Computing the Zeta Function of Two Classes of Singular Curves By

Computing the Zeta Function of Two Classes of Singular Curves

Robert Burko

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto

c Copyright (year of graduation) by Robert Burko Contents

1 Introduction 1 1.0.1 Point Counting and the Zeta Function ...... 1 1.0.2 Weil Cohomology ...... 3 1.1 Algorithmic Approaches ...... 4 1.1.1 `-Adic Methods ...... 4 1.1.2 p-Adic Methods ...... 5 1.1.3 Deformation and Fibration Methods ...... 6 1.1.4 Approaches to Singular Varieties ...... 7 1.2 This Thesis ...... 8 1.3 Applications and Future Work ...... 8 1.3.1 Cryptography ...... 9 1.3.2 Support for Dimca’s Conjecture ...... 10 1.3.3 Potential Generalizations and Improvements ...... 11

2 Cohomology Theories 13 2.1 Preliminaries ...... 13 2.1.1 p-Adic numbers and Witt vectors ...... 13 2.1.2 Useful Properties of Etale´ Maps ...... 14 2.2 Algebraic de Rham Cohomology ...... 15 2.2.1 K¨ahlerDiﬀerentials ...... 15 2.2.2 de Rham Cohomology for Schemes ...... 15 2.2.3 de Rham Cohomology with Logarithmic Singularities ...... 16 2.3 p-Adic Cohomology Theories ...... 20 2.3.1 Monsky-Washnitzer Cohomology ...... 20 2.3.2 Rigid Cohomology and Crystalline Cohomology ...... 21 2.3.3 Comparisons Theorems Between p-Adic and de Rham Cohomology 22 2.4 Exact Sequences ...... 23

ii 3 Superelliptic Curves 25 3.1 Basic Properties ...... 25 3.1.1 The Genus ...... 26 3.1.2 The Zeta Function ...... 30 1 0 − 3.1.3 The Vector Space HMW(CK /K) ...... 31 3.1.4 Some Useful Order-preserving Functions ...... 34 3.2 Computing a Basis for Cohomology ...... 38 3.2.1 The Reduction Process ...... 39 3.2.2 Two Lemmas ...... 43 3.3 The Matrix of Frobenius ...... 51 3.4 Working Within a Crystalline Basis ...... 52 3.5 p-Adic Precision Analysis ...... 59

4 Nodal Curves in P2 62 4.1 Cohomology of the Aﬃne Complement of a Hypersurface in Pn ...... 62 4.2 Basic Properties of Nodal Curves ...... 68 4.2.1 Computing a Lift ...... 68 4.2.2 The Zeta Function of a Nodal Curve ...... 74 4.3 A Crystalline Lattice of the Aﬃne Complement ...... 77 4.4 The Matrix of Frobenius ...... 85 4.5 p-Adic Precision Analysis ...... 88

5 Algorithms and Complexity Estimates 90 5.1 Superelliptic Curve ...... 90 5.1.1 Algorithm ...... 90 5.1.2 Complexity Analysis ...... 92 5.2 Nodal Plane Curve ...... 92 5.2.1 Algorithm ...... 92 5.2.2 Complexity Analysis ...... 93

6 Experiments 95 6.1 Examples of Superelliptic Curves ...... 95 6.2 Examples of Nodal Plane Curves ...... 97

iii Chapter 1

Introduction

1.0.1 Point Counting and the Zeta Function

Let p be a prime, let Fp be a finite field with p elements, and let X be an algebraic variety defined over Fp. For instance, X might be the simultaneous solution of the system of polynomial equations   f1(x1, ..., xn) ≡ 0 mod p   f2(x1, ..., xn) ≡ 0 mod p .  .   fm(x1, ..., xn) ≡ 0 mod p

where the variables x1, ..., xn lie in Fp. If we restrict the variables to take values in a finite extension of Fp, then there are only a finite set of possibilities for each variable, and so this system has a finite number of solutions. An interesting question in number theory, dating back at least of far as Gauss’ Disquisitiones Arithmeticae [1], asks for the number of solutions to such systems.

An almost equivalent but slightly more involved question is to calculate what is known as the zeta function of X. In general, we let X be an algebraic variety defined over a a finite field with q = p elements, and let #X(Fqk ) denote the number of solutions its defining equation has over the finite field Fqk , the so-called “Fqk -rational points” of X. One defines the zeta function, a formal power series associated to X, by

∞ ! X T k Z(X,T ) = exp #X( k ) ∈ [[T ]]. Fq k Q k=1 The zeta function has many interesting properties. For instance, from Galois theory one can determine that the coeﬃcients of its expanded power series are integers. More

1 Chapter 1. Introduction 2 astonishingly, in 1960 it was proven by Dwork that the zeta function is rational, that is, a quotient of two polynomials with integer coeﬃcients [2]. Dwork’s proof is analytic in nature − he shows that the zeta function is meromorphic, both as a function on the complex plane and on the completion of the algebraic closure of the ﬁeld of rational p-adic numbers. The following theorem was conjectured by Weil in 1948 and proven by him in the case of curves, but not proven in full generality until 1974 by Deligne [3].

Theorem 1.0.1. Let X by a smooth projective variety of dimension n deﬁned over Fq. Then

1. Z(X,T ) is a rational function of T , and can be written

P (T )P (T ) ··· P (T ) Z(X,T ) = 1 3 2n−1 P0(T )P1(T ) ··· P2n(T )

with Pi(T ) ∈ 1 + T Z[T ]. Moreover, the polynomials Pi(T ) satisfy the following properties:

n i) P0(T ) = 1 − T and P2n(T ) = 1 − q T .

n ii) The map x → q /x sends the roots of Pi(T ) to the roots of P2n−i(T ), preserving multiplicities. iii) If one writes Y Pi(T ) = (1 − αijT ) j

i/2 then αij is an algebraic integer with complex absolute value q .

P i 2. Let E = i(−1) deg(Pi). There is a functional equation

1 Z X, = ±qnE/2tEZ(X,T ). qnT

3. If R is the integer ring of a number ﬁeld, p is a prime ideal in R lying over p, and

X is the reduction modulo p of a smooth scheme Y over R, then deg(Pi) is the i-th

Betti number of Y ×R C as a topological space

Note that the condition for this theorem is that X is smooth and projective, whereas Dwork’s proof of the rationality of the zeta function is valid for any variety deﬁned over

Fq. Chapter 1. Introduction 3

1.0.2 Weil Cohomology

The proof of Theorem (1.0.1) requires ﬁnding a “good” cohomology theory for X, known as a Weil cohomology. Such a theory is not unique for X and can be deﬁned in the following way.

k Definition 1.0.2. Let = Fq be a finite field of characteristic p, and let K be a field of characteristic 0. A Weil cohomology is a contravariant functor from smooth proper varieties X over k to graded algebras H•(X), where each Hi(X) is a finite dimensional K-vector space satisfying the following properties:

1. If n = dim(X), then Hi(X) = 0 for i∈ / {0, 1, ..., 2n}.

i i 2. (Lefschetz ﬁxed point theorem) There are “Frobenius maps” Fi : H (X) → H (X), i ∈ {0, ..., 2n} such that

2n X i k i #X(Fqk ) = (−1) Tr(Fi |H (X)) i=0

3. (Poincar´eduality) The vector space H2n(X) is one-dimensional, and the map

Hi(X) × H2n−i(X)(n) → H2n(X)(n)

is a perfect pairing for i ∈ {0, 1, ..., 2n}, and equivariant under the Frobenius maps i i −n (here H (X)(n) denotes the vector space H (X) with Fi replaced by q Fi).

