea is to represent a Nielsen class as union of direct products of shorter classes, thereby enabling us to enumerate orbits of magnitude k3 where k is an upper bound for what our standard BRAID algorithm can handle. As an application of our algorithm we classify all braid orbits of Nielsen classes of primitive affine genus zero systems. That is to say that we find the connected components of H(G, C1, ..., Cr)ofG-curves X,whereG is primitive and affine with translation subgroup N and point stabilizer H, such that g(X/H)=0=g(X/G). Recall that G is primitive if and only if H acts irreducibly on the elementary abelian subgroup N via conjugation. Equivalently this means that G acts primitively on the right cosets of N via right multiplication and N acts regularly on them. We compute that there are exactly 939 braid orbits of primitive affine genus zero systems with G = 0. The distribution in terms of degree and number of branch points is given in Table 1. This completes the work of Neubauer on the affine case of the Guralnick-Thompson conjecture. Strictly speaking our new algorithm is not needed to settle the classification of braid orbits of Nielsen classes of primitive affine genus zero systems. However the problem is a good test case for our algorithm both as a debugging tool and as comparison for speed. Indeed our new algorithm shortens run times of BRAID by several orders of magnitude.

176 K. MAGAARD, S. SHPECTOROV, AND G. WANG

The paper is organized as follows. In Section 2 we describe our algorithm and illustrate it with an example that stresses the effectiveness. In Section 3 we discuss how we find all Hurwitz loci of affine primitive genus zero systems, displayed in tables at the end.

2. The Algorithm

We begin by recalling some basic deﬁnitions. The Artin braid group Br has the following presentation in terms of generators

{Q1,Q2,...,Qr−1} and relations QiQi+1Qi = Qi+1QiQi+1;

QiQj = Qj Qi for |i − j|≥2. r The action of Br on G ,orbraid action for short, is deﬁned for all i =1, 2,...,r−1 via: Q :(g ,...,g ,g ,...,g ) → (g ,...,g ,g−1 g g ,...,g ). i 1 i i+1 r 1 i+1) i+1 i i+1 r r Evidently the braid action preserves the product i=1 gi and the set of conju- { } G gacy classes C1,...,Cr where Ci := gi . If the classes Ci are pairwise distinct, then Br permutes the set {C1,...,Cr} like Sr permutes the set {1,...,r} where Qi induces the permutation (i, i + 1). Thus we see that Br surjects naturally onto (r) Sr with kernel B ,thepure Artin braid group . We note that the group B(r) is generated by the elements ··· 2 −1 ··· −1 Qij := Qj−1 Qi+1Qi Qi+1 Qj−1 −1 ··· −1 2 ··· = Qi Qj−2Qj−1Qj−2 Qi, for 1 ≤ i

Lemma 2.1. If {C1,...,Cr} are ordered as above and P is the corresponding o partition, then the set N i (C1,...,Cr):={τ ∈Ni(C1,...,Cr):τi ∈ Ci for all i} is preserved by BP . o This means that the orbits of BP on N i (C1,...,Cr) determine the components in of H (G, C1,...,Cr). As BP -orbits are shorter than the corresponding Br-orbits by a factor of [Sr : SP ], this is a signiﬁcant advantage. For the record we note:

Lemma 2.2. The set of Qij’s such that i and j lie in diﬀerent blocks of P together with the Qi’s such that i and i +1 lieinthesameblockofP is a set of generators of BP .

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 177

2.1. Nodes. As we noted in the introduction, Nielsen classes tend to be very large and thus we need to find ways to handle them effectively. Our algorithm achieves efficiency by interpreting tuples as elements of a Cartesian product. For o this to be compatible with the action of BP on N i (C1,...,Cr), or equivalently, with the action of BP × G on o T (C1,...,Cr):={(τ1,...,τr) ∈Tr(G):τi ∈ Ci for all i}, we need to make some additional definitions. Let 1

Lk := Qi : i ≤ k − 1∩BP ,

Rk := Qi : i ≥ k +1∩BP . o Clearly [Lk,Rk] = 1 and every BP -orbit on N i (C1,...,Cr) is a union of o (Lk × Rk)-orbits. Equivalently every BP × G-orbit on T (C1,...,Cr) is a union of (Lk × Rk × G)-orbits, which we call nodes. We refer to k as the level. Typically we choose k to be close to r/2. If (g1,...,gr) is a representative of a level k node, then we split it into its head (g1,...,gk) and its tail (gk+1,...,gr). Since our package BRAID works with product 1 tuples we will identify the head and the tail with) the product) 1 tuples (g1,...,gk,x)and(y, gk+1,...,gr) respectively, where k r −1 y = i=1 gi, x = i=k+1 gi.Wenotethatx = y and that the actions of Lk and G Rk centralize x and y. Hence the conjugacy class Cx := x is an invariant of the node, which we call the nodal type. The following is clear. Lemma 2.3. For every node the heads of all tuples in the node form an orbit under Lk × G. Similarly the tails form an Rk × G-orbit. We refer to the orbits above as the head (respectively, tail) orbit of the node. With the notation as above we see that the head orbit is of ramification type (C1,...,Ck,Cx) and the tail orbit is of ramification type (Cy,Ck+1 ...,Cr). This observation allows us to determine all possible head and tail orbits independently, using BRAID. Note that subgroups generated by the head or tail may be proper in G. Accordingly, we give the following definitions. For a ramification type {C1,...,Cr}, the partion P as above, and conjugacy classes C and D of G,we define k Lk,C := {(g1,...,gk,x):gi ∈ Ci,x∈ C and1=( gi)x}, i=1

k Rk,D := {(y, gk+1,...,gr):y ∈ D, gi ∈ Ci and 1 = y( gi)}. i=1

Note that Lk ×G acts on Lk,C for all choices of C,andRk ×G acts on Rk,D for all choices of D. By slight abuse of terminology we call (Lk × G)-orbits of ∪C Lk,C heads and (Rk × G)-orbits of ∪DRk,D tails. Clearly, for each node, its head orbit is among the heads and its tail orbit is among the tails. Furthermore the node is a subset of the Cartesian product of its head and its tail. We can now restate our task of ﬁnding nodes as follows. We need to ﬁnd pairs of heads and tails which can correspond to nodes and then identify nodes within the Cartesian product of the head and the tail.

178 K. MAGAARD, S. SHPECTOROV, AND G. WANG

The first of these tasks is achieved with the following definition. A head in −1 −1 Lk,C matches atailinRk,D if D = C := {x : x ∈ C}. Since matching is specified entirely in terms of C and D, we note that either every head in Lk,C matches every tail in Rk,D,orLk,C ×Rk,D contains no matching pairs. The head and tail of a node must necessarily be matching. Experiments show that most pairs of matching heads and tails lead to nodes. So no further restrictions are necessary for our algorithm. −1 Suppose now that H⊂Lk,C and T⊂Rk,D match; i.e. D = C .Ourtask now is to find all nodes in H×T. There are several issues that we need to address. First of all, a pair of representatives (g1,...,gk,x) ∈Hand (y, gk+1,...,gr) ∈R can only give a representative of a node if y = x−1. Therefore H×T is not a union of nodes; in fact most pairs of representative tuples do not work. We address this as follows. Let us select a particular element x0 ∈ C. A natural choice for x0 is, for example, the minimal element of C with respect to the ordering defined in GAP −1 H T H { ∈H [8]. Let y0 = x0 .For and as above, we define 0 := (g1,...,gk,x) : x = x0} and T0 := {(y, gk+1,...,gr) ∈T : y = y0}.WecallH0 and T0 the shadows of H and T .

Lemma 2.4. The shadows H0 and T0 are orbits for Lk × CG(x0) and Rk × CG(x0) respectively.

Our ﬁrst issue is now resolved as the representatives of H0 and T0 automatically combine to give a product 1 tuple. Furthermore for a node N of) type C we can N N { ∈N r } similarly deﬁne the shadow of to be 0 := (g1,...,gr) : i=k+1 gi = x0 .

Lemma 2.5. The shadow N0 is an orbit for Lk × Rk × CG(x0) and furthermore it fully lies in H0 ×T0 where H0 and T0 are the shadows of the head and tail of N . Thus we may work exclusively with shadows of heads, tails and nodes. Our second issue is that combining representatives of matching head and tail shadows may not produce a tuple in T (C1,...,Cr), because it may not generate G. We deﬁne prenodes as Lk × Rk × G-orbits on r {τ ∈ C1 ×···×Cr : τi =1}. i=1 Clearly every node is a prenode. Our terminology, head, tail, type and shadow, extends to prenodes in the obvious way.

Lemma 2.6. If H and T are matching heads and tails, then H0 ×T0 is a disjoint union of prenode shadows.

So now our task is to identify all prenodes within H0 ×T0, that is to ﬁnd a representative for each prenode. To achieve this, we work at the level of Lk-and Rk-orbits of H0 and T0 respectively. Let Oh ⊂H0 be an Lk-orbit and Ot ⊂T0 be and Rk-orbit. We deﬁne normalizers O { ∈ c ∈O ∈O } NCG(x0)( h):= c CG(x0):τ h for all τ h

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 179 and O { ∈ c ∈O ∈O} NCG(x0)( t):= c CG(x0):σ t for all σ t . Because the G-action commutes with that of Lk and Rk, it suﬃces to check the conditions above for just a single τ ∈Oh and a single σ ∈Ot, respectively.

Proposition 2.7. If Oh ⊂H0 is an Lk-orbit and Ot ⊂T0 is an Rk-orbit, then the prenode shadows in H0 ×T0 are in one-to-one correspondence with the double cosets O \ O NCG(x0)( h) CG(x0)/NCG(x0)( t). If {d1,...,ds} is a set of double coset representatives, then a set of representatives { di di ≤ ≤ } for the prenodes can be chosen as (g1,...,gk,gk+1,...,gr ):1 i s ,where (g1,...,gk,x0) and (y0,gk+1,...,gr) are arbitrary representatives of Oh and Ot, respectively.

Proof. Let X be the set of Lk-orbits of H0 and Y be the set of Rk-orbits of T0. X Y O Clearly CG(x0) acts transitively on and on with point stabilizers NCG(x0)( h) O and NCG(x0)( t) respectively. Furthermore, the prenode shadows correspond to the CG(x0)-orbits on X×Y. The latter correspond to the double cosets as above. So to construct all nodes we proceed as follows:

Algorithm: Find all level k nodes

• Input: A group G, conjugacy classes C1,...,Cr and an integer 1 ≤ k ≤ r. • For each type C: – Set D := C−1 and find all heads and tails by using BRAID [19]. – From each head and tail select its shadow. – For each pair of head and tail shadows compute the normalizers and the double coset representatives as in Proposition 2.7. – For each prenode check whether or not its representative generates G. Store the prenodes that pass this test as nodes. • Output all nodes. Nodes are sorted by their type, head, tail, and double coset representative. We close this subsection with the observation that the sum of the lengths of the nodes is computable at this stage. This means that we have calculated |T (C1,...,Cr)| by a method different from that of Staszewski and Völklein [25]. 2.2. Edges. Our next step is to define a graph on our set of nodes whose connected components correspond to the braid orbits on the Nielsen class N i(C1,...,Cr).

Definition 2.8. Let Γk(C1,...,Cr) be the graph whose vertices are the level k nodes of N i(C1,...,Cr). We connect two nodes N1 and N2 by an edge if and only if there exists a tuple τ ∈ N1 and an element Q ∈ BP such that τQ ∈N2.

We remark that it is clear that the connected components of Γk(C1,...,Cr) are complete graphs and are in one-to-one correspondence with the braid orbits on the Nielsen class N i(C1,...,Cr).

Our algorithm for connecting vertices is as follows. Let S be the set of generators of BP as in Lemma 2.1 minus those which are contained in Lk × Rk.Foreach node N we select a random tuple τ ∈N and apply a randomly chosen generator

180 K. MAGAARD, S. SHPECTOROV, AND G. WANG

Q ∈ S to it. Using the head and tail of τQ we find the node N which contains it. If N = N we record the edge. We repeat this until we have s successes at N . Note that this does not mean that we find s distinct neighbors for N .Ifaftera pre-specified number of tries t we have no successes then we conclude that N is an isolated node; i.e. it is a BP -orbit. Once we have gone through all nodes, we obtain a subgraph Γ of Γk(C1,...,Cr). We now find the connected components of Γ and claim that these are likely to be identical to those of Γk(C1,...,Cr). Clearly if Γ is connected, then so is Γk(C1,...,Cr). Hence in this case our conclusion is deterministic. In other cases our algorithm is Monte-Carlo. Based on our experiments, the situation where Γk(C1,...,Cr) is connected is the most likely outcome. It is interesting that even for small values of s we tend to get that Γ is connected whenever Γk(C1,...,Cr) is connected. Also t does not need to be large because if N is not isolated then almost any choice of τ and Q will produce an edge. This, together with the way we represent tuples as products of heads and tails, makes this part of the algorithm very fast. Here is the formal description of the second part of the algorithm.

Algorithm: Finding the braid orbits

• Input: The k-nodes of N i(C1,...,Cr) arranged in terms their type, head, tail and double coset representative. • Initialize the edge set E to the empty set. • For each node N : – Setcounterscanddto0. – Generate a random tuple τ from N by selecting random head and tail. – Apply a randomly chosen Q ∈ S to τ. – Identify the node N containing τQ via its head and tail. – If N = N ,then ∗ Set c to c+1 and set d to d+1. ∗ Add the edge (N , N )toE unless it is already known. – Else, ∗ Set c to c+1. – Repeat this until either d = s or d =0andc = t. • Determine and output the connected components of the graph Γ whose vertices are the input nodes and whose edge set is E.

2.3. Type {1G} nodes. During the development of the algorithm we noticed that a signiﬁcant number of nodes are of type C = {1G}, often more than half of all nodes. This can be explained by the fact that CG(1G)=G is largest among all classes. Furthermore, all computations for these nodes are substantially slower than for nodes of types not equal to C. The next lemma gives a criterion when such nodes can be disregarded.

Lemma 2.9. Suppose N is a prenode of type {1G} and τ is its representative. Let H and T be the subgroups generated by the head and tail of τ, respectively. If H and T do not centralize each other, then the BP -orbit containing N contains also aprenodeN of type not equal to {1G}.

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 181

Proof. Let τ =(g1,...,gr). Then H = g1,...,gk and T = gk+1,...,gr. We can) also take a diﬀerent set of generators for H, namely, the partial products k hi = j=i gj , i =1,...,k.SinceH and T do not centralize each other, some hs does not commute with some gt,wheres ≤ k and t>k. It is now straightforward to see that that the pure braid Qst takes τ to a tuple, whose type is diﬀerent from the type of τ.

This criterion is in fact exact. Indeed, it is clear that if H and T centralize each other then no pure braid (and more generally, no braid that preserves head and tail classes) can change the type. So the type within the BP -orbit can change only if the same conjugacy class is present in the head and in the tail. However, a class that is present both in the head and in the tail must be central, and so the type still cannot change. Hence when H and T centralize each other then then the type (whether identity or not) remains constant on the entire BP orbit. When the prenode N is a node, we have G = H, T, and so the condition in the lemma fails very rarely. Thus, in most cases we need not consider nodes of type {1G}. This turns out to be a signiﬁcant computational advantage.

2.4. An Example. Let G =AGL4(2), the group of affine linear transforma- F4 tions, acting on the 16 points of 2; the vector space of dimension 4 over the field of 2 elements. G has a unique conjugacy class of involutions whose elements have F4 precisely 8 fixed points in their action on 2 (we call this class 2A) and another F4 whose elements have exactly 4 fixed points in their action on 2 (we call this second class 2B). We consider the ramification type C¯ =(2A, 2A, 2A, 2B,2B,2B). The structure constant for C¯ is 21, 267, 671, 040; i.e.

|T (2A, 2A, 2A, 2B,2B,2B)|≤21, 267, 671, 040.

This yields that an upper bound for the size of the corresponding BP -orbit is 65, 934. The available version of our package BRAID ﬁnds an orbit of this size within minutes. However, verifying that there is only one generating orbit takes days. This is due to the fact that BRAID spends most of its time searching for non-generating tuples in order to account for the full structure constant. Staszewski and V¨olklein [25] provided us with the function NumberOfGeneratingNielsenTuples which often helps to get around this problem. However, in this example the function runs out of memory on a 64G computer. On the other hand, after splitting C¯ across the middle into (2A, 2A, 2A, C)and(D, 2B,2B,2B) we compute heads and tails within minutes.

Table 2. Time spent on generating heads and tails

half total number of orbits time spent type with the most orbits (2A,2A,2A,C) 155 2 mins (2A,2A,2A,2A) (D,2B,2B,2B) 619 10 mins (4B,2B,2B,2B)

182 K. MAGAARD, S. SHPECTOROV, AND G. WANG

The step of constructing all nodes also takes little time. The group G has 24 non-identity conjugacy classes and hence we have 24 types of nodes.

Table 3. Results from the function AllMatchingPairs

number of total pairs most pairs least pairs number of types with no pairs 903 3A 4E 11

As shown in the table, our graph Γk(2A, 2A, 2A, 2B,2B,2B) has 903 vertices. Drawing edges and checking that the graph Γ is connected took less than 5 minutes. TheresultisthatΓk(2A, 2A, 2A, 2B,2B,2B) is connected, which means that the Hurwitz space H(AGL4(2), 2A, 2A, 2A, 2B,2B,2B) is connected.