In actuality, Weil cohomologies possess more structure than this, including a cycle class map, K¨unneth formula, and a Lefschetz hyperplane theorem, as well as Frobenius compatibility maps, but for the purposes of this thesis we are interested in the above properties.

When X is not smooth or proper, it is possible that X → Hi(X) is still a well-defined functor, and satisfies some of the above properties. For instance, the rigid cohomology i groups Hrig(X) are defined for any Fq-scheme X, and are finite dimensional and satisfy the Lefschetz fixed point theorem when X is smooth and affine. If X is a singular, projective hypersurface, then Poincaréduality still holds for 0 ≤ i ≤ dim(X) − dim(Y ), where Y is the singular locus of X. From the following lemma, to ensure rationality of the zeta function it is sufficient to have finite dimensionality of each Hi(X) combined with the Lefschetz fixed point formula. Chapter 1. Introduction 4

Lemma 1.0.3. Let V be a ﬁnite dimensional vector space over a ﬁeld K, and let I denote the identity on V . Then for any endomorphism F of V , there is an identity in the power series ring K[[T ]]

∞ ! X T k exp Tr(F |V ) = det(I − TF |V )−1. k k=1

Proof. A simple calculation, see for instance [4, Appx C, Lemma 4.1].

i It follows immediately that if one deﬁnes Pi(T ) = det(I − TFi|H (X)), then the zeta function of X can be written

∞ 2n ! X X T k Z(X,T ) = exp (−1)iTr(F k|Hi(X)) i k k=1 i=0 i 2n ∞ !!(−1) Y X T k = exp Tr(F k|Hi(X)) i k i=0 k=1 2n Y (−1)i+1 = Pi(T ) . i=0 1.1 Algorithmic Approaches

∞ For a variety X deﬁned over Fq, the problem of computing the sequence {#X(Fqk )}k=1 is considered to be a hard problem. The brute force approach of inserting values into the deﬁning equation of X has running time polynomial in q (i.e. exponential in log q) whereas modern approaches (albeit usually only studied for curves and abelian varieties) are expected to run in a number of operations polynomial in log q. Methods for computation of zeta functions of algebraic varieties are commonly placed under two main headings, `-adic and p-adic approaches. In this section we will give short summaries of the various algorithms, as well as work that has been done to broaden their scope.

1.1.1 `-Adic Methods

The ﬁrst polynomial-time algorithm was developed by Schoof [5] and is used to count 2 3 points on an elliptic curve E/Fq given by a Weierstrass model y = x + Ax + B . Schoof computes the trace of the Frobenius endomorphism t = q − #E(Fq) + 1 modulo various primes ` diﬀerent from the characteristic of Fq, by computing the action of Frobenius on √ the `-torsion points of the curve. He then uses the Hasse-Weil bound |t| ≤ 2 q along with the Chinese remainder theorem to calculate t. This algorithm, after improvements Chapter 1. Introduction 5 by Atkins, Elkies and Couveignes, has running time O((log q)4).

In 1990, an attempt to extend Schoof’s algorithm to more general curves and abelian varieties was made by Pila in [6] and improved by Adelman and Huang [7]. Using a similar approach to that of Pila, a randomized algorithm was developed by Huang and Ierardi [8] in which one computes the number of points on a plane curve of degree d which has only ordinary multiple points in time (log q)dO(1) . Practical methods for these “Schoof-like” algorithms have not been implemented nearly as much as in the case of elliptic curves. However, some success in the case of genus 2 curves has been achieved [9].

1.1.2 p-Adic Methods

In 2000, Satoh demonstrated algorithmically [10] that for an elliptic curve E/Fq, one could explicitly construct a unique elliptic curve E/K˜ called the “canonical lift”, defined over a p-adic field K, and satisfying End(E) ∼= End(E˜). One could thereby compute the trace of Frobenius on E directly as the trace of some corresponding endomorphism on E˜. If q = pa and if we consider p to be fixed, then the running time of his algorithm is O(a2+ε). This is much faster than corresponding `-adic methods. However a significant drawback is that Satoh’s algorithm runs exponentially in log p, so one is forced to use curves over fields with small characteristic.

One of the great advantages of Satoh’s algorithm is that it does not depend on the group structure of the curve. Rather one only needs to consider its defining equation. Thus it spawned a number of new techniques for computing zeta functions of more general algebraic varieties. What was required was a sufficiently strong cohomology theory for varieties over Fq that could somehow be represented using the defining equation. One theory that is especially relevant to this thesis originates from the work of Monsky and Washnitzer [11].

In a paper [12] which appeared in 2001, Kedlaya showed in odd characteristic how to explicitly construct the Monsky-Washnitzer cohomology groups of a curve C0, obtained by removing points from a hyperelliptic curve C/Fq given by a nonsingular affine planar equation y2 = f(x). One lifts the coordinate ring of C0 to the integer ring of a p-adic field and “weakly completes” it with respect to the p-adic norm (obtaining what is called a “dagger algebra”). This defines an object with two fibres, one of which is isomorphic to Chapter 1. Introduction 6

C0 (the special fibre), and the other defined over a field of characteristic 0 and possessing analytic properties (the generic fibre). One can then compute the trace of Frobenius from the cohomology of the generic fibre.

Kedlaya’s algorithm has running time O(g4+εa3+ε) where g is the genus of the curve, assuming that the characteristic of the base field is fixed. There has been effort made to show Kedlaya’s algorithm can be implemented effectively for curves over finite field of large characteristic (see [13] and [14]). The algorithm has also been adapted extensively to include more general settings. The algorithm was extended to the characteristic 2 case by Denef and Vercauteren [15], to superelliptic curves by Gaudry and Gürel[16], to

Ca,b curves by Denef and Vercauteren [17], and to smooth projective hypersurfaces by Abbott, Kedlaya, and Roe [18].

1.1.3 Deformation and Fibration Methods

Other p-adic methods for computing zeta functions derive from p-adic analysis, based on the cohomology and deformation theory of Dwork. In 2004, Lauder [19] [20] proposed a method for computing zeta functions of projective hypersurfaces of arbitrary dimension by embedding the surface into a family of hypersurfaces containing a diagonal element1. On such a family Dwork associates a differential equation whose solution parameterizes the matrices of Frobenius for each fibre. One first computes the Frobenius matrix for Dwork cohomology on the diagonal fibre (a relatively simple task), solves the differential equation locally around this specific value, and then evaluates the solution at the intended fibre. This method was later recast by Gerkmann in a framework more akin to algebraic geometry, first for the case of families of elliptic curves [21] and later for families of smooth projective hypersurfaces [22] using Berthelot’s “rigid cohomology” [23] [24]. This is a Weil cohomology theory that has the added advantage of being defined for arbitrary varieties (not necessarily smooth or proper), as well as comparison maps to de Rham cohomology from which it inherits the differential equation of Dwork’s theory.

Yet another novel algorithm for computing zeta functions was proposed by Lauder [25] using a “ﬁbration method”. That is, instead of deforming X to a diagonal hypersurface, one views some aﬃne open subset of X as a parameterized family X → S of lower dimensional subvarieties. The relative cohomology of this family is a vector bundle on a

1 d d A hypersurface of the form X0 + ··· + Xn = 0. Chapter 1. Introduction 7 subset of the projective line, and one computes the cohomology of X as the cokernel of a natural integrable connection associated to this vector bundle. The Frobenius matrix can then be recovered through the commutativity between the connection and Frobenius maps. This method was later improved and implemented by Lauder [26] for the special case of a ﬁbration into hyperelliptic curves, and for more general cases by Walker [27].

1.1.4 Approaches to Singular Varieties

Though methods for computing zeta functions on smooth varieties have been studied extensively, it appears that only a small amount of consideration has been given to varieties which possess singularities, or even smooth varieties which admit known singular models in aﬃne space. This could be for any number of possible reasons: the lack of comparison isomorphisms between de Rham and rigid cohomology, the fact that the Weil conjectures do not hold for singular varieties, that families which possess singular ﬁbres can be completed in various ways, or that they are thought to be less secure from a cryptographic standpoint. In general, computing the zeta function of a singular variety is considered a much harder problem. However the author feels that this study is undervalued and has great potential for the future.

In 2008, Kloosterman [28] published an article reviewing the various p-adic methods for computing zeta functions of hypersurfaces, and identifying the obstructions to extending these methods to hypersurfaces which possess singularities. He proposed a modiﬁed algorithm which attempts to compute the discrepancy between de Rham and rigid cohomology by searching for and discarding eigenvalues of Frobenius which contra- dict a weak form of the Riemann Hypothesis2. It is clear that this tactic is diﬃcult to implement in practice, and only can be prescribed for special situations.

The fundamental obstruction that Kloosterman points out is that for the aﬃne complement of a singular projective hypersurface, the spectral sequence associated to the pole order ﬁltration of the de Rham complex used to compute cohomology does not degenerate at E2. He argues that

“One could try to adjust [Kedlaya’s] algorithm by taking an equisingular lift, and try to identify the extra relations needed to obtain [the cohomology group] as a quotient of [the de Rham complex]. Unfortunately, such a lift

2Contrary to its name, this is in fact a theorem for varieties over ﬁnite ﬁelds. Chapter 1. Introduction 8

might not exist and, except for a few cases, it is not clear at all which relations one needs to add.”

This thesis in part resolves certain cases where these obstructions are manageable, giving some evidence that one could eventually apply these types of algorithms to more general environments.

1.2 This Thesis

Modern methods have yet to provide effective algorithms for counting points on singular varieties. This thesis is comprised of two algorithms for computing the zeta function in specific cases. Chapter 2 reviews the cohomology theories that will be necessary, along with relevant comparison isomorphisms and useful exact sequences. The material in this chapter is not new, however Theorem (2.2.16) is a reformulation of part of Deligne’s work [29] and can be viewed as a generalization of a proposition of Abbott, Kedlaya, and Roe [18, Proposition 2.2.8]. Chapter 3 covers the technical details of a polynomial- time algorithm which computes the zeta function of a superelliptic curve over a finite field whose equation yr = f(x) in affine space has singularities3 along y = 0. This is an extension of the work of Kedlaya [12] as well as Gaudry and Gürel [16]. The main contribution of this chapter is the modified cohomological reduction method when gcd(f, f 0) 6= 1 and the p-adic precision-loss estimates from Lemmas (3.2.3) and (3.2.4). Chapter 4 then deals with the case of computing the zeta function of a nodal curve in P2. The notable parts of this chapter are the construction of a finite equisingular lift of a curve with a small number of singularities (Proposition (4.2.5)), as well as finding a suitable integer k to meet the requirements of Proposition (4.3.2). In Chapter 5, the algorithms are assembled in a step-by-step manner, and estimates on the complexity of each are given. The final chapter displays several results of a simple-minded implementation of both algorithms in MAGMA programming language.

1.3 Applications and Future Work

The algorithms described in this thesis are woven from deep theories that arise from algebraic geometry and p-adic analysis, however the applications admit themselves to various Diophantine questions. For instance, if one desired to know how often a cube could be written as the square of the product of two adjacent numbers modulo a prime, then one

3Equivalently, f(x) is not squarefree. Chapter 1. Introduction 9 could rephrase the question as a point-counting problem on the singular superelliptic 3 2 2 curve y = x (x − 1) over Fp. These algorithms are likely to have industrial applications as well.

1.3.1 Cryptography

Since the late 1980’s, mathematicians and computer scientists have prolifically used algebraic varieties in encryption systems, with the majority of attention given to elliptic curves. This was made possible since elliptic curves over finite fields possess a natural group structure4. The encryption is devised in the following way: Alice and Bob wish to share a secret key. They choose an elliptic curve with a point P . Alice chooses a secret number m, computes mP , and sends the result to Bob. Bob chooses a secret number n, computes nP , and sends the result to Alice. Now Alice computes mnP = m(nP ) and Bob computes mnP = n(mP ), and this is the secret that they share. We assume that no one can determine the secret key mnP without knowledge of either m or n, even if they have intercepted all the communication between Alice and Bob (this is known as the Diffie-Hellman problem).

Elliptic curve cryptography (ECC) was shown to give similar security as RSA5 but used smaller key sizes. In order to ensure the security of an ECC system, one needs to know that the number of rational points on the curve is divisible by a large prime, and it was this necessity that provided the initial motivation for fast point-counting algorithms. The natural question was then to ask whether other geometric objects could be useful in cryptography. A curve C in general does not have a group structure on its points, however one can consider the associated “Jacobian” variety J(C) which does have the structure of an abelian group. Systems based on the Jacobians of hyperelliptic curves,

superelliptic curves, and Ca,b curves have been proposed and are widely studied, and they seem to share many of the advantages of ECC. In the vast majority of such systems, the aﬃne planar representation of the curve has no singularities, which not only limits the number of possible choices for curves, but perhaps also misses some potential beneﬁts.

There have been several attempts to create a secure cryptographic system using singular curves. In 1995 Koyama [30] proposed an RSA-type cryptosystem based on a nodal cubic curve of the form y2 +axy ≡ x3 mod n, where n is the product of two large primes

4One can add two points P and Q together to get a new point P + Q on the curve, or multiply P by an integer n to get a new point nP . 5A standard cryptographic protocol relying on the diﬃcult of factoring large integers. Chapter 1. Introduction 10 p and q, and showed that the decryption speed could be performed twice as fast as RSA. One uses what is known as the “generalized Jacobian”, which in Koyama’s case is a group comprised of the points on the curve minus the point at the origin. Generalized Jacobians of singular curves are quasi-projective group schemes, with some quotient isomorphic to the Jacobian of the normalization of the curve. Algorithms for representing elements and performing group operations on generalized Jacobians have been developed (see for instance [31] or [32]), giving hope for the idea of using singular curves for cryptographic purposes, however it is argued that generalized Jacobians are no more secure than standard Jacobians, and in some cases less secure due the presence of pairing-based attacks [33].