3. Genus Zero Systems and the Guralnick-Thompson Conjecture We now come to our main application. We recall some background. Suppose X is a compact, connected Riemann surface of genus g,andφ : X → P 1C is meromorphic of degree n.LetB := {x ∈ P 1C : |φ−1(x)|

We are interested in the structure of the monodromy group when the genus of X is less than or equal to two and φ is indecomposable in the sense that there do not 1 exits holomorphic functions φ1 : X → Y and φ2 : Y → P C of degree less than the degree of φ such that φ = φ1 ◦ φ2. The condition that X is connected implies that Mon(X, φ) acts transitively on F , whereas the condition that φ is indecomposable implies that the action of Mon(X, φ)onF is primitive. Our first question relates to a conjecture made by Guralnick and Thompson [12] in 1990. By cf(G) we denote the set of isomorphism types of the composition factors of G. In their paper [12] Guralnick and Thompson defined the set ∗ E (g)=( cf(Mon(X, φ))) \{An, Z/pZ : n>4 ,paprime} (X,φ) where X ∈M(g), the moduli space of curves of genus g,andφ : X −→ P 1(C)is meromorphic. They conjectured that E ∗(g) is finite for all g ∈ N. Building on work of Guralnick-Thompson [12], Neubauer [23], Liebeck-Saxl [15], and Liebeck-Shalev [16], the conjecture was established in 2001 by Frohardt and Magaard [4].

The set E ∗(0) is distinguished in that it is contained in E ∗(g) for all g.Moreover the proof of the Guralnick-Thompson conjecture shows that it is possible to compute E ∗(0) explicitly. The idea of the proof of the Guralnick-Thompson conjecture is to employ Riemann’s Existence Theorem to translate the geometric problem to a problem in group theory as follows. If φ : X → P 1C is as above with branch points 1 B = {b1,...,br}, then the set of elements αi ∈ π1(P C\B,b0), each represented by 1 a simple loop around bi, forms a canonical set of generators of π1(P C \ B,b0). Let

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 183 ∼ g denote the genus of X.Wedenotebyσi the image αi in Mon(X, φ) ⊂ SF = Sn. Thus we have that Mon(X, φ)=σ1,...,σr⊂Sn and that r Πi=1σi =1. Moreover the conjugacy class of σi in Mon(X, φ) is uniquely determined by φ. Recall that the index of a permutation σ ∈ Sn is equal to the minimal number of factors needed to express σ as a product of transpositions. The Riemann-Hurwitz formula asserts that r 2(n + g − 1) = ind(σi). i=1 Definition . ∈ 3.1 If τ1,...,τr Sngenerate a transitive subgroup G of Sn such r − r ∈ N that Πi=1τi =1and 2(n + g 1) = i=1 ind(τi) for some g , then we call (τ1,...,τr) a genus g system and G a genus g group. We call a genus g system (τ1,...,τr) primitive if the subgroup of Sn it generates is primitive.

If X and φ areasabove,thenwesaythat(σ1,...,σr)isthegenusg system induced by φ. Theorem 3.2 (Riemann Existence Theorem). For every genus g system 1 (τ1,...,τr) in Sn, there exists a Riemann surface Y and a cover φ : Y −→ P C with branch point set B such that the genus g system induced by φ is (τ1,...,τr).

Definition 3.3. Two covers (Yi,φi), i =1, 2 are equivalent if there exist holomorphic maps ξ1 : Y1 −→ Y2 and ξ2 : Y2 −→ Y1 which are inverses of one another, such that φ1 = ξ1 ◦ φ2 and φ2 = ξ2 ◦ φ1. 1 The Artin braid group acts via automorphisms on π1(P C\B,b0). We have that 1 all sets of canonical generators of π1(P C \ B,b0) lie in the same braid orbit. Also the group G acts via diagonal conjugation on genus g generating sets. The diagonal and braiding actions on genus g generating sets commute and preserve equivalence of covers; that is, if two genus g generating sets lie in the same orbit under either the braid or diagonal conjugation action, then the corresponding covers given by Riemann’s Existence Theorem are equivalent. We call two genus g generating systems braid equivalent if they are in the same orbit under the group generated by the braid action and diagonal conjugation. We have the following result, see for example [26], Proposition 10.14. Theorem 3.4. Two covers are equivalent if and only if the corresponding genus g systems are braid equivalent.

Suppose now that (τ1,...,τr) is a primitive genus g system of Sn.Express) each τi as a product of a minimal number of transpositions; i.e. τi := j σi,j . The system (σ1,1,...,σr,s) is a primitive genus g system generating Sn consisting of precisely 2(n + g − 1) transpositions. By a famous result of Clebsch, see Lemma 10.15 in [26], any two primitive genus g systems of Sn are braid equivalent. Thus we see that every genus g system can be obtained from one of Sn which consists entirely of transpositions. Thus, generically we expect primitive genus g systems in Sn to generate either ∗ An or Sn. We deﬁne P E (g)n,r to be the braid equivalence classes of genus g systems (τ1,...,τr)inSn such that G := τ1,...,τr is a primitive subgroup of

184 K. MAGAARD, S. SHPECTOROV, AND G. WANG

∗ Sn with An ∩ G = An. We also deﬁne GE (g)n,r to be the conjugacy classes of ∗ primitive subgroups of Sn which are generated by a member of P E (g)n,r. We also deﬁne ∗ ∗ P E (g):=∪(n,r)∈N2 P E (g)n,r, and similarly ∗ ∗ GE (g):=∪(n,r)∈N2 GE (g)n,r. We note that the composition factors of elements of GE ∗(g) are elements of E ∗(g). While our ultimate goal is to determine P E ∗(g)whereg ≤ 2, we focus here on the case g =0. Our assumption that G =Mon(X, φ)actsprimitivelyonF isastrongoneand allows us to organize our analysis along the lines of the Aschbacher-O’Nan-Scott Theorem exactly as was done in the original paper of Guralnick and Thompson [12]. We recall the statement of the Aschbacher-O’Nan-Scott Theorem from [12]

Theorem 3.5. Suppose G is a ﬁnite group and H is a maximal subgroup of G such that Hg =1. g∈G Let Q be a minimal normal subgroup of G,letL be a minimal normal subgroup of Q,andletΔ={L = L1,L2,...,Lt} be the set of G-conjugates of L.Then G = HQ and precisely one of the following holds: (A) L is of prime order p. ∗ ∼ (B) F (G)=Q × R where Q = R and H ∩ Q =1. (C1) F ∗(G)=Q is nonabelian, H ∩ Q =1. (C2) F ∗(G)=Q is nonabelian, H ∩ Q =1= L ∩ H. ∗ (C3) F (G)=Q is nonabelian, H ∩ Q = H1 ×···×Ht, where Hi = H ∩ Li =1 , 1 ≤ i ≤ t.

The members of GE ∗(0) that arise in case (C2) were determined by Aschbacher ∗ [1]. In all such examples Q = A5 ×A5. Shih [24] showed that no elements of GE (0) arise in case (B) and Guralnick and Thompson [12] showed the same in case (C1). Guralnick and Neubauer [11] showed that the elements of GE ∗(0) arising in case (C3) all have t ≤ 5. This was strengthened by Guralnick [9]tot ≤ 4andthe ∗ action of Li on the cosets of Hi is a member of GE (0). In case (C3), where Li is of Lie type of rank one, all elements of GE ∗(0) and GE ∗(1) were determined by Frohardt, Guralnick, and Magaard [5], moreover they show that t ≤ 2. In [6] Frohardt, Guralnick, and Magaard showed that if t =1,Li is classical and Li/Hi is a point action, then n =[Li : Hi] ≤ 10, 000. That result together with the results of Aschbacher, Guralnick and Magaard [2] show that if t =1andLi is classical then ∗ [Li : Hi] ≤ 10, 000. In [13] Guralnick and Shareshian show that G ∈ GE (0)n,r = ∗ ∗ if r ≥ 9. Moreover they show that if G ∈ GE (0)n,r with F (G) is alternating of degree d

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 185

In case (A) above, the aﬃne case, we have that F ∗(G) is elementary abelian and it acts regularly on F . Case (A) was ﬁrst considered by Guralnick and Thompson [12]. Their results were then strengthened by Neubauer [23]. After that, case (A) has not received much attention, which is in part due to its computational complexity. The starting point for our investigations is Theorem 1.4 of Neubauer [23].

Theorem 3.6 (Neubauer). If F ∗(G) is elementary abelian of order pe and X = P 1C, then one of the following is true: (1) G =1and 1 ≤ e ≤ 2 (2) p =2and 2 ≤ e ≤ 8, (3) p =3and 2 ≤ e ≤ 4, (4) p =5and 2 ≤ e ≤ 3, (5) p =7or 11 and e =2.

The groups G with G =1and1≤ e ≤ 2 are Frobenius groups and are well understood. Thus we concentrate on the aﬃne groups of degrees

{8, 16, 32, 64, 128, 256, 9, 27, 81, 25, 125, 49, 121}.

Our results are recorded in the tables below. These tables were calculated in several steps which we will now outline.

Algorithm: Enumerating Primitive Genus Zero Systems of Aﬃne Type

• Look up the primitive aﬃne groups G of degree pe using the GAP function AllPrimitiveGroups(DegreeOperation, pe). • For every group G, calculate conjugacy class representatives and permutation indices. • Using the function RestrictedPartions, calculate all possible ramiﬁca- tion types satisfying the genus zero condition of the Riemann-Hurwitz formula. • Fe ∗ Let V = p = F (G). For each conjugacy class representative x calculate dimV (x) and use Scott’s Theorem to eliminate those types from the previous step which can not possibly act irreducibly on V ; i.e. can not generate a primitive group. • Calculate the character table of G and discard those types for which the class structure constant is zero. • For each of the remaining types of length four or more use the old version of BRAID, if possible, or else run our new algorithm. For tuples of length three determine orbits via double cosets.

A few remarks are in order. First of all, the use of Scott’s theorem above is best done in conjunction with a process called translation [4]. In fact, translation was crucial in handling certain types arising in degrees 128 and 256. Secondly, using BRAID on types of length 3 is meaningless as every pure braid orbit has length one. Instead, we can compute possible generating triples using double cosets of centralizers.

186 K. MAGAARD, S. SHPECTOROV, AND G. WANG

Table 4. The Genus Zero Systems for Aﬃne Primitive Groups of Degree 8

group ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit AΓL(1, 8) (3B,3B,6B) 2 1 (3A,3B,7B) 1 1 (3A,3B,7A) 1 1 (3A,3A,6A) 2 1 ASL(3, 2) (4B,3A,7B) 2 1 (4B,3A,7A) 2 1 (4B,3A,6A) 4 1 (4B,3A,4C) 2 1 (4B,4B,7B) 1 1 (4B,4B,7A) 1 1 (4B,4B,6A) 2 1 (4B,4B,4C) 4 1 (2C,4B,7B) 1 1 (2C,4B,7A) 1 1 (2B,7B,7B) 1 1 (2B,7A,7A) 1 1 (2B,6A,7B) 1 1 (2B,6A,7A) 1 1 (2B,3A,3A,3A) 1 120 (2B,4C,7B) 1 1 (2B,4C,7A) 1 1 (2B,4B,3A,3A) 1 84 (2B,4B,4B,3A) 1 66 (2B,4B,4B,4B) 1 36 (2B,2C,3A,3A) 1 30 (2B,2C,4B,3A) 1 24 (2B,2C,4B,4B) 1 24 (2B,2B,3A,7B) 1 21 (2B,2B,3A,7A) 1 21 (2B,2B,3A,6A) 1 30 (2B,2B,3A,4C) 1 24 (2B,2B,4B,7B) 1 14 (2B,2B,4B,7A) 1 14 (2B,2B,4B,6A) 1 24 (2B,2B,4B,4C) 1 24 (2B,2B,2C,7B) 1 7 (2B,2B,2C,7A) 1 7 (2B,2B,2B,3A,3A) 1 864 (2B,2B,2B,4B,3A) 1 648 (2B,2B,2B,4B,4B) 1 456 (2B,2B,2B,2C,3A) 1 216 (2B,2B,2B,2C,4B) 1 192 (2B,2B,2B,2B,7B) 1 147 (2B,2B,2B,2B,7A) 1 147 (2B,2B,2B,2B,6A) 1 216 (2B,2B,2B,2B,4C) 1 192 (2B,2B,2B,2B,2B,3A) 1 6480 (2B,2B,2B,2B,2B,4B) 1 4800 (2B,2B,2B,2B,2B,2C) 1 1680 (2B,2B,2B,2B,2B,2B,2B) 1 48960

Table 5. The Genus Zero Systems for Primitive Groups of Degree 25 and 125

group ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit 52 :3 (3B,3B,3B) 8 1 (3A,3A,3A) 8 1 52 :6 (2A,3B,6B) 4 1 (2A,3A,6B) 8 1 2 5 : S3 (2A,3A,10D) 1 1 (2A,3A,10C) 1 1 (2A,3A,10B) 1 1 (2A,3A,10A) 1 1 52 : D(2 ∗ 6) (2A,2B,2C,3A) 1 12 52 : D(2 ∗ 4) : 2 (2A,2C,2D,4C) 1 1 (2A,2C,2D,4A) 1 1 52 : O +(2, 5) (2C,4F,8A) 1 1 (2C,4E,8B) 1 1 2 5 :((Q8 :3)2) (2B,3B,12B) 1 1 (2B,3B,12A) 1 1 (2B,3A,12D) 1 1 (2B,3A,12C) 1 1 2 5 :((Q8 :3)4) (4F,3A,4E) 1 1 (4D,3A,4G) 1 1 ASL(2, 5) : 2 (2B,3A,20D) 1 1 (2B,3A,20C) 1 1 (2B,3A,20B) 1 1 (2B,3A,20A) 1 1 3 2 5 :4 : S3 (2B,3A,8B) 4 1 (2B,3A,8A) 4 1

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 187

Table 6. The genus zero system of AGL(4, 2) Part 1

Using BRAID ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit (2B,5A,15B) 1 1 (2B,2B,3B,7B) 1 7 (2B,5A,15A) 1 1 (2B,2B,3B,7A) 1 7 (2B,5A,14B) 1 1 (2B,2B,4B,5A) 1 80 (2B,5A,14A) 1 1 (2B,2B,4B,6C) 1 96 (2B,6C,15B) 1 1 (2B,2B,6A,5A) 1 120 (2B,6C,15A) 1 1 (2B,2B,6A,6C) 1 108 (2B,6C,14B) 1 1 (2B,2D,4D,5A) 1 90 (2B,6C,14A) 1 1 (2B,2D,4D,6C) 1 78 (2D,4F,15B) 1 1 (2B,2D,3A,5A) 1 60 (2D,4F,15A) 1 1 (2B,2D,3A,6C) 1 72 (2D,6A,15B) 3 1 (2B,2B,2B,2D,5A) 1 650 (2D,6A,15A) 3 1 (2B,2B,2B,2D,6C) 1 648 (2D,6A,14B) 2 1 (4B,4B,4D) 12 1 (2D,6A,14A) 2 1 (6A,4B,4D) 18 1 (2B,2D,2D,15B) 1 15 (6A,4B,4B) 32 1 (2B,2D,2D,15A) 1 15 (6A,6A,4F) 12 1 (2B,2D,2D,14B) 1 14 (6A,6A,4B) 52 1 (2B,2D,2D,14A) 1 14 (6A,6A,6A) 72 1 (4D,4F,5A) 6 1 (2B,2D,4B,4F) 1 96 (4D,4F,6C) 4 1 (2B,2D,4B,4B) 1 216 (4D,3B,7B) 1 1 (2B,2D,6A,4F) 1 84 (4D,3B,7A) 1 1 (2B,2D,6A,4B) 1 312 (4D,4B,5A) 6 1 (2B,2D,6A,6A) 1 414 (4D,4B,6C) 12 1 (2B,4F,4D,3B) 1 24 (4D,6A,5A) 18 1 (2B,3A,4D,3B) 1 30 (4D,6A,6C) 12 1 (2B,3A,3A,3B) 1 24 (3A,4F,5A) 2 1 (2D,2D,4D,4F) 1 88 (3A,4F,6C) 4 1 (2D,2D,4D,4B) 1 192 (3A,6A,5A) 10 1 (2D,2D,4D,6A) 1 336 (3A,6A,6C) 12 1 (2D,2D,3A,4F) 1 56 (2B,2B,4F,5A) 1 30 (2D,2D,3A,6A) 1 216 (2B,2B,4F,6C) 1 30 (2B,2B,2B,4D,3B) 1 610 (2B,2B,2B,3A,3B) 1 216 (2B,2B,2D,2D,4F) 1 576

Table 7. The genus zero system of AGL(4, 2) Part 2

Using Matching Algorithm ramiﬁcation #of #of orbit ramiﬁcation #of #of orbit type nodes orbits length type nodes orbits length (2B,2B,2D,2D,6A) 170 1 2448 (2B,2B,2B,2B,2B,3B) 107 1 1782 (2B,2D,2D,2D,4D) 63 1 1920 (2B,2B,2D,2D,4B) 151 1 1920 (2B,2B,2B,2D,2D,2D) 903 1 15168 (2B,2D,2D,2D,3A) 56 1 1512