An alternative application, suggested in 2004 by Kohel [34], uses the idea of embedding a finite field problem into the generalized Jacobian group of a singular hyperelliptic curve6 over a smaller base field that is amenable to the index calculus attack, which is subexponential in time. An interesting project for the future would be to see if one could extend this method to the singular curves studied in this thesis. The algorithms for computing zeta functions would then be essential for selecting a curve with an appropriate generalized Jacobian.

1.3.2 Support for Dimca’s Conjecture

In a beautiful paper by Griﬃths [35], the following statement is shown:

If a closed i-form ω in complex projective space with pole order k along a smooth hypersurface V has the property that for some diﬀerential ϕ, ω + dϕ has pole order k −1 along V , then there exists a diﬀerential ψ with pole order k − 1 along V such that ω + dψ also has pole order k − 1 along V .

The same is not true if V is singular, in fact it is always false. Dimca proves [36] that one can make the same assertion for singular curves with ψ having pole order k + 1 + i. In the case that V is a normal crossings divisor in Pn, Proposition (4.3.2) proves that for k large enough7 one can take ψ with pole order k + 1, or pole order k when i = n. Dimca’s general conjecture is that one will be able to ﬁnd ψ with pole order at most k +n along V (see Conjecture (4.1.3)). The reduction algorithm of Chapter 4 could easily be adapted to more general singular hypersurfaces to provide evidence for this conjecture.

6Kohel considers curves of the form y2 = xf(x)2 where f is a squarefree polynomial. 7For nodal curves we see this is k ≥ 2, for higher dimensional normal crossings divisors one would suspect the requirement to be k ≥ n. Chapter 1. Introduction 11

1.3.3 Potential Generalizations and Improvements

The methods used in this thesis should be immediately extendable to more general situations. For instance, one would expect that an algorithm similar to the one described in

Chapter 3 should be extendable to Ca,b curves with singular planar equations. For the algorithm of Chapter 4, one could try to extend this method to the case of curves in P2 with ordinary multiple points. Since every irreducible projective plane curve is birational to a curve with ordinary multiple points [37, Appx. A], one would then have an algorithm to compute the zeta function of a wide selection of curves. With additional eﬀort, an extension of the algorithm to certain classes of higher dimensional singular varieties should be feasible, particularly in the case of normal crossings divisors.

For projective curves with worse singularities the situation may be more dire since an equisingular lift may not exist, however it seems highly likely that the zeta function may be recovered through different types of lifting. For instance, Kloosterman [28] considers the case where one can define a sequence of lifts to smooth varieties Xk where the singular locus is lifted mod pk. Lauder [38] considers projective singular hypersurfaces that can be embedded into a family of smooth varieties and lifted to the integer ring of a number field. He then performs a base change of the parameterizing curve t 7→ te so that the family extends to “semistable degeneration”, and shows that one can uniquely define a computable “limiting Frobenius structure” for the new family. It seems, at least experimentally, that Frobenius action on the cohomology of the semistable fibre can be computed from the limiting Frobenius structure. Unfortunately it is unclear for now what the relationship is to the zeta function of the original hypersurface. However, the theory behind limiting Frobenius structures suggests another possible application. If one could reverse the process, and use the Frobenius action on the degenerate fibre to compute the limiting Frobenius structure, there would immediately exist a deformation algorithm to compute the zeta function of a smooth variety from a singular one. This has the potential of leading to faster algorithms, since one expects at least heuristically that the cohomology of the singular fibre will have lower dimension.

Lastly, it should be mentioned that the author feels that the algorithm in Chapter 4 is not yet optimal. In particular, as opposed to the explicit reduction method of Chapter 3, the algorithm of Chapter 4 requires ﬁnding solutions to seemingly random systems of linear equations in a vector space of dimension roughly d2, where d is the degree of the curve. This is the limiting step of that algorithm (see Step 4 of the complexity analysis) and might be possible to avoid using local data at the singular points of the curve in Chapter 1. Introduction 12 conjunction with the Poincar´eresidue map (see Theorem (2.2.16)). Chapter 2

Cohomology Theories

In this section we give an overview of the cohomology theories that will be used in this thesis, relevant theorems, as well as the comparison maps between them.

2.1 Preliminaries

2.1.1 p-Adic numbers and Witt vectors

Let p be a prime, a a positive integer, put q = pa, and let k be the ﬁnite ﬁeld with q elements.

Proposition 2.1.1. There is a unique local domain W (k) having characteristic 0, maximal ideal (p), complete with respect to its p-adic topology, and such that W (k)/pW (k) ∼= k. Proof. See [39, Section 8.10].

Deﬁnition 2.1.2. The ring W (k), which we will henceforth denote by V , is called the ring of Witt vectors of k.

One can construct V explicitly: Let θ ∈ k be a primitive element with minimal polynomial P (x). Let P (x) ∈ Z[x] be the monic polynomial with coeﬃcients in {0, 1, ..., p} such that its reduction modulo p is P . Then P (x) is irreducible, by irreducibility of P . Fix an algebraic closure Q of Q, and let θ ∈ Q be an algebraic integer with minimal polynomial P (x). Then the ring Zp[θ] is then a local domain, complete with respect to its maximal ideal (p), and with residue ﬁeld

Zp[θ] ∼ ∼ k = Fp[θ] = . pZp[θ]

Therefore Zp[θ] = V .

13 Chapter 2. Cohomology Theories 14

Deﬁnition 2.1.3. Let Z(p) denote the subring of Q consisting of elements whose denominators are not divisible by p. Then we will call Z(p)[θ] ⊂ V the ring of Witt vectors of

ﬁnite length, and denote it Vﬁn.

Deﬁnition 2.1.4. Let K denote the quotient ﬁeld of V . The p-adic valuation on K, i denoted ordp : K → Z, is the function which sends b ∈ K to max{i ∈ Z : b ∈ p V }.

Deﬁne the p-adic absolute value, denoted | · |p, to be the absolute value on K given by

−ordp(b) |b|p = p .

Remark 2.1.5. In the the case that a = 1, one has V = Zp and K = Qp. For a > 1, K is the unique unramiﬁed extension of Qp of degree a. Deﬁnition 2.1.6. The p-power Frobenius on k, which sends x to xp, induces an endomorphism on V which extends to an automorphism on K. Either of these maps will be called the p-power Frobenius map, denoted by σ. If V is a K-vector space, and φ is an additive map on V , we will say that φ is σ-linear if φ(cv) = σ(c)φ(v) for all c ∈ K, v ∈ V .

Remark 2.1.7. Note that σa is the identity on K, so if φ is a σ-linear map on a K-vector space V , then φa is linear. If V is ﬁnite dimensional, and M is the matrix of φ with respect to some basis B, then the matrix for φa is MM σ ··· M σa−1 , where M σi is the matrix with entries equal to the entries of M acted upon by σi.

2.1.2 Useful Properties of Etale´ Maps

We will use this section to recall a few results about ´etalemorphisms. These will later allow us to collect information near singularities of particular curves by performing an ´etalebase change.

Proposition 2.1.8. Suppose S is a locally Noetherian scheme, and f : X → S be an ´etalemorphism of schemes over an algebraically closed ﬁeld. Fix a point x ∈ X. Then

the map between formal completions of the local rings ObS,f(x) → ObX,x is an isomorphism. Proof. See [40, Proposition 3.26].

Definition 2.1.9. Let I be coherent sheaf of ideals on a locally Noetherian scheme X, n defining a closed subscheme Y . The X-scheme Proj(⊕n≥0I ) → X is called the blowing- up of X along Y , and is denoted XeY . Proposition 2.1.10. Let X and Z be locally Noetherian schemes, and let Y be a closed subscheme of X. Let g : Z → X be a flat morphism of schemes. Then there is a canonical isomorphism ∼ Zeg−1(Y ) −→ XeY ×X Z Chapter 2. Cohomology Theories 15

Proof. See [40, Proposition 1.12 (c)].

2.2 Algebraic de Rham Cohomology

Algebraic de Rham Cohomology, introduced by Grothendieck, is the algebraic analogue of de Rham theory on smooth manifolds. Given a morphism X → S of objects in a i category C, deﬁne the i-th (algebraic) relative de Rham cohomology group HdR(X/S).

2.2.1 K¨ahlerDiﬀerentials

Let A be a commutative ring with identity, let B be an A-algebra, and let M be a B-module.

Deﬁnition 2.2.1. An A-derivation of B into M is an A-linear map d : B → M which 0 satisﬁes da = 0 for all a ∈ A, as well as the Leibniz rule: for b1, bb2 ∈ B, d(bb ) = bdb0 + b0db.

Deﬁnition 2.2.2. A module of relative diﬀerential forms of B over A is a B-module 1 1 ΩB/A equipped with an A-derivation d : B → ΩB/A, satisfying the following universal property: For any B-module M, and for any A-derivation d0 : B → M, there exists a 1 0 unique B-module homomorphism f :ΩB/A → M such that d = f ◦ d.

1 The universal property ensures that ΩB/A is unique. One way to prove existence of 1 ΩB/A is to give an explicit construction: One takes the free B-module generated by the symbols {db : b ∈ B} and quotients by the submodule generated by d(bb0) − bdb0 − b0db and d(b + b0) − db − db0 for b, b0 ∈ B, and da for a ∈ A. Deﬁne the module of relative 1 diﬀerential i-forms to be i-th exterior power of the B-module ΩB/A

i i 1 ΩB/A := ∧ ΩB/A

i i+1 with induced maps di :ΩB/A → ΩB/A satisfying di+1 ◦ di = 0.

• Deﬁnition 2.2.3. The complex (ΩB/A, d) is called the de Rham complex of B over A, i and an element ω ∈ ΩB/A is called a relative i-form. We call ω closed if di(ω) = 0 and i−1 exact if ω = di−1ν for some ν ∈ ΩB/A.

2.2.2 de Rham Cohomology for Schemes

Definition 2.2.4. Given affine schemes X = Spec(B) and Y = Spec(A), and a morphism X → Y , we define the (algebraic) relative de Rham cohomology of X over Y , denoted Chapter 2. Cohomology Theories 16

i • HdR(X/Y ), to be the cohomology of the complex (ΩB/A, d). i To be more precise, we deﬁne HdR(X/Y ) to be the closed i-forms modulo exact i-forms

i i+1 ker{di :Ω → Ω } Hi (X/Y ) := B/A B/A . dR i−1 i im{di−1 :ΩB/A → ΩB/A}

This definition can be generalized to arbitrary schemes. Proposition 2.2.5. Let f : X → Y be a morphism of schemes. Then there exists a 1 unique quasi-coherent sheaf ΩX/Y on X such that for any affine open subset V of Y , any affine open subset U of f −1(V ), and any x ∈ U, we have

Ω1 ∼= Ω1 (Ω1 ) ∼= Ω1 X/Y U OX (U)/OY (V ) X/Y x OX,x/OY,f(x)

Proof. see [40, Proposition 6.1.17].

1 Definition 2.2.6. The sheaf ΩX/Y is called the sheaf of relative differential 1-forms of X over Y . The sheaf Ωi := ∧i Ω1 is called the sheaf of relative differential i-forms of X/Y OX X/Y i i+1 X over Y . Exterior differentiation then induces maps di :ΩX/Y → ΩX/Y . If Y = Spec(A) i is affine, we also denote these sheaves ΩX/A. Definition 2.2.7. Let f : X → Y be a morphism of schemes. We define the (algebraic) i relative de Rham cohomology of X over Y , denoted HdR(X/Y ), to be the global sections i • of the sheaf R f∗ΩX/Y on Y . i • Remark 2.2.8. If Y is affine, and X is quasi-compact and separated, then R f∗ΩX/Y (Y ) = i • i • H (X, ΩX/Y ) := R ΓΩX/Y , where Γ is the global sections functor [40, Proposition 2.28]. i i • In this case, ΩX/Y (U) is acyclic for any open affine subset U of X, and H (X, ΩX/Y ) may i • • be computed as H (X, ΩX/Y ), i.e. the hypercohomology of ΩX/Y . In particular, if X is affine, we arrive at the previous definition. Remark 2.2.9. It can be shown (e.g. [40, Proposition 6.2.2]) that if f : X → Y is smooth, i then ΩX/Y is a locally free OX -module.

2.2.3 de Rham Cohomology with Logarithmic Singularities

Here we will review the constructions involving differentials with logarithmic poles along a subscheme. First we require several preliminary definitions. Definition 2.2.10. Let f : X → S be a morphism of finite type. Then f is smooth of relative dimension n if it is smooth, and all of its non-empty fibres are equidimensional of dimension n. We say f is étale if it is smooth of relative dimension 0. Chapter 2. Cohomology Theories 17

Definition 2.2.11. A smooth pair (resp. a smooth proper pair) of relative dimension n over a scheme S, is a pair of S-schemes (X,Z) in which f : X → S is smooth (resp. smooth proper) of relative dimension n, and Z is a relative reduced normal crossings divisor on X. That is, étale-locally on X, one can find an S-isomorphism of X with n a Zariski open subset of AS under which Z is carried to an open subset of a union of coordinate hyperplanes.

Remark 2.2.12. In the deﬁnition of a smooth pair (X,Z) we include the possibility of Z being empty.

Definition 2.2.13. Let (X,Z) be a smooth pair of relative dimension n over a scheme S, such that U := X \ Z is affine with inclusion map ι : U → X. We define the sheaf of rel- 1 ative differential 1-forms on X with logarithmic singularities along Z, denoted Ω(X,Z)/S, 1 to be the sub-OX -module of ι∗ΩU/S defined on affine étaleopen sets V in the following 1 way: If t1, ..., tn are local coordinates for V on AS, and Z|V is defined by an equation 1 t1 ··· tr = 0, then Γ(V, Ω(X,Z)/S) is defined to be the free Γ(V, OX )-module generated by the elements dt1/t1, ..., dtr/tr, dtr+1, ..., dtn.