188 K. MAGAARD, S. SHPECTOROV, AND G. WANG

Table 8. Genus zero systems for other primitive aﬃne groups of degree 16

Group ramiﬁcation #of largest Group ramiﬁcation #of largest type orbits orbit type orbits orbit 4 4 2 : D(2 ∗ 5) (2A,5B,4C) 1 1 2 .A6 (2B,5B,5B) 4 1 (2A,5B,4B) 1 1 (2B,5A,5A) 4 1 (2A,5B,4A) 1 1 (3A,3B,5B) 2 1 (2A,5A,4C) 1 1 (3A,3B,5A) 2 1 (2A,5A,4B) 1 1 (2B,2B,2B,5B) 2 30 (2A,5A,4A) 1 1 (2B,2B,2B,5A) 2 30 (2A,2A,2A,4C) 1 12 (2B,2B,3A,3B) 1 36 (2A,2A,2A,4B) 1 12 (2B,2B,2B,2B,2B) 2 864 (2A,2A,2A,4A) 1 12 4 (A4 × A4):2 (2A,6B,6C) 1 1 2 : S5 (2C,5A,12A) 1 1 (2A,6A,6D) 1 1 (2C,5A,8A) 1 1 (2A,2A,3E,3A) 1 1 (2E,6C,12A) 1 1 (2A,2A,3D,3B) 1 1 (2E,6C,8A) 1 1 (24 :5).4 (2A,4B,8B) 1 1 (2E,4E,12A) 1 1 (2A,4A,8A) 1 1 (2C,2E,2E,12A) 1 6 4 2 : S3 × S3 (2E,6B,6C) 3 1 (2C,2E,2E,8A) 1 8 (2D,2E,2E,6C) 1 12 (2D,6C,5A) 3 1 (2C,2E,2E,6B) 1 12 (2D,4E,5A) 3 1 (2C,2D,2E,6A) 1 3 (2C,2E,2D,5A) 1 15 (2C,2D,2E,2E,2E) 1 48 (2E,2E,2D,6C) 1 18 24.32 :4 (2C,4D,8B) 2 1 (2E,2E,2D,4E) 1 24 (2C,4C,8A) 2 1 (2C,2E,2E,2E,2D) 1 120 (3A,4C,4D) 3 1 4 (S4 × S4):2 (2E,6B,8A) 1 1 2 : A5 (2C,5A,5B) 3 1 (2C,4F,12A) 1 1 (2C,6C,5B) 1 1 (2C,6C,8A) 1 1 (2C,6C,5A) 1 1 (2E,2C,2D,8A) 1 4 (2C,6B,5B) 1 1 (2E,2C,2C,12A) 1 2 (2C,6B,5A) 1 1 (2F,4F,6B) 3 1 (2C,6A,5B) 1 1 (2E,2C,2F,6B) 1 6 (2C,6A,5A) 1 1 (2E,2C,3A,4F) 1 6 (2C,2C,2C,5B) 1 30 (2C,2D,2F,4F) 1 12 (2C,2C,2C,5A) 1 30 (2C,2C,2F,6C) 1 6 (2C,2C,2C,6C) 1 18 representatives for the prenodes. (2E,2E,2C,2C,3A) 1 12 (2C,2C,2C,6B) 1 18 (2E,2C,2C,2D,2F) 1 24 (2C,2C,2C,6A) 1 18 (2C,2C,2C,2C,2C) 1 576 4 AΓL(2, 4) (2C,4C,5A) 1 1 2 .A7 (2B,4A,14B) 2 1 (2C,4C,15A) 1 1 (2B,4A,14A) 2 1 (3B,4C,6C) 4 1 (2B,7B,6A) 2 1 (2B,2C,3B,4C) 1 20 (2B,7A,6A) 2 1 ASL(2, 4) : 2 (2C,5A,6A) 2 1 (2B,5A,7B) 2 1 (2B,6A,6A) 2 1 (2B,5A,7A) 2 1 (2B,2C,2C,5A) 1 10 (3B,3A,7B) 1 1 (2B,2B,2C,6A) 1 12 (3B,3A,7A) 1 1 (4A,4A,3A) 2 1 (3B,4A,6A) 6 1 (2B,2B,2B,2C,2C) 1 80 (3B,4A,5A) 10 1 4 2 .S6 (2B,2B,3B,5A) 1 10 (2B,2B,2B,7B) 2 21 (6B,4D,3B) 2 1 (2B,2B,2B,7A) 2 21 (6B,6B,3B) 6 1 (4A,4A,4A) 24 1 (2B,2D,4D,3B) 1 12 (2B,2B,3B,4A) 1 192 (2B,2D,6B,3B) 1 24 (2B,2B,2D,2D,3B) 1 108

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 189

Table 9. The Genus Zero Systems for Aﬃne Primitive Groups of Degree 32

group ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit ASL(5, 2) (2D,3B,31A) 1 1 (2D,8A,6F) 16 1 (2D,3B,31B) 1 1 (2D,12A,6F) 16 1 (2D,3B,31C) 1 1 (2D,6E,6F) 22 1 (2D,3B,31E) 1 1 (2D,5A,6F) 18 1 (2D,3B,31D) 1 1 (4A,4A,6F) 12 1 (2D,3B,31F) 1 1 (4A,3B,8A) 12 1 (2D,3B,30A) 1 1 (4A,3B,12A) 12 1 (2D,3B,30B) 1 1 (4A,3B,6E) 24 1 (2D,4J,21B) 2 1 (4A,3B,5A) 18 1 (2D,4J,21A) 2 1 (4I,3B,4J) 18 1 (6C,3B,4J) 12 1 (2D,2D,2D,6F) 1 720 (2B,2D,3B,4J) 1 84 (2D,2D,4A,3B) 1 624

Table 10. The Genus Zero Systems for Affine Primitive Groups of Degree 64 group ramification #of largest ramification #of largest type orbits orbit type orbits orbit 6 2 2 :3 : S3 (2E,3F,12A) 1 1 (2E,6C,12B) 1 1 26 :7:6 (2E,3B,12B) 1 1 (2E,3A,12A) 1 1 6 2 2 :(3 :3):D8 (2G,4D,6D) 3 1 (2F,4D,6E) 3 1 6 2 2 :(3 :3):SD16 (2E,4G,8D) 1 1 (2E,4G,8B) 1 1 26 :(6× GL(3, 2)) (2F,3C,14A) 1 1 (2F,3C,14B) 1 1 6 2 : S7 (2I,4N,6K) 4 1 (2I,4D,7A) 3 1 6 2 :(GL(2, 2) S3) (2L,4N,6I) 4 1 26 :(GL(3, 2) 2) (2J, 4Q, 14H) 1 1 (2J, 4Q, 14G) 1 1 (2I, 2J, 2J, 7B) 1 1 (2I, 2J, 2J, 7A) 1 1 6 2 2 :7 : S3 (2C,3A,14C) 1 1 (2C,3A,14D) 1 1 (2C,3A,14E) 1 1 (2C,3A,14F) 1 1 (2C,3A,14G) 1 1 (2C,3A,14H) 1 1 6 2 : A7 (2D,4F,7A) 2 1 (2D,4F,7B) 2 1 26 :GL(3, 2) (2G,4F,8D) 1 1 (2G,4F,8B) 1 1 (2G,4D,6C) 2 1 6 2 : S8 (2C,6L,6K) 4 1 (2C,4O,7A) 6 1 26 : GO − (6, 2) (4H,6C,12I) 2 1 (2C,8E,12I) 6 1 AGL(6, 2) (2B,3B,15D) 4 1 (2B,3B,15E) 4 1

190 K. MAGAARD, S. SHPECTOROV, AND G. WANG

Table 11. The Genus Zero Systems for Primitive Groups of De- gree 9

group ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit 32 :4 (2A,4A,4A) 2 1 (2A,4B,4B) 2 1 32 : D(2 × 4) (2C,4A,6A) 1 1 (2A,4A,6B) 1 1 (2A,2C,2C,6A) 1 2 (2A,2A,2C,6B) 1 2 (2A,2C,2B,4A) 1 4 (2A,2A,2C,2C,2B) 1 8 2 3 :(2A4) (3B,4A,3E) 1 1 (3B,6B,4A) 1 1 (3B,6A,3D) 1 1 (3A,4A,3D) 1 1 (3A,6B,3E) 1 1 (3A,6A,4A) 1 1 (3B,3B,3B,2A) 1 1 (3A,3A,3A,2A) 1 1 AΓL(1, 9) (2A,4A,8A) 1 1 (2A,4A,8B) 1 1 AGL(2, 3) (2A,3C,8B) 1 1 (2A,3C,8A) 1 1 (2A,6A,8B) 1 1 (2A,6A,8A) 1 1 (2A,2A,2A,8B) 1 16 (2A,2A,2A,8A) 1 16 (2A,2A,3A,3C) 1 12 (2A,2A,3A,4A) 1 12 (2A,2A,3A,6A) 1 12 (2A,2A,2A,2A,3A) 1 216

Table 12. The Genus Zero Systems for Primitive Groups of De- gree 27

group ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit 3 3 .A4 (2A,3B,9D) 1 1 (2A,3B,9B) 1 1 (2A,3A,9C) 1 1 (2A,3A,9A) 1 1 3 3 (A4 × 2) (2B,3D,12B) 1 1 (2B,3D,12A) 1 1 (2A,2B,2B,3D) 1 24 3 3 .S4 (2B,4A,9B) 1 1 (2B,4A,9A) 1 1 3 3 (S4 × 2) (2E,4A,6G) 4 1 (2B,2E,2E,4A) 1 16 ASL(3, 3) (2A,3F,13D) 2 1 (2A,3F,13C) 2 1 (2A,3F,13B) 2 1 (2A, 3F,13A) 2 1 AGL(3, 3) (2C,4A,13D) 1 1 (2C,4A,13C) 1 1 (2C,4A,13B) 1 1 (2C,4A,13A) 1 1 (3E,6E,4A) 8 1

Table 13. The Genus Zero Systems for Primitive Groups of De- gree 49

group ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit 72 :4 (2A,4B,4B) 12 1 (2A,4A,4A) 12 1 72 :3:D(2 ∗ 4) (2A,4A,6C) 3 1 (2A,4A,6B) 3 1

GENERATING SETS OF AFFINE GROUPS OF LOW GENUS 191

Table 14. The Genus Zero Systems for Primitive Groups of De- gree 81

group ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit 4 3 :(GL(1, 3) S4) (6S,4C,6K) 2 1 4 3 :(2× S5) (6K,4A,6M) 1 1 4 3 : S5 (6E,12A,3G) 1 1 AGL(4, 3) (2C,5A,8E) 1 1 (2C,5A,8F) 1 1

Table 15. The Genus Zero Systems for Primitive Groups of De- gree 121

group ramiﬁcation #of largest ramiﬁcation #of largest type orbits orbit type orbits orbit 112 :3 (3A,3A,3A) 40 1 (3B,3B,3B) 40 1 112 :4 (2A,4A,4A) 30 1 (2A,4B,4B) 30 1 112 :6 (2A,3B,6B) 20 1 (2A,3A,6A) 30 1 2 11 :(Q8 : D6) (2B,3A,8A) 5 1 (2B,3A,8B) 5 1

References [1] M. Aschbacher, On conjectures of Guralnick and Thompson, J. Algebra, 135 (1990), no. 2, 277 – 343. MR1080850 (91m:20007) [2] M. Aschbacher, R. Guralnick, K. Magaard, Rank 3 permutation characters and primitive groups of low genus, In preparation. [3] M. Fried, Alternating groups and the moduli space lifting invariants, Israel J. Math. 179 (2010), 57 – 125. MR2735035 (2012a:14055) [4] D. Frohardt and K. Magaard, Composition Factors of Monodromy Groups, Annals of Mathematics, 154 (2001), 327–345. MR1865973 (2002j:20005) [5] D. Frohardt, R. Guralnick, K. Magaard, Genus 0 actions of group of Lie rank 1, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), 449 – 483, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., Providence, RI, 2002. MR1935417 (2003j:20019) [6] D. Frohardt, R. Guralnick, K. Magaard, Genus 2 point actions of classical groups, In preparation. [7] M. Fried, H. Volklein¨ , The inverse Galois problem and rational points on moduli spaces, Math. Ann. 290 (1991), 771 – 800. MR1119950 (93a:12004) [8] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.12; 2008. (http://www.gap-system.org) [9] R. Guralnick, Monodromy groups of coverings of curves. Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge, 2003, 1–46. MR2012212 (2004j:14029) [10] R. Guralnick, K. Magaard, On the Minimal Degree of a Primitive Permutation Group, J. Algebra, 207, (1998), 127 – 145. MR1643074 (99g:20014) [11] R. Guralnick, M. Neubauer, Monodromy groups of branched covering: the generic case, Recent developments in the inverse Galois problem (Seattle, WA,1993), 325 – 352, Contemp. Math., 186, Amer. Math. Soc., Providence, RI, 1995. MR1352281 (96h:20007) [12] R. Guralnick, J. Thompson, Finite groups of genus zero, J. Algebra 131 (1990), no. 1, 303 – 341. MR1055011 (91e:20006) [13] R. Guralnick, J. Shareshian, Symmetric and alternating groups as monodromy groups of Riemann surfaces. I. Generic covers and covers with many branch points. With an appendix by Guralnick and R. Staﬀord. Mem. Amer. Math. Soc. 189 (2007). MR2343794 (2009b:14055) [14] A. James, K. Magaard, S. Shpectorov, The GAP package MAPCLASS, accepted by the GAP council Nov 2011.

192 K. MAGAARD, S. SHPECTOROV, AND G. WANG

[15] M. Liebeck, J. Saxl, Minimal degrees of primitive permutation groups, with an application to mondromy groups of covers of Riemann surfaces, Proc. London Math. Soc. (3) 63 (1991), no. 2, 266 – 314. MR1114511 (92f:20003) [16] M. Liebeck, A. Shalev, Simple groups, permutation groups, and probability, J. Amer. Math. Soc. 12 (1999), no. 2, 497 – 520. MR1639620 (99h:20004) [17] F. Liu, B. Osserman, The irreducibility of certain pure-cycle Hurwitz spaces. American J. of Mathematics, Volume 130, Number 6, (2008), 1687–1708. MR2464030 (2009h:14052) [18] K. Magaard, S. Shpectorov, T. Shaska and H. Volklein¨ , The locus of curves with prescribed automorphism group, Communications in arithmetic fundamental groups (Kyoto, 1999/2001). S¯urikaisekikenky¯usho K¯oky¯uroku No. 1267 (2002), 112 –141. MR1954371 [19] K. Magaard, S. Shpectorov and H. Volklein¨ , A GAP package for braid orbit computation, and applications, Experiment. Math. 12 (2003), no. 4, 385 –393. MR2043989 (2005e:12007) [20] K. Magaard, H. Volklein¨ , The monodromy group of a function on a general curve, Israel Journal of Math. 141 (2004), 355–368. MR2063042 (2005e:14047) [21] K. Magaard, H. Volklein,¨ G. Wiesend, The Combinatorics of Degenerate Covers and an Application to General Curves of Genus 3, Albanian J. Math. 2 (2008), no. 3, 145–158. MR2495806 (2009k:14053) [22] M. Neubauer, On solvable monodromy groups of ﬁxed genus, PhD Thesis University of Southern California, (1989). MR2716273 [23] M. Neubauer, On monodromy groups of ﬁxed genus, J. Algebra 153 (1992), no. 1, 215–261. MR1195412 (93m:20003) [24] T. Shih, A note on groups of genus zero, Comm. Algebra 19 (1991), no. 10, 2813 – 2826. MR1129542 (93b:20009) [25] R. Staszewski, H. Volklein¨ , An Algorithm for Calculating the Number of Generating Tuples in a Nielsen Class, private communication. [26] H. Volklein¨ , Groups as Galois Groups: An Introduction, Cambridge University Press (1996). MR1405612 (98b:12003)

School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K E-mail address: [email protected] School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K E-mail address: [email protected] School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11361

Classiﬁcation of algebraic ODEs with respect to rational solvability

L. X. Chˆau Ngˆo, J. Rafael Sendra, and Franz Winkler

Abstract. In this paper, we introduce a group of aﬃne linear transformations and consider its action on the set of parametrizable algebraic ODEs. In this way the set of parametrizable ODEs is partitioned into classes with an invariant associated system, and hence of equal complexity in terms of rational solvability. We study some special parametrizable ODEs: some well-known and obviously parametrizable classses of ODEs, and some classes of ODEs with special geometric shapes, whose associated systems are characterized by classical ODEs such as separable or homogeneous ones.

Contents 1. Introduction 2. Preliminaries 3. A group of aﬃne linear transformations 4. Solvable AODEs and their associated systems 5. Parametrizable ODEs with special geometric shapes 6. Conclusion References

1. Introduction Solving algebraic ordinary differential equations (AODEs) is still a challenge in symbolic computation. After the work by J.F. Ritt [Rit50] and later by E.R. Kol- chin [Kol73] in differential algebra, the theory of differential equations has been rapidly developed from the algebraic point of view. In particular, most of the studies of AODEs can be seen as a differential counterpart of the one of algebraic equations. In this paper, we first recall the notion of a general solution of an AODE from the point of view of differential algebra. Then we describe a geometric approach to decide the existence of a rational general solution of a parametrizable ODE of

2010 Mathematics Subject Classiﬁcation. Primary 35A24, 35F50; Secondary 14E05, 14H50, 68W30. First and third authors partially supported by the Austrian Science Fund (FWF) via the Doc- toral Program “Computational Mathematics” (W1214), project DK11 and project DIFFOP (P20336- N18), second and third authors partially supported by [Ministerio de Econom´ıa y Competitividad, proyecto MTM2011-25816-C02-01].

c 2012 American Mathematical Society 193

194 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER order 1, i.e., an AODE whose solution surface is rational. In the affirmative case this decision method can be turned into an algorithm for actually computing such a rational general solution. A rational parametrization of the solution surface allows us to reduce the given differential equation to a system of autonomous AODEs of order 1 and of degree 1 in the derivatives. This often turns out to be an advantage because the original differential equation is typically of higher degree in the derivative. In fact, we can solve the associated system in a generic situation and therefore in most cases. Obviously, some equations (or their associated systems) are easier to solve than others. So, the natural question is whether a given equation can be transformed into an easier one, and thus is of the same low complexity. Such a classification is the main goal of this paper. Since we are interested in rational solutions, the natural transformations are birational maps (i.e., invertible rational maps with rational inverse). However, since we are working in a differential frame, we cannot expect all birational transformations to be suitable. Indeed, we investigate birational transformations preserving certain characteristics of the rational solutions of the corresponding equations. In this paper, which is the starting point of our strategy, we focus on linear transformations preserving rational solvability. We characterize them showing that they form a group whose orbits yield a decomposition of the set of parametrizable ODEs into classes with an invariant associated system, and hence of equal complexity in terms of rational solvability. Now, intuitively speaking, the easiest solvable AODE in a class will be seen as a normal representative. The goal is then twofold: on the one hand to find interesting classes in this quotient set, on the other to determine normal representatives; all from a computational point of view. We demonstrate this strategy by treating some special and interesting parametrizable ODEs in Section 4 and Section 5.