Setting Ωi = ∧i Ω1 , the exterior diﬀerentiation operator inherited from (X,Z)/S OX (X,Z)/S i ι∗ΩU/S induces maps i i+1 di :Ω(X,Z)/S → Ω(X,Z)/S. The resulting complex is called the de Rham complex of (X,Z) over S. We deﬁne the i i-th algebraic de Rham cohomology group of (X,Z) over S, denoted HdR((X,Z)/S), to i • be the global sections of R f∗Ω(X,Z)/S.

i Remark 2.2.14. As in the remark of the previous section, if S is aﬃne then HdR((X,Z)/S) i • can be computed as H (X, Ω(X,Z)/S), the hypercohomology of the de Rham complex of (X,Z).

i Definition 2.2.15. Using the notation from the previous definition, denote by Wk(Ω(X,Z)/S) i the submodule of Ω(X,Z)/S generated on affine étaleopen sets V by differentials of the form dtj(1) dtj(l) ∧ · · · ∧ ∧ dtj(l+1) ∧ · · · ∧ dtj(i), l ≤ k. tj(1) tj(l) • • Then Wk(Ω(X,Z)/S) is a subcomplex of Ω(X,Z)/S. Let

W • • • Grk (Ω(X,Z)/S) = Wk(Ω(X,Z)/S)/Wk−1(Ω(X,Z)/S) denote the associated graded complex. Chapter 2. Cohomology Theories 18

We will make use of the following construction of Deligne from [29, section 3.I], adapted from the language of complex analytic geometry to our present environment. Let (X,Z) be a smooth pair over a scheme S of relative dimension n. Then around each point P ∈ X, there is an ´etaleneighbourhood i : U → X, and an open immersion n −1 j : U → AS, such that i (Z) is a union H1 ∪ · · · ∪ Hr where each Hi is the intersection n T of a distinct hyperplane in AS with U. For a subset J ⊂ {1, ..., r}, let HJ = i∈J Hi, k S k F and deﬁne ZU = |J|=k HJ and ZeU = |J|=k HJ , where the union is taken over all subsets of {1, ..., r} of length k. These glue together to form S-schemes Zk and Zek. Put Z0 = Ze0 = X. By construction, Zek is a smooth scheme, Z1 = Z, and for k > 1, Zk is k−1 F S the singular locus of Z . Additionally, the projection |J|=k HJ → |J|=k HJ gives a k k k k map Ze → Z which is in fact the normalization of Z . Let ik : Ze → X denote the compositions of the normalization with the inclusion map.

We now deﬁne a sheaf of Z-modules εk on the small ´etalesite of Zek as follows. Let k Vk L ε (HJ ) = ( i∈J ZHi ), where ZHi denotes a copy of the integers corresponding to Hi. These glue together to form a locally free sheaf of rank 1. For a sheaf F on Zek, let F(εk) k denote the tensor product F ×Z ε .

Theorem 2.2.16 (Poincar´eResidue Theorem). Suppose (X,Z) is a smooth pair of rela-

tive dimension n over S. Then for each 1 ≤ k ≤ n there is an isomorphism of OX -modules

W • • k Res : Gr (Ω ) → ιk∗Ω (ε )[−k]. k (X,Z)/S Zek

´ n Proof. Etale-locally on X, Z is given in some Zariski-open set U ⊂ AS as a union

of hyperplanes H1, ..., Hr. Localizing further, we may assume that U = Spec(R) and

S = Spec(A) are affine, and the hyperplanes are defined by local coordinates t1, ..., tr ∈ R. Then by definition • k M • k ιk∗Ω (ε )(U) = Ω ⊗ ε (HJ ) Zek RJ /A |J|=k

where RJ = R/(tj1 , ..., tjk ) for J = {j1, ..., jk} ⊂ {1, .., r}. We can deﬁne a surjective map

• • Wk(Ω(X,Z)/S)(U) → ιk∗ΩZk (ε)(U)

dt dt by ﬁrst deﬁning it on sections ω = α ∧ j1 ∧ · · · ∧ jk to be tj1 tjk

ω 7→ (α mod t , ..., t ) ⊗ (1 ∧ · · · ∧ 1 ) j1 jk Hj1 Hjk Chapter 2. Cohomology Theories 19 and extending linearly over A. We define Res to be the above map modulo its kernel, • which is equal to Wk−1(Ω(X,Z)/S)(U), thus defining an isomorphism on the graded complex. To see that Res is well-defined globally, it is enough to see that it is independent of the choice the local parameters defining Z. This is immediate from the following observations:

1. If σ is a permutation of J, then

dt dt dt dt ω = α ∧ j1 ∧ · · · ∧ jk = sgn(σ)α ∧ σ(j1) ∧ · · · ∧ σ(jk) , tj1 tjk tσ(j1) tσ(jk)

so the map

ω 7→ (sgn(σ)α mod tj , ..., tj ) ⊗ (1H ∧ · · · ∧ 1H ) 1 k σ(j1) σ(jk) = (α mod t , ..., t ) ⊗ (1 ∧ · · · ∧ 1 ) j1 jk Hj1 Hjk

is the same as the one deﬁned previously.

∗ • 2. If u ∈ R is a unit and α ∈ Wk−1(Ω(X,Z)/S)(U), then

d(utji ) dtji du • α ∧ − α ∧ = α ∧ ∈ Wk−1(Ω(X,Z)/S)(U), utji tji u

so choosing a new parameter for Hji gives an equivalent element in the graded complex.

Remark 2.2.17. It is clear that εk is isomorphic to the constant sheaf on Zek for k = 1. The same is true for k = n, since Zen is a disjoint union of schemes isomorphic to S, on which εn is constant. Therefore for n = 2 we can ignore εk entirely, however it is worth noting that the map W • • Res : Gr (Ω ) → i2∗Ω [−2] 2 (X,Z)/S Ze2 is noncanonical, i.e. it depends on a choice of sign at each crossing.

Deﬁnition 2.2.18. Let (X,Z) be a smooth pair over a scheme S, let F be a sheaf of

OX -modules, and let m be a nonnegative integer. Let OX (mZ) denote the invertible sheaf on X corresponding to the divisor mZ. Deﬁne the m-th twist of F along Z to be

F(mZ) := F ⊗OX OX (mZ). Chapter 2. Cohomology Theories 20

Remark 2.2.19. On affine étaleopen sets V of X, where V |Z is defined by t1 ··· tr = 0, i i Ω(X,Z)/S is the sub-OX -module of ΩX/S(Z) generated freely by dt1/t1, ..., dtr/tr, dtr+1, ..., dtn i and ΩX/S is the submodule generated by dt1, ..., dtn. Thus for any integer m there are

natural inclusions of OX -modules

i i i i ΩX/S(mZ) ⊂ Ω(X,Z)/S(mZ) ⊂ ΩX/S((m + 1)Z) ⊂ ι∗ΩU/S,

i i i and ι∗ΩU/S is equal to the direct limit of either ΩX/S(mZ) or Ω(X,Z)/S(mZ) as m increases. On aﬃne subsets V ⊂ X, we can write

i i i+1 Γ(V, Ω(X,Z)/S) = {s ∈ Γ(V, ΩX/S(Z)) : dis ∈ Γ(V, ΩX/S(Z))}.

We have the following useful fact about the homology sheaves of diﬀerential forms and their twists.

Theorem 2.2.20. Let (X,Z) be a smooth pair over a scheme S. For each nonnegative integer m, the natural map of complexes of sheaves

• • Ω(X,Z)/S → Ω(X,Z)/S(mZ)

induces maps on the homology sheaves whose kernels and cokernels are killed by lcm(1, ..., m).