2. Preliminaries Let K be an algebraically closed field of characteristic zero. Let F (u, v, w)bea trivariate polynomial over K.Thealgebraic ordinary differential equation (AODE) of order 1 defined by F is of the form (1) F (x, y, y)=0, where y is an indeterminate over the differential field of rational functions K(x) d with the derivation = dx . Let {F } be the radical differential ideal generated by F in the differential ring K(x){y}. Then one can prove ([Rit50], II, Section 14) that (2) {F } =({F } : S) ∩{F, S}, ∂F where S = is the separant of F .({F } : S) is a prime differential ideal. So the ∂y set of solutions of F = 0 consists of two components: the general component on which the separant does not vanish, and the singular component which also requires vanishing of S. Of course, almost all the solutions of F = 0 belong to the general component. This decomposition is valid for differential polynomials of any order. Definition 2.1. A generic zero of the prime differential ideal {F } : S is called a general solution of F (x, y, y) = 0. A common zero of F and S is called a singular solution of F (x, y, y)=0.

CLASSIFICATION OF ALGEBRAIC ODEs 195

We are interested in computing a rational general solution of F (x, y, y)=0, i.e., a general solution of the form m m−1 amx + am−1x + ···+ a0 (3) y = n n−1 , bnx + bn−1x + ···+ b0 where ai,bj are constants in a transcendental extension ﬁeld of K.Inthesequel, by an arbitrary constant we mean a transcendental constant over K. We now give a geometric approach to compute an explicit rational general solution of F (x, y, y) = 0 provided that the solution surface in A3(K), deﬁned by (4) F (u, v, w)=0, is rationally parametrizable; that is, it admits a rational parametrization

(5) P(s, t)=(χ1(s, t),χ2(s, t),χ3(s, t)), where χ1,χ2,χ3 are bivariate rational functions over K and the Jacobian of P(s, t) has generic rank 2. Definition 2.2. An AODE F (x, y, y) = 0 is called a parametrizable ODE if it admits a rational parametrization of the form (5). In the sequel, we denote by AODE the set AODE = {F (x, y, y)=0| F ∈ K[x, y, z]} and by PODE the set PODE = {F ∈AODE|the surface F = 0 is rationally parametrizable}. A solution y = f(x)ofF (x, y, y) = 0 generates a curve C(x)=(x, f(x),f(x)) on the solution surface F (u, v, w)=0.Herex is viewed as the parameter of the space curve. If f(x) is a rational function, the parametric curve C(x) is then rational. Definition 2.3. Let y = f(x) be a rational solution of F (x, y, y)=0.The curve C(x)=(x, f(x),f(x)) is called a rational solution curve of F (x, y, y)=0. The rational solution curve generated by a rational general solution of F (x, y, y)= 0 is called a rational general solution curve. Assume that the solution surface parametrization P(s, t) in (5) is proper, i.e., it has an inverse and its inverse is also rational or, equivalently, K(P(s, t)) = K(s, t). Then a rational general solution curve can be determined by computing (s(x),t(x)) such that P(s(x),t(x)) = C(x). In order to satisfy this condition, it turns out that (s(x),t(x)) must be a rational general solution of the system ⎧ − · ⎨⎪ χ2t χ3 χ1t s = · − · , (6) χ1s χ2t χ1t χ2s ⎪ χ1s · χ3 − χ2s ⎩t = , χ1s · χ2t − χ1t · χ2s provided that χ1s · χ2t − χ1t · χ2s =0.Here χis,χit denote the partial derivatives of χi w.r.t. s and t, respectively. Definition 2.4. The system (6) is called the associated system of the AODE F (x, y, y) = 0 w.r.t. the parametrization P(s, t).

196 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER

The associated system (6) is constructed in such a way that if (s(x),t(x)) is a rational solution of the associated system and P(s(x),t(x)) is well deﬁned, then P(s(x),t(x)) = (x + c, χ2(s(x),t(x)),χ2(s(x),t(x)) ) for some constant c. Therefore,

y = χ2(s(x − c),t(x − c)) is a rational solution of the corresponding differential equation F (x, y, y)=0.In fact, the correspondence also holds for rational general solutions. Of course, we have to specify the notion of a general solution of the associated system (6) in the differential algebra context. Hence we have the following theorem, whose proof can be found in [NW10]. Theorem 2.1. If the parametrization P(s, t) is proper, then there is a one-to- one correspondence between rational general solutions of the parametrizable ODE F (x, y, y)=0and rational general solutions of its associated system w.r.t. P(s, t). The associated system (6) is an autonomous system in two differential indeter- minates s and t; and the degrees w.r.t. s and t are 1. Beside these advantages, in [NW10]and[NW11] the authors provide an algorithm for determining the rational general solution of the associated system in a generic case; later in the next paragraphs, we clarify the meaning of generality. Note that one can derive from the associated system a single rational ODE of order 1 and of degree 1 in the derivative, namely: dt χ · χ − χ (7) = 1s 3 2s . ds χ2t − χ3 · χ1t This type of differential equation is well-known in the literature [Jou79, PS83, Lin88, Sin92, Car94, CLPZ02]. In fact, Darboux’s theory on invariant algebraic curves studies the algebraic solutions of this type of differential equations and we apply that theory to the associated system (6) in order to find a rational solution.

Definition 2.5. Let M1,M2,N1,N2 be polynomials in K[s, t]and gcd(M ,N )=1fori =1, 2. An invariant algebraic curve of the rational system i i ⎧ ⎪ M (s, t) ⎨⎪s = 1 , N1(s, t) (8) ⎪ M (s, t) ⎩⎪t = 2 , N2(s, t) is an algebraic curve G(s, t) = 0 such that

GsM1N2 + GtM2N1 = GK, where Gs and Gt are the partial derivatives of G w.r.t. s and t,andK is some polynomial. An invariant algebraic curve of the system is called a general invariant algebraic curve if it contains an arbitrary constant in its coefficients. One can think of a general invariant algebraic curve as an infinite family of invariant algebraic curves over K. A rational general solution of the system (8) parametrizes a general invariant algebraic curve of the system. Assume that we have found an irreducible invariant algebraic curve G(s, t) = 0 of the system (8) containing an arbitrary constant c in its coefficients, and assume that it is rational when seen as a curve over the algebraic closure of K(c). Then we can obtain a

CLASSIFICATION OF ALGEBRAIC ODEs 197 rational general solution of the system (8) from a proper rational parametrization of that general invariant algebraic curve. Namely, we take any proper rational parametrization of the invariant algebraic curve and use system (8) to define a reparametrization for the invariant algebraic curve itself. This new parametrization is a rational solution of the system (8). For a complete description of this step we refer to [NW11]. Of course, the main problem is computing an irreducible invariant algebraic curve of the system; for that we use the undetermined coefficients method based on thedegreeboundgivenby[Car94] for systems having no dicritical singularities, which is the generic case. Example 2.1. We illustrate this approach by considering the differential equation (9) F (x, y, y) ≡ y2 +3y − 2y − 3x =0. The corresponding algebraic surface z2 +3z − 2y − 3x = 0 can be parametrized by t 2s + t2 1 2s + t2 t P (s, t)= + , − − , . 0 s s2 s s2 s This is a proper parametrization and the corresponding associated system is # s = st, t = s + t2. We compute the set of irreducible invariant algebraic curves of the system and obtain {s =0,t2 +2s =0,s2 + ct2 +2cs =0| c is an arbitrary constant}. The general invariant algebraic curve s2 + ct2 +2cs = 0 can be parametrized by 2cx2 2cx Q(x)= − , − . x2 + c x2 + c By the algorithm RATSOLVE in [NW11], we have to solve an auxiliary differential equation for the reparametrization, namely: 1 Q Q − 2 T = 1(T ) 2(T )= T . Q1(T ) 1 Hence, T (x)= . So the rational general solution of the associated system is x 2c 2cx s(x)=Q (T (x)) = − ,t(x)=Q (T (x)) = − . 1 1+cx2 2 1+cx2 We observe that 1 χ (s(x),t(x)) = x − . 1 c Therefore, the rational general solution of (9) is 1 1 1 1 1 3 y = χ s x + ,t x + = x2 + x + + , 2 c c 2 c 2c2 2c which, after a change of parameter, can be written as 1 y = ((x + c)2 +3c). 2

198 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER

3. A group of affine linear transformations Up to now, we have considered parametrizable ODEs of order 1 independently. We have mentioned in the introduction that some equations (or their associated systems) are easier to solve than others. So, the natural question is whether a given equation can be transformed into an easier one. As a first step in this direction, we develop in this section a family of birational transformations preserving certain characteristics of the rational solutions of the corresponding equations. Precisely, we define a group of affine linear transformations on K(x)3 mapping an integral curve of the space to another one. By an integral curve of the space we mean a parametric curve of the form C(x)=(x, f(x),f(x)). So this group can act on the set of all AODEs of order 1 and it is compatible with the solution curves of the corresponding differential equations. Therefore, the group orbits partition the set of all AODEs of order 1. Most of the observations in this section are elementary but we prove them for the sake of completeness. Let L : K(x)3 −→ K(x)3 be an affine linear transformation defined by

L(v)=Av + B, where A is an invertible 3 × 3matrixoverK, B is a column vector over K and v is a column vector over K(x). We want to determine A and B such that for any f ∈ K(x), there exists g ∈ K(x) with ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x x x L ⎝ f(x) ⎠ = A ⎝ f(x) ⎠ + B = ⎝ g(x) ⎠ , f (x) f (x) g(x) i.e., L maps an integral curve to an integral curve. By choosing some special rational functions for f(x), we see that A and B must be of the forms ⎛ ⎞ ⎛ ⎞ 100 0 A := ⎝ ba0 ⎠ ,B:= ⎝ c ⎠ , 00a b where a, b and c are in K and a =0.Let G be the set of all such aﬃne linear transformations. We represent the elements of G by a pair of matrices [A, B]. Let ⎡⎛ ⎞ ⎛ ⎞⎤ 100 0 ⎣⎝ ⎠ ⎝ ⎠⎦ Li := bi ai 0 , ci ,i=1, 2, 00ai bi be two elements in G. The usual composition of maps deﬁnes a multiplication on G as ⎡⎛ ⎞ ⎛ ⎞⎤ 100 0 ⎣⎝ ⎠ ⎝ ⎠⎦ L1 ◦ L2 = b1 + a1b2 a1a2 0 , c1 + a1c2 00a1a2 b1 + a1b2 and an inverse operation as ⎡⎛ ⎞ ⎛ ⎞⎤ 100 0 − L 1 = ⎣⎝ − b1 1 0 ⎠ , ⎝ − c1 ⎠⎦ . 1 a1 a1 a1 001 − b1 a1 a1

CLASSIFICATION OF ALGEBRAIC ODEs 199

Hence G is a group with the unit element (the identity map) ⎡⎛ ⎞ ⎛ ⎞⎤ 100 0 I = ⎣⎝ 010⎠ , ⎝ 0 ⎠⎦ . 001 0 This group can be naturally generalized to higher dimensional spaces; i.e., to the case of higher order AODEs. Lemma 3.1. The group G deﬁnes a group action on AODE by G×AODE → AODE b 1 c b 1 (L, F ) → L · F =(F ◦ L−1)(x, y, y)=F x, − x + y− , − + y , a a a a a where ⎡⎛ ⎞ ⎛ ⎞⎤ 100 0 L := ⎣⎝ ba0 ⎠ , ⎝ c ⎠⎦ . 00a b Proof. We have ◦ · ◦ ◦ −1 ◦ −1 ◦ −1 (L1 L2) F = F (L1 L2) =F (L2 L1 ) ◦ −1 ◦ −1 =(F L2 ) L1

=L1 · (L2 · F ), and I · F = F . Therefore, this is an action of the group G on the set AODE.

Remark 3.1. Let F ∈PODE and P(s, t) be a proper parametrization of the solution surface, then (L◦P)(s, t) is a proper parametrization of the solution surface of (L · F ), because (L · F )((L ◦P)(s, t)) = F (L−1((L ◦P)(s, t))) = F (P(s, t)) = 0. Therefore, (L · F ) ∈PODE. Moreover, the group G also deﬁnes a group action on PODE. The action of G on PODE induces a partition of PODE into classes for which the solvability, and in particular the rational solvability, is an invariant property. In the next theorem we state that the associated system is also invariant. Theorem 3.1. Let F ∈PODE,andL ∈G. For every proper rational parametrization P of the surface F (x, y, z)=0, the associated system of F (x, y, y)=0 w.r.t. P and the associated system of (L · F )(x, y, y)=0w.r.t. L ◦P are equal.

Proof. Let P(s, t)=(χ1(s, t),χ2(s, t),χ3(s, t)) be a proper rational parametrization of F (x, y, z)=0.ThenL · F can be parametrized by (L ◦P)(s, t). The f f associated system of F (x, y, y) = 0 w.r.t. P(s, t)is s = 1 ,t = 2 where g g 2 2 2 2 2 2 2 2 2 2 2 2 2 1 χ1t 2 2 χ1s 1 2 2 χ1s χ1t 2 f1 = 2 2 ,f2 = 2 2 , and g = 2 2 . χ3 χ2t χ2s χ3 χ2s χ2t We have

(L ◦P)(s, t)=(χ1,bχ1 + aχ2+c, b + aχ3),

200 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER where a, b and c are constants and a#= 0. So the associated$ system of f˜ f˜ (L · F )(x, y, y) = 0 w.r.t. (L ◦P)is s = 1 ,t = 2 where g˜ g˜ 2 2 2 2 2 2 2 2 ˜ 2 1 χ1t 2 ˜ 2 χ1s 1 2 f1 = 2 2 = af1, f2 = 2 2 = af2 b + aχ3 bχ1t + aχ2t bχ1s + aχ2s b + aχ3 and 2 2 2 2 2 χ1s χ1t 2 g˜ = 2 2 = ag. bχ1s + aχ2s bχ1t + aχ2t Therefore, the associated system of F (x, y, y) = 0 w.r.t. P and the associated system of (L · F )(x, y, y) = 0 w.r.t. L ◦P are equal.

Clearly, specially interesting classes of PODE are those containing autonomous parametrizable ODEs. Algorithmically, if we are given an equation in PODE and we want to check whether it is in the autonomous class, we may apply to the equation a generic element in G (i.e., introducing undetermined elements in the description of L ∈G) and afterwards require the coeﬃcients of the resulting equation not to depend on x. In the next corollary we describe the type of associated system we get for these equations. Corollary 3.1. Let F ∈PODE and L ∈G such that (L · F )(x, y, y)=0 is an autonomous AODE. There exists a proper rational parametrization P(s, t) of F (x, y, z)=0such that its associated system is of the form M(t) (10) s =1,t = . N(t) Proof. Since (L · F )(x, y, y) = 0 is an autonomous parametrizable ODE, the plane algebraic curve (L · F )(y, z) = 0 is rational, and for every proper rational parametrization (f(t),g(t)) of (L · F )(y, z) = 0 the associated system of (L · F )(x, y, y) = 0 w.r.t. P(s, t)=(s, f(t),g(t)) is of the form {s =1,t = g(t)/f (t)}.

Remark 3.2. The converse of Corollary 3.1 is not true. Indeed, we consider the equation F (x, y, y)=y − y2 − y − yx =0. It belongs to PODE and it can be properly parametrized as

2 P1(s, t)=(s, t + t + ts, t). The associated system w.r.t. P1(s, t)is{s =1,t =0} that is of the form (10). Let us see that the class of F (x, y, y) = 0 does not contain any autonomous equation. A generic transformation yields 1 1 b 1 1 b b2 c (L · F )(x, y, y)=− y2 − y +2 y − xy + y + − − , a2 a a2 a a a a2 a and from here the conclusion is clear.

CLASSIFICATION OF ALGEBRAIC ODEs 201

Example 3.1. As in Example 2.1 we consider the diﬀerential equation

F (x, y, y) ≡ y2 +3y − 2y − 3x =0.

We ﬁrst check whether in the class of F there exists an autonomous AODE. For this, we apply a generic L to F to get

1 3 2b 2 2b 3b b2 2c (L · F )(x, y, y)= y2 + y − y − y + x − 3x − + + . a2 a a2 a a a a2 a Therefore, for every a =0and b such that 2b − 3a = 0, we get an autonomous AODE. In particular, for a =1,b =3/2andc =0weget ⎡⎛ ⎞ ⎛ ⎞⎤ 100 0 ⎢⎜ 3 ⎟ ⎜ ⎟⎥ L = ⎣⎝ 10⎠ , ⎝ 0 ⎠⎦ , 2 3 001 2 i.e., we obtain 9 F (L−1(x, y, y)) ≡ y2 − 2y − =0. 4 t2 9 The last equation can be parametrized by P (s, t)= s, − ,t . Its associated 2 2 8 system is {s =1,t =1}. Therefore, this is also the associated system of the given diﬀerential equation w.r.t. the parametrization t2 3 9 3 (L ◦P )(s, t)= s, − s − ,t− . 2 2 2 8 2 The general invariant algebraic curve of this associated system is s−t+˜c =0,where c˜ is an arbitrary constant. Again using the algorithm RATSOLVE in [NW11] we obtain a rational general solution of this associated system, namely: s(x)= x, t(x)=x +˜c. Therefore, we see that the rational general solution of the given diﬀerential equation is ! " t(x)2 3 9 1 3 2 3 y = − s(x) − = x +˜c − +3 c˜ − . 2 2 8 2 2 2

Now, it is clear that this rational general solution is equivalent to the rational general solution computed in Example 2.1 up to a change of the arbitrary constant. In fact, we have (see P0(s, t)inExample2.1) t 2s + t2 t 3 ((L ◦P )−1 ◦P )(s, t)= + , + . 2 0 s s2 s 2 2c 2cx This birational mapping transforms the planar curve − , − in 1+cx2 1+cx2 1 3 Example 2.1 into the planar curve x − ,x+ , whose deﬁning equation is c 2 1 3 s − t + + =0. c 2

202 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER

4. Solvable AODEs and their associated systems Based on these observations, the study of parametrizable ODEs can be reduced to the study of their normal forms with respect to, for instance, an affine linear transformation in G. In this section, we describe some of the special parametrizable AODEs that are good candidates for normal forms. They are classified in [Pia33], Chapter V; and in [Mur60], Chapter A2, Part I as those solvable for y, those solvable for y and those solvable for x. One can derive from these special AODEs new differential equations of order 1 and of degree 1, which are of the same complexity in terms of rational solvability. In fact, the three special types are, under minimal requirements, in PODE and they have an obvious proper parametrization. So we can interpret the results in the light of our algebraic geometric approach.

4.1. Equations solvable for y. We consider a differential equation solvable for y, i.e., y = G(x, y), where G(x, y) is a rational function. Then we need not change the variable because it is already in the desired form for applying Darboux’s theory (see equation (7)). Since G(x, y) is rational, (s, t, G(s, t)) is a parametrization of the solution surface, and hence the equation belongs to PODE; moreover, it is proper because K(s, t, G(s, t)) = K(s, t). If we apply an affine linear transformation L ∈Gto F = y − G(x, y), then b 1 b 1 c (L · F )(x, y, y)=− + y − G x, − x + y− . a a a a a Therefore, the new differential equation is of the same form. In other words, the property of being solvable for y is invariant in the class, and we do not enlarge this class by applying the transformations in G. The associated system, via the parametrization (s, t, G(s, t)), is {s =1,t = G(s, t)} and the single rational ODE derived from the system (see equation 7) is the original equation dt = G(s, t). ds 4.2. Equations solvable for y. Let the differential equation be of the form y = G(x, y), where G(x, y) is a rational function. A typical example is Clairaut’s equation in Example 4.1. Let us assume that G is a rational function. Clearly this type of equations belongs to PODE since (s, G(s, t),t) is a proper parametrization of the solution surface y = G(x, z). In this class, if we apply an affine linear transformation L ∈Gto F = y − G(x, y), then b 1 c b 1 (L · F )(x, y, y)=− x + y− − G x, − + y . a a a a a Therefore, this class is also closed under the group action of G, i.e., we do not enlarge this class by applying the transformations in G. The associated system, via the parametrization (s, G(s, t),t), is t − G (s, t) s =1,t = s , Gt(s, t)

CLASSIFICATION OF ALGEBRAIC ODEs 203 where Gs and Gt are the partial derivatives of G(s, t) w.r.t. s and t, respectively. Moreover, the single rational ODE derived from the system (see equation (7)) is dt t − G (s, t) = s , ds Gt(s, t) which is of the desired form. Let us see that one gets the same equation using the classical reasoning. One can differentiate the equation w.r.t. x to obtain y = Gx(x, y )+Gy (x, y ) · y , where Gx and Gy are the partial derivatives of G(x, y ) w.r.t. x and y , respectively. Denoting y byy ˜, one can rewrite the above differential equation in the form dy˜ y˜ = G (x, y˜)+G (x, y˜) · , x y˜ dx or equivalently, dy˜ y˜ − G (x, y˜) = x . dx Gy˜(x, y˜) Example 4.1. [Clairaut’s equation]Letf be a smooth function of one variable. We consider Clairaut’s differential equation y = yx + f(y). This is a differential equation solvable for y and it can be parametrized by

P3(s, t)=(s, st + f(t),t).

If f is rational, then P3(s, t) is a proper rational parametrization of the differential equation. The associated system w.r.t. P3(s, t)is{s =1,t =0}.Thesetof irreducible invariant algebraic curves is {t − c =0| c is an arbitrary constant}. Using the algorithm RATSOLVE in [NW11], we obtain (s(x),t(x)) = (x, c)asa rational general solution of the associated system. So we get the rational general solution of Clairaut’s differential equation, namely: y = cx + f(c). 4.3. Equations solvable for x. We consider a differential equation of the form x = G(y, y). Assuming that G is rational, this AODE belongs to PODE, because (G(s, t),s,t) is a proper parametrization of the solution surface. If we apply an affine linear transformation L ∈Gto F = x − G(y, y), then b 1 c b 1 (L · F )(x, y, y)=x − G − x + y− , − + y . a a a a a Thedegreeofx in this equation is no longer linear. So this class is not closed under the action of the group G. The associated system, via the parametrization (G(s, t),s,t), is 1 − tG (s, t) s = t, t = s Gt(s, t)

204 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER where Gs and Gt are the partial derivatives of G w.r.t. s and t, respectively. Moreover, the single rational ODE derived from the system (see equation (7)) is dt 1 − tG (s, t) = s . ds tGt(s, t) Let us see that one gets the same equation using the classical reasoning. One can differentiate the equation w.r.t. y to obtain dx dy = G (y, y )+G (y, y ) · . dy y y dy Lety ˜ = y,thenwehave 1 dy˜ = G (y, y˜)+G (y, y˜) · . y˜ y y˜ dy So we have transformed the differential equation x = G(y, y)toanewdifferential equation of order 1 and of degree 1 in the desired form, namely dy˜ 1 − yG˜ (y, y˜) = y . dy yG˜ y˜(y, y˜) We summarize the three classes, and their geometric interpretation, in the following table:

Solvable for y Solvable for y Solvable for x AODE y = G(x, y) y = G(x, y) x = G(y, y) Proper (s, t, G(s, t)) (s, G(s, t),t) (G(s, t),s,t) Parametrization ⎧ ⎧ # ⎨s =1 ⎨s = t Associated s =1 t − G (s, t) 1 − tG (s, t) System t = G(s, t) ⎩t = s ⎩t = s Gt(s, t) Gt(s, t) dt dt t − G (s, t) dt 1 − tG (s, t) Equation (7) = G(s, t) = s = s ds ds Gt(s, t) ds tGt(s, t)

Example 4.2. As we have already mentioned, if F ∈PODEis solvable for y, then all elements in the class are solvable for y; similarly if F ∈PODE is solvable for y. However, this is not the case for equations solvable in x. So, if we are given F ∈PODEwe may try to check whether there exists L ∈Gsuch that (L · F ) is solvable for x. For this purpose, we apply a generic transformation in G,and afterwards require that (L · F ) be linear in x. For instance, let us consider the equation F (x, y, y) ≡−3x − 4x2 +4xy − y2 +2xy +2y − yy +8− 8y +2y2 =0, which belongs to PODE.Notethat (s2 + st − 2t2, 2s2 +2st − 4t2 + s, 2+t) is a proper parametrization of F (x, y, z) = 0. Applying a generic transformation in G one gets a quadratic polynomial in x,andthecoeﬃcientofx2 is −(2a + b)2 . a2

CLASSIFICATION OF ALGEBRAIC ODEs 205

So if we take, for instance, a =1,b= −2andc = 0 we get an equation in the class solvable for x; indeed, we get x − y2 − yy +2y2 =0.

5. Parametrizable ODEs with special geometric shapes In [FG04, FG06], an autonomous AODE is associated to a plane algebraic curve. Accordingly, an autonomous AODE possessing a rational general solution is associated to a rational plane curve. In fact, these are special AODEs in PODE, whose solution surfaces are cylindrical surfaces over a rational plane curve. Observe that the action of an element in G on an autonomous AODE typically results in a non-autonomous one. Hence, the resulting AODE has the same associated system and the same rational solvability. Therefore, autonomy is not an intrinsic property of an AODE with respect to rational solvability. In this section, we consider some classes in PODE having special geometric shapes and one of the classes is a generalization of autonomous AODEs. 5.1. Differential equations of pencil type. We first consider parametrizable ODEs whose solution surface is a pencil of rational curves. More precisely, we assume that F (x, y, z) = 0 is the defining equation of an algebraic curve over the algebraic closure K(x)ofK(x) and that it is K(x)-parametrizable; i.e., K(x)isthe optimal field of the parametrization of the curve. The latter assumption is always fulfilled if the degree of the curve is odd (cf. [SWPD08], Chapter 5). With these assumptions, the surface F (x, y, z) = 0 has a proper parametrization of the form

(11) P4(s, t)=(s, f(s, t),g(s, t)), where f and g are rational functions in s and t. Indeed, letting (f(s, t),g(s, t)) ∈ K(s)(t)2 be a proper parametrization of the curve (recall that Lüroth’s theorem is valid over every field), then K(s)(f(s, t),g(s, t)) = K(s)(t). P4 parametrizes the surface F (x, y, z)=0andK(s, f(s, t),g(s, t)) = K(s, t); hence it is proper. The surface parametrized by (11) is called a pencil of rational curves.Inthis case, the associated system of F (x, y, y) = 0 w.r.t. P (s, t)is 4 −f (s, t)+g(s, t) (12) s =1,t = s , ft(s, t) where fs and ft are the partial derivatives of f w.r.t. s and t, respectively. The derived differential equation from the associated system (see equation (7)) is dt −f (s, t)+g(s, t) (13) = s . ds ft(s, t) In fact, there are several cases, in which the associated system (12) and the derived ODE (13) are simple: it can be separable or homogeneous. For instance, if f(s, t) and g(s, t) are homogeneous polynomials of degree m +1 and m, respectively, then the derived differential equation (13) is homogeneous. In this case, we can write t t f(s, t)=sm+1f 1, ,g(s, t)=smg 1, . s s t So the birational change of parameters s∗ = s, t∗ = transforms (s, f(s, t),g(s, t)) s into the parametrization m+1 m (s, s f1(t),s f2(t)). We consider, in the next subsections, the following two cases:

206 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER

• [Cylindrical type] f(s, t)=λs + f1(t)andg(s, t)=f2(t); m+1 m • [Quasi-cylindrical type] f(s, t)=s f1(t)andg(s, t)=s f2(t); where f1,f2 are non-constant rational functions such that (f1(t),f2(t)) is proper; i.e., K(f1(t),f2(t)) = K(t). 5.1.1. Diﬀerential equations of cylindrical type. Definition 5.1. Let F ∈PODE. F is of cylindrical type iﬀ F (x, y, z)=0has a proper rational parametrization of the form

(14) P5(s, t)=(0,f1(t),f2(t)) + s(1,λ,0) = (s, λs + f1(t),f2(t)), where λ is a constant and f1(t) is non-constant; i.e., F canbewrittenas (15) G(y − λx, y)=0, where G(u, v) = 0 is a rational curve. It is clear that an autonomous AODE with rational solutions is a special case of cylindrical type, corresponding to λ =0. Note that the properness of P5(s, t) is equivalent to the properness of (f1(t),f2(t)) because K(s, λs + f1(t),f2(t)) = K(s)(f1(t),f2(t)) = K(s)(t). If an AODE can be parametrized by a proper rational parametrization of the form

(16) P6(s, t)=(f1(t),f2(t),f3(t)) + s(1,λ,0), − where λ is a constant and f2(t) λf1(t) = 0, then by a change of parameters we can bring it to the standard cylindrical type. Indeed, one can apply the birational ∗ ∗ transformation {s = f1(t)+s, t = t}. Theorem 5.1. Every parametrizable ODE of cylindrical type is transformable into an autonomous AODE by the transformation ⎡⎛ ⎞ ⎛ ⎞⎤ 100 0 L := ⎣⎝ −λ 10⎠ , ⎝ 0 ⎠⎦ . 001 −λ As a consequence (see Theorem 3.1 and Corollary 3.1), every parametrizable ODE of cylindrical type has a parametrization w.r.t. which its associated system is of the M(t) form s =1,t = ,whereM,N are polynomials in one variable over K. N(t) Proof. We have L · F = F (L−1(x, y, y)) = G(y, y + λ), which is an autonomous AODE. The associated system w.r.t. the parametrization in (14) is f2(t) − λ (17) s =1,t= . f1(t) A rational general solution of this system, if it exists, is of the form αx + β (s(x),t(x)) = x + c, , γx + δ

CLASSIFICATION OF ALGEBRAIC ODEs 207 where α, β, γ, δ are constants and c is an arbitrary constant. Here we use the fact that the second diﬀerential equation in the associated system is autonomous. So from [FG04, FG06] we know the exact degree of a possible rational solution, which in this case is 1. This exact degree bound is derived from an exact degree bound for curve parametrizations in [SW01]. In this case, a rational general solution of G(y − λx, y)=0is α(x − c)+β y(x)=f (t(x − c)) + λx = f + λx, 1 1 γ(x − c)+δ where c is an arbitrary constant. Remark 5.1. If the integral & f (t) P (t) ϕ(t)= 1 dt = 1 f2(t) − λ P2(t) is a rational function, then the general irreducible invariant algebraic curve of the system (17) is

P1(t) − sP2(t) − cP2(t)=0, where c is an arbitrary constant. Hence, the system (17) has a general solution of the form (x, t(x)), where t(x) is an algebraic function satisfying the equation

P1(t(x)) − xP2(t(x)) − cP2(t(x)) = 0. So a general solution of G(y − λx, y) = 0 is an algebraic solution given by

y = f1(t(x)) + λx. By Theorem 5.1, the autonomous AODEs are the representatives of parametrizable ODEs of cylindrical type. In order to check whether an F (x, y, y)=0in PODE is equivalent to a parametric ODE of cylindrical type, we proceed as follows. First we apply a generic transformation L ∈G,sayG(a, b, x, y, y)=(L·F )(x, y, y). Then we consider the differential equation (18) G(a, b, c, x, y, y)=0 and determine a, b, c such that the new differential equation is an autonomous AODE; i.e., the coefficients of G(a, b, c, x, y, y) w.r.t. x must be all zero except for the coefficient of degree 0. 5.1.2. Differential equations of quasi-cylindrical type. Definition 5.2. Let G(u, v) = 0 be a rational plane curve. A differential equation of the form y y (19) G , =0, xm+1 xm is called of quasi-cylindrical type. Of course, there are other differential equations which are transformable into this type via linear affine transformations. These are of the form ay + bx + c ay + b (20) G , =0, xm+1 xm where a, b and c are constants and a =0.

208 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER

Suppose that (f1(t),f2(t)) is a proper rational parametrization of G(u, v)=0. Then the solution surface of (19) can be properly parametrized by m+1 m (21) P7(s, t)=(s, s f1(t),s f2(t)).

Note that P7(s, t) is proper, because it is a special case of the parametrization considered in (11). With respect to P (s, t) the associated system is separable 7 −(m +1)f1(t)+f2(t) (22) s =1,t= . sf1(t) Therefore, we can always decide whether the differential equation (19) has a rational general solution or not. 5.2. Differential equations of cone type. A rational conical surface (say with vertex at the origin) can be parametrized as s E(t) where E(t) is a space curve parametrization; if the curve E(t) is contained in a plane passing through the origin, the surface is that plane. This motivates the following definition. Definition 5.3. A parametrizable ODE is of cone type if its solution surface has a parametrization of the form

m1 m2 m3 (23) P8(s, t)=(s f1(t),s f2(t),s f3(t)), where f1,f2,f3 are rational functions and m1,m2,m3 are integers.

• When m1 =1,f1(t)=1and m2 = m3 +1, we obtain a quasi-cylindrical surface. The associated system w.r.t. (23) is ⎧ − − ⎪ s1 m1 f (t) − s1 m2+m3 f (t)f (t) ⎪ 2 3 1 ⎪s = − , ⎨ m1f1(t)f2(t) m2f1(t)f2(t) (24) ⎪ − − ⎪ − m1 m2+m3 ⎩⎪ m2s f2(t)+m1s f3(t)f1(t) t = − . m1f1(t)f2(t) m2f1(t)f2(t) • In fact, we consider the case m2 = m1 + m3, i.e., the parametrization is

m1 m1+m3 m3 (25) P9(s, t)=(s f1(t),s f2(t),s f3(t)),

in which P7(s, t) is a special case. Then the derived diﬀerential equation is separable, namely:

dt −m2f2(t)+m1f1(t)f3(t) (26) = − . ds (f2(t) f3(t)f1(t))s By integration we can decide if the associated system has a general invariant algebraic curve and proceed as in the algorithm RATSOLVE ([NW11]) to check the existence of a rational general solution of the system (24) with m2 = m1 + m3. The diﬀerential equation corresponding to the parametrization (25) is of the form ym1 y m1 (27) G , =0, xm1+m3 xm3 where G(u, v) = 0 is a rational planar curve.

CLASSIFICATION OF ALGEBRAIC ODEs 209

In general, from the form (27) we do not know whether the surface is rational or not. However, in some special cases, we can decide this property. For instance, if the rational curve G(u, v) = 0 has a rational parametrization of the form (28) (g(t)m1 ,h(t)m1 ), then the surface deﬁned by (27) can be parametrized by

m1 m1+m3 m3 (29) P10(s, t)=(s ,s g(t),s h(t)).

m1 m1 This parametrization is proper if (g(t) ,h(t) ) is proper and gcd(m1,m3)=1. Then we can continue applying our method for deciding the existence of a rational general solution and computing it in the aﬃrmative case.