Proof. For the assertion about the cokernel, see [18, Theorem 2.2.5]. For the assertion about the kernel, see [41, Proposition 3]

2.3 p-Adic Cohomology Theories

If a variety is defined over a field of characteristic p > 0, then de Rham cohomology is not a suitable cohomology theory, for it does not offer finite dimensionality in even the 1 simplest of cases. For instance, the HdR(Spec(Fp[x])) contains the linearly independent pi−1 ∞ set {x dx}i=1. It is therefore necessary to develop cohomology theories over a field of characteristic 0.

2.3.1 Monsky-Washnitzer Cohomology

Deﬁnition 2.3.1. For a V -algebra A, we will denote by Aˆ its completion with respect to the ideal pA, that is, the universal object of the inverse system · · · → A/p2A → A/pA. Chapter 2. Cohomology Theories 21

ˆ Remark 2.3.2. If A is ﬁnitely generated by elements {x1, ..., xn}, then an element of A can be written as power series in multi-index notation

X α cαx

with cα ∈ V satisfying |cα|p → 0 as |α| → ∞. Here α = a1, .., an is a multi-index of

α a1 an nonnegative integers, x = x1 ··· xn , and |α| = a1 + ··· + an.

Definition 2.3.3. Let A be a finitely generated V -algebra, and let {x1, ..., xn} be a set of generators. We define the dagger algebra of A, denoted A†, to be the subalgebra of Aˆ consisting of the overconvergent power series. That is, A† consists of elements representable as X α cαx

with cα ∈ V , such that there exist real numbers C, r with C > 0, 0 < r < 1, satisfying |α| |cα|p ≤ Cr for all α.

Definition 2.3.4. Let X be a smooth affine variety over k with coordinate ring A. By a theorem of Elkik [42], there exists a V -algebra A such that A/pA ∼= A and X = Spec(A) is a smooth affine -scheme. The i-th cohomology group of the complex Ω• is denoted V A†/V i HMW(X/V ). We define the i-th Monsky-Washnitzer cohomology group of X, denoted i i HMW(X/K), to be HMW(X/V ) ⊗V K. Remark 2.3.5. Monsky and Washnitzer prove [11, Theorem 5.6] that the map sending i X to HMW(X/K) is a contravariant functor. In particular, it is independent of the choices that were made (namely, the choice of the V -algebra A and its generators). Additionally, σ can be extended to a σ-linear map on A†, which induces a σ-linear map i F on HMW(X/K).

2.3.2 Rigid Cohomology and Crystalline Cohomology

We now discuss the notions and relevant theorems of rigid cohomology and crystalline cohomology. We do not give the technical background, as it is lengthy and unnecessary for putting the theory to practical use. The explicit constructions may be found in a work of Shiho [43] for crystalline cohomology, and Berthelot [23] for rigid cohomology.

Deﬁnition 2.3.6. Let (X,Z) be a smooth proper pair over k. One can deﬁne a V -module i Hcrys((X,Z)/V ), contravariantly functorial in the pair (X,Z), called the i-th crystalline cohomology group of (X,Z). Moreover, the Frobenius on k induces a σ-linear map on i i Hcrys((X,Z)/V ). If Z is empty, we will write the crystalline cohomology as Hcrys(X/V ). Chapter 2. Cohomology Theories 22

Definition 2.3.7. Let X be a smooth k-scheme. One can define a finite-dimensional K- i vector space Hrig(X/K), contravariantly functorial in X, called the i-th rigid cohomology group of X, with a σ-linear map induced by the Frobenius on k.

From Berthelot [24, Propositions 1.9 and 1.10], we have the following comparison isomorphisms, connecting rigid, crystalline, and Monsky-Washnitzer cohomology groups.

Proposition 2.3.8. Let X be a smooth, proper k-scheme. There is a functorial, Frobenius- equivariant isomorphism

i ∼ i Hrig(X/K) = Hcrys(X/V ) ⊗V K

Proposition 2.3.9. Let X be a smooth, aﬃne k-scheme. There is a functorial, Frobenius- equivariant isomorphism i ∼ i Hrig(X/K) = HMW(X/K)

The following is a result of Shiho [43, Theorem 2.4.4]

Theorem 2.3.10. Let (X,Z) be a smooth pair over k, and set U = X \ Z. There is a functorial, Frobenius-equivariant isomorphism

i ∼ i Hcrys((X,Z)/V ) ⊗V K = Hrig(U/K)

2.3.3 Comparisons Theorems Between p-Adic and de Rham Co- homology

Fundamental to all p-adic point counting algorithms is the ability to compute a p-adic cohomology group as the de Rham cohomology of some scheme. Additionally, the following comparison isomorphisms can be used to give eﬀective p-adic bounds for the algorithms.

Theorem 2.3.11. [44, Cor 2.6] Let (X, Z) be a smooth proper pair over V . Put U =

X \ Z, and let U and UK denote the special and generic ﬁbres of U. Then there is an isomorphism i ∼ i Hrig(U/K) = HdR(UK /K)

In particular, when Z is empty the theorem reduces to an isomorphism

i ∼ i Hrig(X/K) = HdR(XK /K). Chapter 2. Cohomology Theories 23

Proposition 2.3.12. Let (X, Z) be a smooth proper pair over V . Let X and Z be the special ﬁbres of X and Z, respectively. Then there is an isomorphism

i ∼ i Hcrys((X,Z)/V ) = HdR((X, Z)/V )

Proof. See [45, Theorem 6.4].

2.4 Exact Sequences

Of particular use to us will be several be exact sequences which have analogues in each cohomology theory. We begin with an algebraic version of the Gysin sequence from the theory of complex manifolds.

Theorem 2.4.1. Let X be a smooth variety over a ﬁeld of characteristic 0, and let Z be a smooth closed subvariety of pure codimension r. Then there is a long exact sequence

i i δ i+1−2r i+1 · · · → HdR(X) → HdR(X \ Z) −→ HdR (Z) → HdR (X) → · · ·

The map δ we will refer to as the Poincar´eresidue map.

Proof. Combine Theorem 3.3 and Proposition 3.4 from [46].

As in most cohomology theories, there is a notion of cohomology with compact support i i for both de Rham and rigid cohomology, denoted HdR,c(X) and Hrig,c(X), respectively, which are isomorphic to cohomology without compact support for X proper. One has the following excision sequences for these objects, which carries the added beneﬁt of allowing us to forgo smoothness assumptions.

Proposition 2.4.2. Let X be variety over a ﬁeld of characteristic 0, let Z be closed subvariety, and let U denote the complement. Then there is a long exact sequence

i i i i+1 · · · → HdR,c(U) → HdR,c(X) → HdR,c(Z) → HdR,c(U) → · · ·

Proof. See [47, Proposition 1.11]

Proposition 2.4.3. Let X be a separated k-scheme of ﬁnite type, let Z be closed subscheme, and let U denote the complement. Then there exists a Frobenius-equivariant long exact sequence

i i i i+1 · · · → Hrig,c(U/K) → Hrig,c(X/K) → Hrig,c(Z/K) → Hrig,c(U/K) → · · · Chapter 2. Cohomology Theories 24

Proof. See [48, 3.1 (iii)].