6. Conclusion We have described an algebraic geometric approach to classify parametrizable ODEs of order 1 w.r.t. their rational solvability. These classes are the orbits generated by a group of affine linear transformations acting on AODEs. AODEs in the same equivalence class share important characteristics, such as the associated system, and the complexity of determining general rational solutions. We have pointed out some interesting classes in this equivalence relation. This is the first step towards classifying AODEs w.r.t. a more general group of birational transformations preserving certain characteristics of the rational solutions of AODEs. Finally, we have analyzed some classes of AODEs having general rational solutions. It turns out that being autonomous is not a characteristic property of such a class. Some geometric properties of differential equations carry over to representatives of their corresponding classes, which can obviously be solved rationally.

References [Car94] M. M. Carnicer, The Poincaré problem in the nondicritical case, Annals of Mathe- matics 140(2) (1994), 289–294. MR1298714 (95k:32031) [CLPZ02] C. Christopher, J. Llibre, C. Pantazi, and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems, J. Physics A: Mathematical and General 35 (2002), 2457–2476. MR1909404 (2003c:34037) [FG04] R. Feng and X.-S. Gao, Rational general solutions of algebraic ordinary differential equations, Proc. ISSAC 2004. ACM Press, New York (2004), 155–162. MR2126938 (2005j:34002) [FG06] , A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs, J. Symbolic Computation 41(7) (2006), 739–762. MR2232199 (2006m:65135) [Jou79] J. P. Jouanolou, Equations de pfaff algébriques, Lecture Notes in Mathematics, 1979. MR537038 (81k:14008) [Kol73] E. R. Kolchin, Differential algebra and Algebraic groups, Academic Press, 1973. MR0568864 (58:27929) [Lin88] A. Lins Neto, Algebraic solutions of polynomial differential equations and foliations in dimension two, vol. 1345, Holomorphic Dynamics, Lecture Notes in Mathematics, Springer Berlin/Heidelberg, 1988. MR980960 (90c:58142) [Mur60] G. M. Murphy, Ordinary differential equations and their solutions,VanNostrand Reinhold Company, 1960. MR0114953 (22:5762) [NW10] L. X. C. Ngô and F. Winkler, Rational general solutions of first order non- autonomous parametrizable ODEs, J. Symbolic Computation 45(12) (2010), 1426– 1441. MR2733387 (2012c:34013) [NW11] , Rational general solutions of planar rational systems of autonomous ODEs, J. Symbolic Computation 46(10) (2011), 1173–1186. MR2831479

210 L. X. CHAUˆ NGO,ˆ J. RAFAEL SENDRA, AND FRANZ WINKLER

[Pia33] H. T. H. Piaggio, An elementary treatise on differential equations, London, G. Bell and Sons, Ltd, 1933. [PS83] M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Transactions of the American Mathematical Society 279(1) (1983), 215–229. MR704611 (85d:12008) [Rit50] J. F. Ritt, Differential algebra, vol. 33, Amer. Math. Society. Colloquium Publications, 1950. MR0035763 (12:7c) [Sin92] M. F. Singer, Liouvillian first integrals of differential equations, Transactions of the American Mathematical Society 333(2) (1992), 673–688. MR1062869 (92m:12014) [SW01] J. R. Sendra and F. Winkler, Tracing index of rational curve parametrizations, Comp.Aided Geom.Design 18 (2001), 771–795. MR1857997 (2002h:65022) [SWPD08] J. R. Sendra, F. Winkler, and S. Pérez-D´ıaz, Rational algebraic curves - a computer algebra approach, Springer, 2008. MR2361646 (2009a:14073)

DK Computational Mathematics, Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria E-mail address: [email protected] Dpto. de Matematicas,´ Universidad de Alcala,´ Alcala´ de Henares/Madrid, Spain. Member of the Research Group ASYNACS (Ref. CCEE2011/R34) E-mail address: [email protected] Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11365

Circle packings on conformal and aﬃne tori

Christopher T. Sass, Kenneth Stephenson, and G. Brock Williams

Abstract. In this note we survey recent advances in the study of circle packings on conformal and affine tori. For conformal tori, packings are rigid, and this is used in conjunction with Brooks parameters to produce combinatorial coordinates on their moduli space. For affine tori, in contrast, packings are flexible, and a two-parameter family of affine packings is demonstrated.

1. Introduction The theory of circle packings has deep connections to many branches of mathematics from function theory to geometry to probability. In recent years there have been applications to conformal and quasiconformal mapping, Teichmüller theory, and computer imaging [13, 15, 17, 4]. Packings on conformal tori are characterized by their rigidity: given a triangulation of a topological torus, there is a unique conformal torus that supports a packing for that triangulation. Packings on affine tori, on the other hand, are noted for their flexibility: given this same triangulation, there are two degrees of freedom in the choice of an affine torus that supports a packing for the triangulation. For conformal tori, we will exploit this rigidity to give combinatorial coordinates on the moduli space of conformal tori. That is, we can describe every conformal torus in terms of the combinatorics of triangulations [18]. Special cases of circle packings for affine tori have been studied in [10, 9, 11]. Our approach to such circle packings is rooted in considerations of actually computing the packings. We demonstrate their flexibility by showing that given a triangulation there is a certain two parameter family of affine tori supporting a packing for the triangulation.

2. Tori 2.1. Conformal Tori. Conformal tori may be represented as the plane C modulo a discrete group of the form z +1,z+ τ with τ in the upper half plane H ∈ H [8]. Conversely, each τ determines a torus Rτ .TwotoriRτ1 and Rτ2 are conformally equivalent iﬀ τ1 and τ2 diﬀer by an element of the group PSL2(Z). The space M of conformal equivalence classes of tori is called the moduli space of tori and is given by H/PSL2(Z). For our purposes, the most convenient

2010 Mathematics Subject Classiﬁcation. Primary 52C26, 30F60. Key words and phrases. Circle packing, tori.

c 2012 American Mathematical Society 211

212 CHRISTOPHER T. SASS, KENNETH STEPHENSON, AND G. BROCK WILLIAMS

x = −1/2 x = 1/2

0 1

Figure 1. A fundamental region for PSL2(Z).Alltoriarecon- formally equivalent to Rτ for some τ in this region. fundamental region for M is the portion of H bounded below by the unit circle and ± 1 on the left and right by the vertical lines x = 2 . See Figure 1. 2.2. Affine Tori. ATeichmüller parameter ω ∈ H determines a parallelogram fundamental domain having corners 0, 1, ω,andω +1foratorusT (ω). Given a Teichmüller parameter ω ∈ H, an affine parameter c ∈ C determines an affine torus T (ω, c) by a developing map f : C → C,wheref(z)=z if c = 0 (flat torus), and f(z)=ecz if c = 0. The side-pairing maps for the developed image of a fundamental domain for T (ω, c)(wherec =0)are z → ecz and z → ecωz. Observe that in either case, the side-pairing maps for the torus are affine maps of the plane.

3. Circle Packings 3.1. Definitions. After their introduction by William Thurston, circle packings have been widely studied, especially in connection with analytic functions [16]. Definition 3.1. A circle packing is a configuration of circles with a specified pattern of tangencies. In particular, if K is a triangulation of an orientable topological surface, then a circle packing P for K is a configuration of circles such that

(1) P contains a circle Cv for each vertex v in K, (2) Cv is externally tangent to Cu if v, u is an edge of K, (3) Cv,Cu,Cw forms a positively oriented mutually tangent triple of circles if v, u, w is a positively oriented face of K We restrict attention here to complexes K which triangulate tori, hence are necessarily ﬁnite, and to circle packings P which are locally univalent, meaning that the circles neighboring any circle wrap once around it.

CIRCLE PACKINGS ON CONFORMAL AND AFFINE TORI 213

A vertex label R for a complex K is an assignment of a positive number R(u) to each vertex u ∈K. We will think of these numbers as putative radii and often use the notation R(u)=ru. Given a face u, v, w∈Kand label entries ru,rv,rw, we can always lay out a triple of mutually tangent circles having these radii. Their centers determine a triangle, and therefore the label entries ru,rv,rw determine a euclidean angle at the vertex u. Adding the angle at u relative to the label R over all faces containing u yields an angle sum θR(u). Definition 3.2. AlabelR for K is said to satisfy the packing condition at vertex v if θR(u)=2π. R will be called a packing label if it satisfies the packing condition at all vertices of K. The packing condition is the key to circle packing. It is a necessary local condition: if R represents the radii of circles in a (locally univalent) packing P for K,thenR is necessarily a packing label. If K is simply connected, then the converse also holds. Thus, the ability to compute packing labels is tantamount to the ability to create circle packings. The reader should note, however, that the packing condition is local; when K is not simply connected, topology presents obstructions, and not every packing label R gives rise to an actual circle packing P . This is part of the challenge for packings of tori. 3.2. Packings on Conformal Tori. Alan Beardon and Kenneth Stephenson proved a discrete version of the Uniformization Theorem which implies the existence of packings on Riemann surfaces [1]. Discrete Uniformization Theorem. For every (locally finite) triangulation K of an orientable surface, there is a unique Riemann surface SK which supports a circle packing filling SK and having K as its underlying triangulation. Consequently, the combinatorial structure of K uniquely determines the conformal structure of SK. Notice here the rigidity of packings on Riemann surfaces - the combinatorial structure completely determines the conformal structure. As we will see, this stands in sharp contrast to the situation for affine surfaces. One consequence of the rigidity of packings on conformal tori is the fact that most tori do not support a packing. There are only countably many combinatorially distinct triangulations of a topological torus, but the moduli space M is uncount- able. However Phil Bowers and Kenneth Stephenson showed that the packable tori (that is, conformal tori which support a circle packing for some triangulation) are dense in M [2, 3]. 3.3. Brooks Parameterization of Quadrilateral Interstices. A key component of Bowers and Stephenson’s proof was the application of Robert Brooks’s parameterization of quadrilateral interstices, that is, regions bounded by 4 tangent circles [5, 6, 7]. Suppose the sides of such an interstice have been labeled as “left,” “right,” “top,” and “bottom” (for our purposes, it doesn’t matter which direction we consider “up” so long as the left and right sides and the top and bottom sides are opposite one another). Now place a small circle in the upper left corner, tangent to the top and left circles. Increase its radius until it hits either the bottom or the right circle. Identify such a circle as “H” (horizontal) if it intersects the top and bottom and “V” (vertical) if it intersects the left and right sides. Notice this new circle will form triangular interstices with the existing circles and, unless the new circle happens to hit all 4 sides of the original interstice, it will form a new

214 CHRISTOPHER T. SASS, KENNETH STEPHENSON, AND G. BROCK WILLIAMS

Figure 2. A quadrilateral interstice before (left) and after (right) filling with a Brooks packing. Here HHVVVHHHVVHHHH is the filling patter, giving Brooks parameter β =5.290322581. quadrilateral interstice in the bottom right corner. We repeat, adding circles as required toward the bottom right corner. See Figure 2. Suppose h1 is the number of horizontal circles which can be added to our original interstice before a vertical circle must be added. Let v1 be the number of vertical circles which can then be added before the Brooks procedure requires another horizontal circle. Repeat this process to define sequences h2,h3,... and v2,v3,.... The Brooks parameter is defined as the continued fraction 1 (3.1) β = h1 + 1 . v1 + 1 h2+ v2+... Note that the Brooks parameter will be rational if and only if at some stage in the procedure, the next circle intersects all four sides of the remaining quadrilateral interstice (as happens in Figure 2). Then only triangular interstices will remain, both hn and vn will be zero after that stage, and a finite circle packing will have been formed inside the original interstice. If the Brooks parameter is irrational, on the other hand, then the construction gives an infinite collection of circles accumulating in the bottom-right corner of the original interstice, thus failing to be locally finite. The correspondence between quadrilateral interstices and parameters β goes in both directions. That is, given an interstice, the Brooks procedure above produces a positive parameter β. Conversely, every β>0 can be uniquely written as a continued fraction of the form (3.1). Reversing the Brooks procedure, the continued fraction then defines an abstract triangulation of a topological quadrilateral. Gluing together infinitely many copies of this triangulation (matching top to bottom and left to right) produces an infinite simply connected triangulation with a Z × Z symmetry group, which we will denote as K(β). If β is rational, then the Discrete Uniformization Theorem guarantees the existence of a (suitably normalized) packing P (β)forK(β), filling the entire plane. We call such a packing a Brooks packing for β. The circles corresponding to vertices of the original topological quadrilateral now form quadrilateral interstices with Brooks parameter β.

CIRCLE PACKINGS ON CONFORMAL AND AFFINE TORI 215

0 1

Figure 3. The packing P4,8(β) lifts to a Brooks packing on the torus Rτ , with τ = τ(2,β). Here the Brooks packing is that of Figure 2, so β =5.290322581.

If β is irrational, the symmetry of K(β) and the continuity of Brooks parameters with respect to their defining interstices imply K(β) can be realized by a configuration P (β) of circles which is not locally finite (and hence not a true circle packing) at the corners of each quadrilateral. Such a configuration is called a generalized Brooks packing with singularities [18].

3.4. Packing Coordinates. The rigidity of packings on conformal tori thus produces a correspondence between Brooks parameters β and (generalized) Brooks packings that can be exploited to describe coordinates on the moduli space M [18]. Rigidity also forces the Z × Z symmetry of K(β) to be reﬂected in a corresponding symmetry of P (β). Thus if Km,n(β)isanm × n rectangular piece of the triangulated grid K(β), P (β) can be normalized so that the circles corresponding to the lower left and lower right corners of Km,n(β) are centered at 0 and 1, respectively. Denote this portion of P (β) corresponding to Km,n(β)byPm,n(β) and let τ(m/n, β) be the center of the circle at the upper left corner of Pm,n(β). Because of the Z × Z symmetry of P (β), each of circles in Pm,n(β) corresponding to vertices of the original quadrilaterals has diameter 1/n.Thusm circles of diameter 1/n connect 0 to τ(m/n, β), and Pm,n(β) lifts to a Brooks packing on the torus Rτ(m/n,β). See Figure 3, where m =8,n =4,andβ =5.290322581. Now notice as β varies, τ(m/n, β) sweeps out an arc of a circle centered at 0 of radius m/n.Asβ tends to 0 and inﬁnity, the packings will slide to the left and right, respectively. In the limit, each circle will have 6 neighbors, resulting in the

216 CHRISTOPHER T. SASS, KENNETH STEPHENSON, AND G. BROCK WILLIAMS regular hex packing. Thus as β varies from 0 to inﬁnity, τ(m/n, β) sweeps out an arc of measure π/3from(m/n)ei2π/3 to (m/n)eiπ/3. We can extend the deﬁnition of τ(m/n, β)toτ(α, β) for real numbers α ≥ 1 by continuity. Notice that for α ≥ 1, and 0 <β<∞, the arcs formed by τ(α, β) cover the fundamental region for PSL2Z pictured in Figure 1. As a result, we have the following theorem [18]: Theorem 3.3. The construction described above provides coordinates (α, β) for the moduli space M of tori. Moreover, Rτ supports a Brooks packing if τ = τ(α, β) for α, β ∈ Q and a generalized Brooks packing with periodic singularities if α ∈ Q but β/∈ Q. Thus the conformal tori which support Brooks packings are dense in M.

4. Packing Labels for Combinatorial Tori: New Existence Proof The following theorem is a consequence of Beardon and Stephenson’s Discrete Uniformization Theorem, cited above, but the proof is new and will generalize to the setting of aﬃne tori. Theorem 4.1. If K is a combinatorial torus, then there exists a packing label for K. Proof. The argument uses three key tools.

(1) Continuity of angle sums: The angle sum θR(v)iscontinuousintheentries of the label R. (2) Monotonicity of angle sums: If R(v)=R(v) for all v ∈K\{u},andif R(u)

Fix a vertex v0 ∈Kand deﬁne a label R0 ≡ 1. Deﬁne a subset of vertices S { ∈K\{ } ≤ } = v v0 : θR0 (v) 2π .

Define a set Φ of labels R such that θR(v) ≤ 2π for v ∈S, R(v)=R0(v)for v ∈K\(S∪{v0}), and R(v0)=1.LetR(v):=infR∈Φ R(v). By continuity and monotonicity, if R is nondegenerate (that is, R(v) > 0 for all vertices), then ∈S θR(v)=2π for v . Suppose towards a contradiction that R degenerates. Let K0 denote the set of vertices at which R =0,F0, the number of faces having all three vertices in K0, and Fm, the number of “mixed” faces having a vertex in K0 and a vertex not in K0. ∈ → →∞ ∈ ChooseRn Φ such that Rn(v) R(v)asn for all v K.Itiseasytocheck → →∞ | | that w∈K θRn (w) π(F0 + Fm)asn . By counting, F0 + Fm > 2 K0 . 0 ∈S Hence there exist a number N and a vertex w0 such that θRN (w0) > 2π,a ∈S contradiction. We conclude that θR(v)=2π for v . We have thus corrected the angle sums that were too small by decreasing the radii at those vertices (except v0). However, this also caused neighboring angle sums to decrease. So some angle sums that had been too large might now be too small. Consequently, we iterate the process: decrease radii for larger sets of vertices (leaving the radius at v0 fixed) to fix angle sums that are too small. After finitely many iterations (K is finite), all angle sums are too large except at v0. A final

CIRCLE PACKINGS ON CONFORMAL AND AFFINE TORI 217 iteration, this time increasing radii at which angle sums are too large, will force angle sums to be 2π at all vertices in K\{v0}. But the total angle sum added over all faces of K is πF, and hence by counting the angle sum at v0 is πF − 2π(V − 1) = 2πV − 2π(V − 1) = 2π. 5. Packings on Affine Tori 5.1. Affine Packing Labels. Let K be a combinatorial torus. The concate- nation of a pair of simple closed edge paths Γ1 and Γ2 having a single vertex in common (the corner)isafundamental pair for K if a combinatorial cut along Γ=Γ1 ∗ Γ2 produces a combinatorial closed disc. This combinatorial closed disc is a combinatorial fundamental domain for K. A face label S for K is a tuple of positive real numbers, one for each pair (v, f) of vertices v ∈Kand faces f ∈Kcontaining the vertex v. Face label entries will be denoted by Sf (v). A face label S for K is strongly consistent if whenever faces f and g share an edge u, v,then S (u) S (u) f = g . Sf (v) Sg(v) Let Γ be a simple closed edge path and let A>0. A face label S is “Γ(A)” if whenever faces f and g share an edge u, v∈Γ with f to the left of g, A · Sf (u)= Sg(u)andA · Sf (v)=Sg(v). The number A is the affine factor for Γ. A face label S provides a geometry on each face of K, and hence an angle at each vertex of each face. The angle sum θS(v) is defined as before; namely, as the sum of the angles at v in all faces containing v.WesaythatS satisfies the packing condition if θS(v)=2π for all vertices v ∈K. Definition 5.1. A face label S for K is an affine packing label for K with affine factors A and B (positive real numbers) if S is strongly consistent, S satisfies the packing condition, and there is a fundamental pair Γ = Γ1 ∗ Γ2 for K such that S is Γ1(A)andΓ2(B). If R is a vertex label and S is a face label, let the product R · S be the face label given by (R · S)f (v)=R(v) · Sf (v) for all vertices v and faces f containing v. 5.2. Existence of Affine Packing Labels. With these definitions, we can produce packing labels on affine tori [12]. Theorem 5.2. If K is a combinatorial torus and S is a face label for K,then there is a vertex label R for K such that the face label R · S satisfies the packing condition.

Proof. Reinterpret angles αR and angle sums θR in theorem 4.1 as being relative to the face label R · S. The argument remains valid under this reinterpre- tation.

Lemma 5.3. The face label properties strong consistency, Γ1(A),andΓ2(B) are preserved under multiplication by vertex labels. Proof. Clear from the deﬁnitions.

Theorem 5.4. If K is a combinatorial torus, Γ=Γ1 ∗Γ2 is a fundamental pair for K,andA, B > 0, then there is an aﬃne packing label for K that is Γ1(A) and Γ2(B).

218 CHRISTOPHER T. SASS, KENNETH STEPHENSON, AND G. BROCK WILLIAMS

Α Α Α Α Α ΑΒ Α Α Α Α 1 1 Β 1 Β 1 1 1 1 1 1 Β 1 1 1 1 1 1 1 Γ 2 1 1 1 Β Β 1 Β 1 1 1 1 1 1 Β Β 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Β Γ 1

Figure 4. Face label that is strongly consistent, Γ1(A), and Γ2(B).

Proof. By theorem 5.2 and lemma 5.3, it suﬃces to produce a face label S for K that is strongly consistent, Γ1(A), and Γ2(B).Butasseenin[12], such face labels may be easily and routinely constructed. See Figure 4.

5.3. From Affine Packing Labels to Packings. Given K,Γ=Γ1 ∗Γ2,and A, B > 0, we have a vertex label R and a face label S such that R · S is an affine packing label. A combinatorial cut along Γ results in a combinatorial closed disc K. From the face label R·S we can construct a vertex packing label R for K. R satisfies the boundary conditions: angle sums for two paired non-corner boundary vertices add to 2π, angle sums for the four corner vertices add to 2π, R(v)=A · R(v)for vertices paired along Γ1,andR(w )=B · R(w) for vertices paired along Γ2.See Figure 5 for two examples based on the combinatorics of Figure 4. On the right in Figure 5, the paired circles along the top and bottom edges are the same radii (A =1.0), and the circles along the right edge have radii one half the radii of the paired circles along the left edge (B =0.5). The boundary conditions satisfied by such a packing guarantee that the side-pairing maps are affine maps of the form F (z)=αz + γ, G(z)=βz + σ,where|α| = A and |β| = B. Conformal tori T (ω) have canonical metrics, but affine tori T (ω, c), with c =0, do not. This raises the issue of what a “circle” is. If π : T (ω, c) → C is the covering projection, then a circle on T (ω, c) refers to a homotopically trivial simple closed curve τ such that the developed image of each component of π−1(τ) is a euclidean circle. It follows that circle packings for the combinatorial fundamental domains K constructed above may be considered to be circle packings on affine tori. See [12] for more on this approach. The flexibility of circle packings for affine tori is reflected

CIRCLE PACKINGS ON CONFORMAL AND AFFINE TORI 219

Figure 5. Circle Packings for K: on the left, A =1/2andB =2, while on the right, A =1andB =1/2.

Figure 6. A more complicated example with aﬃne parameters A =0.2andB =2.5.

in the freedom in the choice of combinatorics and the aﬃne factors A and B.An example having richer combinatorics is displayed in Figure 6.

Note: images in this paper were produced in CirclePack,see[14].

220 CHRISTOPHER T. SASS, KENNETH STEPHENSON, AND G. BROCK WILLIAMS

References [1] Alan F. Beardon and Kenneth Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), 1383–1425. MR1087197 (92b:52038) [2] Philip L. Bowers and Kenneth Stephenson, The set of circle packing points in the Teichmüller space of a surface of finite conformal type is dense, Math. Proc. Camb. Phil. Soc. 111 (1992), 487–513. MR1151326 (93a:30050) [3] , Circle packings in surfaces of finite type: An in situ approach with application to moduli, Topology 32 (1993), 157–183. MR1204413 (94d:30083) [4] Phillip L. Bowers and Monica K. Hurdal et al, Quasi-conformally flat mapping the human cerebellum, Medical Image Computing and Computer-Assisted Intervention (C. Taylor and A. Colchester, eds.), Lecture Notes in Computer Science, vol. 1679, Springer, 1999, pp. 279– 286. [5] Robert Brooks, On the deformation theory of classical Schottky groups, Duke Math. J. 52 (1985), 1009–1024. MR816397 (87g:32024) [6] , Circle packings and co-compact extensions of Kleinian groups, Inventiones Mathe- maticae 86 (1986), 461–469. MR860677 (88b:32050) [7] , The continued fraction parameter in the deformation theory of classical Schottky groups, Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 41–54. MR1188193 (93j:52029) [8] Gareth Jones and David Singerman, Complex functions: An algebraic and geometric view- point, Cambridge University Press, Cambridge, 1987. MR890746 (89b:30001) [9] Sadayoshi Kojima, Shigeru Mizushima, and Ser Peow Tan, Circle packings on surfaces with projective structures, J. Differential Geom. 63 (2003), no. 3, 349–397. MR2015468 (2004k:52024) [10] , Circle packings on surfaces with projective structures and uniformization,PacificJ. Math. 225 (2006), no. 2, 287–300. MR2233737 (2007h:57021) [11] Shigeru Mizushima, Circle packings on complex affine tori,OsakaJ.Math.37 (2000), no. 4, 873–881. MR1809910 (2003b:52011) [12] Christopher T. Sass, Circle packings on affine tori, Ph.D. thesis, University of Tennessee, Knoxville, August 2011. [13] Kenneth Stephenson, Circle packings in the approximation of conformal mappings,Bul- letin, Amer. Math. Soc. (Research Announcements) 23, no. 2 (1990), 407–415. MR1049434 (92c:30009) [14] , CirclePack open software, (1992-2011), http://www.math.utk.edu/∼kens. [15] Kenneth Stephenson, Introduction to circle packing, Cambridge University Press, Cambridge, 2005, The theory of discrete analytic functions. MR2131318 [16] William Thurston, The geometry and topology of 3-manifolds, Princeton University Notes, preprint. [17] G. Brock Williams, A circle packing measureable Riemann mapping theorem,Proc.AMS 134 (2006), no. 7, 2139–2146. MR2215785 (2006m:52043) [18] , Circle packing coordinates for the moduli space of tori,Proc.AMS139 (2011), no. 7, 2577–2585. MR2784827 (2012c:52043)

Mathematics Department, Young Harris College E-mail address: [email protected] Department of Mathematics, University of Tennessee, Knoxville E-mail address: [email protected] Department of Mathematics and Statistics, Texas Tech University E-mail address: [email protected]

Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11360

Eﬀective radical parametrization of trigonal curves

Josef Schicho and David Sevilla

Abstract. Let C be a non-hyperelliptic algebraic curve. It is known that its canonical image is the intersection of the quadrics that contain it, except when C is trigonal (that is, it has a linear system of degree 3 and dimension 1) or isomorphic to a plane quintic (genus 6). In this context, we present a method to decide whether a given algebraic curve is trigonal, and in the aﬃrmative case to compute a map from C to the projective line whose ﬁbers cut out the linear system.

1. Introduction In the context of symbolic computation for algebraic geometry, an unsolved problem (at least from a computational perspective) is the parametrization of algebraic curves by radicals. Allowing radicals rather than just rational functions greatly enlarges the class of parametrizable functions. For example, one class of curves which are clearly parametrizable by radicals is that of hyperelliptic curves. 2 Every such curve can be written as y = P (x) for some polynomial P (x), and we can quickly write the parametrization x = t, y = P (t). This can be taken further: the roots of univariate polynomials of degree ≤ 4 can be written in terms of radicals. Therefore, curves which can be expressed as f(x, y) = 0 where one of the variables occurs with degree ≤ 4 can also be parametrized by radicals. The minimum degree which can be obtained by is called the gonality of the curve; hyperelliptic curves are precisely those of gonality two. It is thus interesting to characterize the curves of gonality three (or trigonal) and four, and further to produce algorithms that detect this situation and can even compute a radical parametrization. The description of such an algorithm for trigonal curves is the purpose of this article. In this article, the coefficient field always has characteristic zero, and it will generally be assumed to be algebraically closed although we will point out the necessary modifications for the non-algebraically-closed case when they arise. Our algorithm is based on the Lie algebra method introduced in [dGHPS06] (see also [dGPS09]). We use Lie algebra computations (which mostly amount to linear algebra) to decide if a certain algebraic variety associated to the input curve is a rational normal scroll, which is the case precisely when the curve is trigonal.

2010 Mathematics Subject Classiﬁcation. Primary 14H51, 68W30; Secondary 17B45. Partially supported by the Austrian FWF project P 22766-N18 “Radical parametrizations of algebraic curves”.

c 2012 American Mathematical Society 221

222 JOSEF SCHICHO AND DAVID SEVILLA

Further, we can compute an isomorphism between that variety and the scroll when it exists. Algorithm 1 sketches the classiﬁcation part of the algorithm (that is, the detection of trigonality as opposed to the calculation of a 3 : 1 map). It is based on Theorem 2.1.

Algorithm 1: Sketch of algorithm to detect trigonality Input: a non-hyperelliptic curve C of genus g ≥ 3 Output: true if C is trigonal, false otherwise Compute the canonical map ϕ: C → Pg−1 and its image ϕ(C) Compute the intersection D of all the quadrics that contain ϕ(C) if D = C then return false else Determine which type of surface is D if D = P2 then return true // g =3 else if D is a rational normal scroll then return true else return false // Veronese end end

The article is structured as follows. Section 2 recalls the classical theoretical background on trigonal curves and rational scrolls needed. Section 3 is a quick survey of the relevant concepts of Lie algebras and their representations. Section 4 describes the method proper. Our computational experiences with it are reported in Section 5.

2. Classical results on trigonality Let C be an algebraic curve of genus g ≥ 4 and assume that C is not hyperelliptic, so that it is isomorphic to its image by the canonical map ϕ: C → Pg−1. Enriques proved in [Enr19]thatϕ(C) is the intersection of the quadrics that con- 1 tain it, except when C is trigonal (that is, it has a g3) or isomorphic to a plane quintic (g = 6); the proof was completed in [Bab39]. In those cases, the corresponding varieties are minimal degree surfaces, see [GH78, p. 522 and onwards]. Although not relevant in our case it is worth mentioning that [Pet23]provesthat the ideal is always generated by the quadrics and cubics containing the canonical curve. We exclude from our study the curves with genus lower than 3 since they are 1 1 hyperelliptic, thus they have a g2 which can be made into a g3 by adding a base point; the problem is then to find a point in the curve over the field of definition. Also, if the curve is non-hyperelliptic of genus 3, it is isomorphic to its canonical image which is a quartic in P2, and the system of lines through any point of the 1 curve cuts out a g3.

EFFECTIVE RADICAL PARAMETRIZATION OF TRIGONAL CURVES 223

There exist efficient algorithms for the computation of the canonical map, determination of hyperellipticity, and calculation of the space of forms of a given degree containing a curve, for example in Magma [BCP97] and at least partially in Maple. The following theorem summarizes the classification of canonical curves according to the intersection of the quadric hypersurfaces that contain them. Theorem 2.1 ([GH78, p. 535]). For any canonical curve C ⊂ Pg−1 over an algebraically closed field, either (1) C is entirely cut out by quadric hypersurfaces; or (2) C is trigonal, in which case the intersection of all quadrics containing C is isomorphic to the rational normal scroll swept out by the trichords of C;or (3) C is isomorphic to a plane quintic, in which case the intersection of the quadrics containing C is isomorphic to the Veronese surface in P5,swept out by the conic curves through five coplanar points of C. We recall the definition of rational normal scroll (from this point, simply scroll): given two nonnegative integers m ≥ n with m+n ≥ 2, the scroll Sm,n is the Zariski closure of the image of (s, t) → (1 : s : s2 : ...: sm : t : st : s2t : ...: snt) ⊂ Pm+n+1. It is defined by equations of degree two involving four terms each. It is a ruled surface, its pencil of lines being given by the fibers s = constant. This ruling is unique except when m = n = 1 in which case there are two rulings. Any map 1 Sm,n → P whose fibers are lines is called a structure map. If we work over a non-algebraically closed field k, one may get surfaces S which are isomorphic to scrolls only over k (called twists). The structure map S → P1 over k is given by a divisor class, which may be not defined over k,ormaybe defined over k but have no divisors in it over k. The first case can only occur for genus 4, see Section 4.2 for the details. In the second case (the divisor class is defined over k but has no elements over it), one can define a map S → E whose fibers are lines, where E is a conic over k with no points defined over it (see [SWPD08]). This is done by taking the structure map S → P1 and symmetrizing it with its Galois conjugates over k.Since E has no points defined over k, neither does S or the trigonal curve C. However, one can always go to a degree 2 extension of k where E hasapointandworkon that extension. Let M be a chosen algebraic variety (a “model”) and X ∈ PN be any given variety. In some cases, we can use Lie algebra representations (section 3) to decide if X is projectively isomorphic to M (we call this recognition of M) and furthermore to compute a projective isomorphism between them in the affirmative case (we call this constructive recognition). We will use the same terms for Lie algebras. Strictly speaking, we are not interested in constructive recognition of scrolls, since we only need the structure map whose fibres will cut out the trigonal linear system. However, it is possible to use the method described below to construct isomorphisms to the models of the scrolls, as we will comment in each case.

3. Lie algebras The Lie algebra of a projective variety is an algebraic invariant which is relatively easy to calculate when the variety is generated by quadrics (it is often

224 JOSEF SCHICHO AND DAVID SEVILLA cheaper than a Gröbner basis of the defining ideal, if only generators are given). We offer here a quick summary of relevant properties of Lie algebras in general, see [dG00, FH91] for a general overview. Most definitions and basic results can be found in the aforementioned references, we limit ourselves to what is relevant for our purposes. Definition 3.1. Let X ⊂ PN be an embedded projective variety. The group N of automorphisms of P is PGLN+1, the group of all invertible matrices of size N + 1 modulo scalar matrices. Let PGLN+1(X) be the subgroup of all projective transformations that map X to itself (this is always an algebraic group). The Lie algebra L(X)ofX is defined as the tangent space of PGLN+1(X) at the identity, together with its natural Lie product. Example 3.2. N (1) The Lie algebra of P is the tangent space of PGLN+1 at the identity matrix;thisisdenotedslN+1 and its elements are trace zero matrices. 1 (2) In particular, L(P )=sl2, whose elements are 2 × 2 trace zero matrices. It has dimension 3 and it has a basis 10 01 00 h := ,x:= ,y:= 0 −1 00 10 This particular basis is an instance of the so-called Chevalley basis or canonical basis,see[dG00, Section 5.11]. Remark 3.3. Note that the Lie algebra of any X ⊂ PN is a subalgebra of slN+1, since PGLN+1(X) is a subgroup of PGLN+1 so the same relation holds for their tangent spaces.

For varieties of general type (in particular curves of genus at least 2), PGLN+1(X) is ﬁnite and therefore the Lie algebra is zero. On the other hand, the Veronese surface and the rational scrolls have Lie algebras of positive dimension. This allows us to reduce the recognition problem for these surfaces to Lie algebra computations (see Section 4). The next theorem provides a fast way to compute the Lie algebra of the varieties we are interested in. Theorem 3.4. Let X ⊂ PN such that I(X) is generated by quadrics, and G be a set of quadratic generators. Then 2 2 d 2 L(X)= M ∈ slN+1 : ∀f ∈ G, f(IN + tM)2 ∈ I dt t=0 d d where by dt f(IN +tM) we mean the function dt f applied to the image of the vector of variables by the transformation IN + tM. Proof. See [dGPS09, Theorem 5]. Remark 3.5. The previous theorem may be true without the hypothesis on the generators, but we do not know a proof; however this restricted version is all that we need for the upcoming discussion. 3.1. Representations of Lie algebras. As in the case of groups, one can understand a lot of things about Lie algebras by thinking of them as spaces of matrices. This is the meaning of the concept of representation that we introduce now.

EFFECTIVE RADICAL PARAMETRIZATION OF TRIGONAL CURVES 225

Definition 3.6. A representation of a Lie algebra L is a Lie algebra homomorphism L → gl(V ) for some vector space V . This is equivalent to a bilinear action L × V → V which turns V into a Lie module over L.Thedimension of a representation is the dimension of V .

Definition 3.7. (1) A linear subspace W of V is a submodule iﬀ L · W ⊆ W , that is, if the action can be restricted to W . (2) A module is irreducible iﬀ it only has trivial submodules (the trivial and total subspaces).

An important class of Lie algebras, which are basic building blocks in the classiﬁcation theory of Lie algebras and the study of their representations, is that of semisimple ones. The following convenient result holds:

Theorem 3.8 (Weil’s theorem). If L is semisimple, every ﬁnite-dimensional module over L is a direct sum of irreducible modules.

Proof. See [dG00, Section 4.4].

3.2. The Lie algebra sl2. A particularly important Lie algebra is sl2 (Ex- ample 3.2). Since it will feature often in this article, we highlight here some of the properties that we will use. Computationally speaking, it is easy to recognize sl2. First we need a definition. Definition 3.9. Let L be a Lie algebra. For each x ∈ L, define a Lie endomorphism adx : L → L as adx(y)=[x, y]. Then the Killing form on L is a bilinear map BL : L × L → k given by BL(x, y)=Trace(adX ◦ adY ). Theorem 3.10. [dGPS09, Proposition 10] Let L be a semisimple Lie algebra of dimension 3. Then L is isomorphic to sl2 iff its Killing form is isotropic.

In particular, over an algebraically closed field sl2 is the only semisimple algebra of dimension 3. ∼ Once we have recognized L = sl2, the construction of a Lie algebra isomorphism can be done by finding a Chevalley basis of L, see the proof of the previous theorem. In general, the procedure amounts to finding a rational point on a conic over the field of definition, which in the case of Q needs factorization of integers. The irreducible representations of sl2 can be described as follows: for every nonnegative integer n, there exists a unique irreducible representation (up to isomorphism) of dimension n +1. For n = 0 this is just the trivial representation. For positive n, we describe it as an action of sl2 on the n + 1-dimensional vector space of homogeneous polynomials of degree n in the variables x, y: ab P (x, y) · = P (ax + by, cx + dy) cd

In terms of modules, they consist of a 1-dimensional module with the zero action, a 2-dimensional module N,andthe(n + 1)-dimensional symmetric powers Symn(N) for n ≥ 2.

226 JOSEF SCHICHO AND DAVID SEVILLA

Remark 3.11. By choosing an adequate basis of eigenvectors of the image of h by the representation, the images of the Chevalley basis can be taken to be ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ n 01 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ n − 2 ⎟ ⎜ 02 ⎟ ⎜n 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ → ⎜ .. ⎟ → ⎜ .. .. ⎟ → ⎜ .. .. ⎟ h ⎜ . ⎟ ,x ⎜ . . ⎟ ,y ⎜ . . ⎟ ⎝ −n +2 ⎠ ⎝ 0 n⎠ ⎝ 20 ⎠ −n 0 10 See [dG00, Section 5.1] and [FH91, Lecture 11].

4. The Lie algebra method for trigonal curves

Consider the problem of recognizing Sm,n. In fact, since we normally have only avarietyX but not m, n, we want to decide if X is isomorphic to Sm,n for some unknown m, n. The knowledge of the representations of the Lie algebras of the scrolls, in fact of semisimple parts of them, will allow us to decide the answer and even to compute such an isomorphism. Definition 4.1. Every finite-dimensional Lie algebra L canbewrittenasa semidirect sum of two parts called a solvable part and a semisimple part. The latter is called a Levi subalgebra of L, and it is unique up to conjugation, so we will speak of “the” Levi subalgebra of L and denote it as LSA(L). For a variety X, we will denote LSA(L(X)) simply by LSA(X). As mentioned before, the Lie algebra of a curve of genus 2 or higher is zero since its automorphism group is finite. The rest of the cases that arise in Theorem 2.1 are studied in the next result. Theorem 4.2. Let k be an algebraically closed field of characteristic zero. As above, let Sm,n be the the rational normal scroll with parameters m, n,andletV be the image of the Veronese map P2 → P5. ∼ (1) LSA(S ) = sl if m = n. m,n ∼ 2 (2) LSA(S ) = sl + sl (a direct sum of two Lie algebras) m,m∼ 2 2 (3) LSA(V ) = sl3. Proof. See [Oda88, Section 3.4]. Additionally, for any fixed pair m, n one can easily check the claim, for example in Magma. It is clear now that just by looking at the dimension of the Levi subalgebra we can discard the two cases where the curve is not trigonal. In other words, we can recognize a trigonal curve by the dimension of its Levi subalgebra. Corollary 4.3. Let k be any field of characteristic zero, let C be a canonical curve and X be the intersection of the quadrics that contain it. Then one of the following occurs: • If dim LSA(X)=0then X = C and C is not trigonal. • If dim LSA(X)=3then X is a twist of Sm,n with m = n and C is trigonal. • If dim LSA(X)=6then X is a twist of S and C is trigonal. ∼ m,m • If dim LSA(X)=8then X = V and C is not trigonal. Proof. Pass to k and apply the previous theorem.

EFFECTIVE RADICAL PARAMETRIZATION OF TRIGONAL CURVES 227

If the surface is the Veronese surface, the algorithm terminates and reports that the curve is not trigonal. Nevertheless, for the sake of completeness we must mention that an analysis similar to the scrolls below can be performed and results in the constructive recognition of the Veronese surface; this provides an isomorphism to a plane quintic curve, and knowing a point the pencil of lines through that point 1 will give a g4. This only occurs for genus 6. 4.1. The case m = n. Since the Levi subalgebras of these scrolls are always sl2, and thanks to the classification above, we have a necessary and sufficient condition for to the recognition problem. Following Section 3.2 we assume that we have constructed an isomorphism σ : sl2 → LSA(X). The representation of sl2 given by the inclusion of Sm,n, m = n into projective space is known. Theorem 4.4. Consider S ⊂ Pm+n+1, m = n. The module sl -module m,n ∼ 2 induced on the underlying vector space V = km+n+2 decomposes into irreducible modules as Symm(k2) ⊕ Symn(k2). Proof. See [Oda88, Section 3.4]. Additionally, for any fixed pair m, n one can easily check the claim, for example in Magma.

Remark 4.5. Consider the representation of sl2 corresponding to the module m+n+2 structure, Rep: sl2 → gl(k ). With respect to a suitable basis, the matrices of the elements h, x, y in a Chevalley basis will consist on two blocks of dimensions m +1 and n + 1 having the form given in Remark 3.11. Thus Rep(h) is a diagonal matrix with eigenvalues m, m − 2,...,−m, n, n − 2,...,−n,and ⎛ ⎞ 0 ⎜ ⎟ ⎜ m 0 ⎟ ⎜ ⎟ ⎜ .. .. ⎟ ⎜ . . ⎟ ⎜ ⎟ ⎜ 20 ⎟ ⎜ ⎟ ⎜ 10 ⎟ Rep(y)=⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ n 0 ⎟ ⎜ ⎟ ⎜ . . ⎟ ⎜ .. .. ⎟ ⎝ 20 ⎠ 10 m+n+1 Theorem 4.6. Let X ∈ P be a variety isomorphic to Sm,n with m> n and let v, w ∈ km+n+2 such that v is an eigenvector of Rep(h) with largest eigenvalue and w = Rep(y) · v. Then the function γ : X → P1 : x → (w · x)/(v · x), where · is the scalar product, has as its fibers the lines of X. Proof. We prove this for the scroll and the result will follow since the isomorphism will respect the construction. So we consider Sm,n which is the image of (s : t) → (1 : s : s2 : ... : sm : t : st : s2t : ... : snt) ⊂ Pm+n+1.ByRe- mark 4.5 we must have v =(λ, 0,...,0) and w =(0,mλ,0,...,0). In terms of the parametrization, γ on Sm,n is just mλs/λ = ms and its fibres are clearly lines. It is worth noting that one can extend this procedure to calculate a basis of eigenvectors of km+n+2. One can determine the eigenvalues of Rep(h)andread off m and n. Then we just need an eigenvector for m and n, but if m, n have

228 JOSEF SCHICHO AND DAVID SEVILLA the same parity there is an ambiguity since the eigenspace for n has dimension 2; it suffices to intersect it with the kernel of Rep(x) in order to isolate the correct unidimensional eigenspace for n. By successive application of Rep(y)wecomplete the basis of eigenvectors, and the conversion from the canonical basis to the new basis produces a linear isomorphism of Pm+n+1 which restricts to an isomorphism between X and Sm,n. 4.2. The case m = n. The idea is similar to the previous case but the details are somewhat different. One can obtain representations of 2sl2 from representations of sl2,orinother words 2sl2-modules from sl2-modules. Indeed, in general, if V1 is a L1-module and V2 is a L2-module, then V1 ⊗ V2 is a (L1 + L2)-module via the action (l1 + l2) · (v1 ⊗ v2)=(l1 · v1) ⊗ v2 + v1 ⊗ (l2 · v2). We saw in Section 3.1 that the irreducible representations of sl2 are classified by their dimension. Likewise, every irreducible representation of 2sl2 is the tensor product of two irreducible representations of sl2 and determined by the dimensions of the two parts. This is due to the characterization of a representation of a semisimple algebra being irreducible precisely when it has a highest weight (see [dG00, Chapter 8] for definitions and properties), and the fact that the tensor product of two irreducible representations of sl2 has a highest weight given by the highest eigenvalues of the matrices corresponding to the respective h elements in their Chevalley bases (see Remark 3.11). 2m+1 Thus, the inclusion Sm,m ⊂ P induces two representations of sl2, corresponding to the two summands of the Levi subalgebra. Each one of them is the tensor product of an irreducible and a trivial representation, of dimensions 2 and m +1(see[Oda88, Section 3.4], or calculate for particular values of m). In terms of matrices, this is given by the Kronecker products (see [HJ91, Section 4.2]) of ⎛ ⎞ m ⎜ − ⎟ ⎜ m 2 ⎟ 10 ⎜ . ⎟ and ⎝ .. ⎠ 0 −1 −m with I2 and Im+1, respectively. They are ⎛ ⎞ m ⎛ ⎞ ⎜ ⎟ ⎜ m − 2 ⎟ 1 ⎜ ⎟ ⎜ −1 ⎟ ⎜ .. ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ −m ⎟ ⎜ − ⎟ ⎜ ⎟ or ⎜ 1 ⎟ ⎜ m ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ m − 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 1 ⎠ ⎝ .. ⎠ . −1 −m

In short, we have two copies of sl2 acting on the underlying vector space, one of them decomposes into irreducibles as a sum of m two-dimensional representations and the other as a sum of two m-dimensional representations. Once we decompose 2sl2 into two copies of sl2, how to discern the two tensor product representations? Note that it is only needed if m>1. We can decide this by considering the matrices for h. Note that the square of a matrix similar to

EFFECTIVE RADICAL PARAMETRIZATION OF TRIGONAL CURVES 229 the right hand side is the identity, which is not the case for the matrix on the left. So we can identify this case, for example, by picking one of the two matrices and checking if the degree of its minimal polynomial is 2. Once we have distinguished the two representations, we concentrate on the left hand side representation. Call 2m+2 it Rep: sl2 → gl(k ). We obtain a result very similar to that of the case m = n.

2m+1 Theorem 4.7. Let X ∈ P be a variety isomorphic to Sm,m and let v, w ∈ k2m+2 such that v is an eigenvector of Rep(h) with eigenvalue m and w = Rep(y)·v. Then the function γ : X → P1 deﬁned by γ(x)=(w · x)/(v · x) has as its ﬁbers the lines of X.

Proof. It suﬃces to prove this for the scroll Sm,m. This is the image of (s : t) → (1 : s : s2 : ... : sm : t : st : s2t : ... : smt) ⊂ P2m+1.ThenbyRemark 4.5 any such v must be equal to (λ1, 0,...,0,λ2, 0,...,0). But the coordinates of w are the result of a right-shift and multiplication by m. As a result, γ on Sm,m is just (mλ1s + mλ2st)/(λ1 + λ2t)=ms. Clearly its ﬁbres are lines.

5. Computational experiences We have tested our implementation in Magma V2.14-7 against random examples of trigonal curves over the ﬁeld of rational numbers. The computer used is a 64 Bit, Dual AMD Opteron Processor 250 (2.4 GHZ) with 8 GB RAM. We have generated trigonal curves in the following two ways:

(1) Let C : f(x, y, z) = 0 homogeneous with degy f = 3. Then the projection (x : y : z) → (x : z)isa3:1maptoP1. The genus of a curve of degree 3 in y and degree d in x is 2(d − 1) generically. The size of the coefficients is controlled directly. (2) Let C be defined by the affine equation Resultantu(F, G)=0where

3 0=x − a1(u)x − a2(u)=:F, 2 0=y − a3(u) − a4(u)x − a5(u)x =: G

for some polynomials a1,...,a5. This clearly gives a field extension of degree 3, thus there is a 3 : 1 map from C to the affine line. Examples show that the degree and coefficient size for a given genus are significantly larger than for the previous construction. It is important to remark that we need not compute either the genus or the hyperellipticity of the curve beforehand, since both things are detected by the algorithm: the genus is a byproduct of the computation of the canonical image, and the canonical image of a hyperelliptic curve is a rational normal curve, a situation detected by the Lie algebra computation. For the same reason, we have only performed our timing tests on trigonal curves: when the intersection of the quadrics containing the canonical curve is unidimensional or isomorphic to the Veronese surface, this will be detected by inspecting the dimension of the whole Lie algebra or of the Levi subalgebra respectively. These are the time results (in seconds) for samples of fifty random curves of degree 3 in one of the variables, for various values of total degree, genus and

230 JOSEF SCHICHO AND DAVID SEVILLA coefficient size. Note the effect of the coefficient size on the computing time.

genus degx bit height min avg max 43 8 0.020 0.030 0.080 4 3 200 0.140 2.606 36.240 18 10 8 26.880 27.843 28.410 30 16 2 1690.340 2099.167 2435.630 30 16 8 out of memory

For the second method, we choose a1,...,a5 randomly of degree d and integer coeﬃcients between −e and e. These are the time results (in seconds) for samples of ﬁfty random curves, for various values of d, e. (d, e) genus deg bit height min avg max (4, 2) 3 − 417− 20 10 − 18 0.030 0.207 0.980 (4, 2000) 42085− 95 0.630 0.847 1.490 (5, 2) 3 − 620− 25 16 − 21 0.050 1.939 2.890 (5, 200) 62576− 86 6.680 8.975 11.370 Our Magma implementation can be obtained by contacting us directly.

References [Bab39] D. W. Babbage. A note on the quadrics through a canonical curve. J. London Math. Soc., 14:310–315, 1939. MR0000496 (1:83b) [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993). MR1484478 [dG00] Willem A. de Graaf. Lie algebras: theory and algorithms,volume56ofNorth-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 2000. MR1743970 (2001j:17011) [dGHPS06] Willem A. de Graaf, Michael Harrison, Jana P´ılniková, and Josef Schicho. A Lie algebra method for rational parametrization of Severi-Brauer surfaces. J. Algebra, 303(2):514–529, 2006. MR2255120 (2007e:14058) [dGPS09] Willem A. de Graaf, Jana P´ılniková, and Josef Schicho. Parametrizing del Pezzo surfaces of degree 8 using Lie algebras. J. Symbolic Comput., 44(1):1–14, 2009. MR2474200 (2009k:14118) [Enr19] Federigo Enriques. Sulle curve canoniche di genere p dello spazio a p − 1 dimensioni. Rend. Accad. Sci. Ist. Bologna, (23):80–82, 1919. [FH91] William Fulton and Joe Harris. Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in Math- ematics. MR1153249 (93a:20069) [GH78] Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. Wiley- Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR507725 (80b:14001) [HJ91] Roger A. Horn and Charles R. Johnson. Topics in matrix analysis. Cambridge Uni- versity Press, Cambridge, 1991. MR1091716 (92e:15003) [Oda88] Tadao Oda. Convex bodies and algebraic geometry,volume15ofErgebnisse der Math- ematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties, Trans- lated from the Japanese. MR922894 (88m:14038) [Pet23] K. Petri. Uber¨ die invariante Darstellung algebraischer Funktionen einer Veränderlichen. Math. Ann., 88(3-4):242–289, 1923. MR1512130 [SWPD08] J. Rafael Sendra, Franz Winkler, and Sonia Pérez-D´ıaz. Rational algebraic curves, volume 22 of Algorithms and Computation in Mathematics. Springer, Berlin, 2008. A computer algebra approach. MR2361646 (2009a:14073)

EFFECTIVE RADICAL PARAMETRIZATION OF TRIGONAL CURVES 231

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria E-mail address: [email protected] Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria E-mail address: [email protected]

CONM 572 opttoa leri n nltcGeometry Analytic and Algebraic Computational

This volume contains the proceedings of three AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties held January 8, 2007, in New Orleans, LA; January 6, 2009, in Washington, DC; and January 6, 2011, in New Orleans, LA. Algebraic, analytic, and geometric methods are used to study algebraic curves and Rie- mann surfaces from a variety of points of view. The object of the study is the same. The methods are different. The fact that a multitude of methods, stemming from very different mathematical cultures, can be used to study the same objects makes this area both fascinat- ing and challenging. • epl ta. Editors al., et Seppälä

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