For a space M that has an action of Frobenius, given an integer l, we denote by M(l) the space M with the action of Frobenius multiplied by q−l. For X a smooth k-scheme of dimension n, Poincar´eduality in rigid cohomology gives a Frobenius-equivariant K- pairing i 2n−i Hrig,c(X) × Hrig (X)(n) → K (2.1) which leads to the following proposition (for details of these constructions see [49]).

Proposition 2.4.4. Let (X,Z) be a smooth pair over k with Z also smooth. Put U = X \ Z. Then there exists a Frobenius-equivariant long exact sequence

i i i−1 i+1 · · · → Hrig(X/K) → Hrig(U/K) → Hrig (Z/K)(−1) → Hrig (X/K) → · · ·

Proof. See [18, Proposition 2.4.1 b] Chapter 3

Superelliptic Curves

3.1 Basic Properties

Definition 3.1.1. We define a superelliptic curve C → S of genus g to be a smooth projective morphism of schemes such that the fibre over each closed point η → S is connected, has dimension 1, genus g, and is birationally equivalent to an affine plane curve C0 given by an equation

n r d d−1 Y ei y = x + ad−1x + ··· + a0 = (x − αi) ∈ k(η)[x] i=1 where r is a ﬁxed prime and d is a positive integer such that (r, d) = 1. We require the singular points of the planar equation to be rational over k(η)(αi ∈ k(η) if ei > 1) and if char(k(η)) = p > 0, we require (p, r) = 1 and (p, ei) = 1 for each i.

A superelliptic curve C over a ﬁeld F comes equipped with an automorphism ρ of order r induced by the map (x, y) 7−→ (x, ζy), where ζ is a primitive r-th root of unity. Letting C/hρi be the quotient variety, there is an isomorphism

1 C/hρi → PF given by [(x, y)] 7→ x on C0. Suppose C is a superelliptic curve over a field F , birationally equivalent to an affine plane curve C0 defined by the equation

n r Y ei y = f(x) = (x − αi) . i=1

25 Chapter 3. Superelliptic Curves 26

For simplicity assume that αi ∈ F for all i, and write the division of ei by r with remainder 0 as ei = δir + λi. Let D be the curve deﬁned by the equation

n r Y λi y = (x − αi) . i=1

For a point (a, b) ∈ D0 one has

n !r n Y δi r Y rδi b (a − αi) = b (a − αi) i=1 i=1 n ! n ! Y λi Y rδi = (a − αi) (a − αi) i=1 i=1 n Y ei = (a − αi) i=1

0 0 Qn so we can deﬁne a morphism ϕ : D → C as the map which sends (a, b) to (a, b i=1(a− δi 0 0 αi) ). This gives a birational equivalence between C and D over F . We will therefore assume from now on that 1 ≤ ei ≤ r − 1 for all i.

3.1.1 The Genus

a k Let p > 0 be a prime, p 6= 2. Fix a positive integer a, and put q = p and = Fq. Let V denote the ring of Witt vectors over k, Vfin the Witt vectors of finite length, and K the fraction field of V .

0 Suppose Cek is a superelliptic curve over k with associated planar curve Ck defined by an equation n r Y ei y = f(x) = (x − αi) . i=1 0 Suppose that the singular points of Ck are ordered (α1, 0), ..., (αm, 0) for some m ≤ n. k Qn k Then by definition αi ∈ for 1 ≤ i ≤ m. Let τ(x) = i=m+1(x−αi) ∈ [x]. Let α1, ..., αm be lifts of the elements α1, ..., αm to Vfin[x], and let τ(x) ∈ Vfin[x] be a polynomial obtained by lifting the coefficients of τ(x) to Vfin[x]. We then define

m Y ei f(x) = τ(x) (x − αi) . i=1 Chapter 3. Superelliptic Curves 27

r 0 0 The equation y = f(x) defines a V -scheme C with special fibre isomorphic to Ck . Let 0 2 C denote the closure of C in PV , and let CK and Ck denote the generic and special fibres 0 0 0 0 of C, respectively. Put C = C \{y = 0} and let CK and Ck denote the generic and special fibres of C0.

Proposition 3.1.2. There exists a smooth V -scheme Ce over C, isomorphic to C0 outside a ﬁnite number of points, and admitting a unique point lying above each singular point of

C. If P∞ is the closed point in Ce lying above ∞, then ordP∞ (x) = −r and ordP∞ (y) = −d.

If Pi is the closed point lying above (αi, 0), then ordPi (x − αi) = r and ordPi (y) = ei.

Proof. To prove the proposition we compute a sequence of blowups of the closed singular points of C restricted to affine neighbourhoods. Fix an algebraic closure K of K, and let V denote its ring of integers. Then V is flat over V . Suppose Ce → C is a blowup of C along a closed subvariety Z. To check that Ce is smooth, from [40, Ch. 4, Definition 3.35] it suffices to check smoothness at the closed points of Ce ×Spec(V ), which by Proposition (2.1.10) is isomorphic to the blowup of C × Spec(V ) along Z. Therefore we can assume K = K and V = V .

r d−r d X Suppose r < d. Then C has equation Y Z = Z f( Z ) in projective coordinates, which is the union of C0 with an additional singular point at inﬁnity ∞ = [0 : 1 : 0]. 1 Restricting to Y 6= 0 and letting (S,T ) be projective coordinates for PV , the blowup at 2 1 the closed point of ∞ gives equations for the total transform of the curve in AV × PV as x zd−r = zdf( ) z T x = Sz

where z := Z/Y and x := X/Y . Note that any point above ∞ occurs at x = z = 0.

Let U1 and U2 be the aﬃne spaces corresponding to T 6= 0 and S 6= 0 respectively. We

choose s := S/T to be an aﬃne coordinate on U1, and t := T/S to be an aﬃne coordinate d−r d on U2. The local equation for the total transform of C on U1 is z = z f(s) so the coordinates of the strict transform satisfy the equation zrf(s) = 1. Thus there is no

point lying above ∞ on U1, and the strict transform of the curve is contained entirely in

U2.

d−r d−r d d 1 The total transform of C in U2 is defined in affine coordinates by t x = x t f( t ) = d d x f−1(t) where we have set f−1(t) = t f(1/t). Note that f−1(0) = 1 6= 0. The strict trans- d−r r form W satisfies the equation t = x f−1(t), which contains a single point lying above Chapter 3. Superelliptic Curves 28

∞, at x = t = 0. Let OW denote the coordinate ring of W , and deﬁne

1 O [u, ] W u B = r . (u − f−1(t))

Let g : Spec(B) → W denote the induced morphism of schemes. Since (p, r) = 1, we find d that (ur − f (t)) = rur−1 is a unit, so g is étale.Letting v = ux, we can rewrite the du −1 ring B 1 1 OW [u, ] V [t, v, u, ] u u B = r = d−r r r . (u − f−1(t)) (t − v , u − f−1(t)) −1 Since f−1(0) 6= 0, the set g ((0, 0)) consists of r distinct closed points, corresponding r to the roots of u − f−1(0). Let V ⊂ Spec(B) be an open affine set containing exactly one of these points, which we will denote by Q∞. The map

1 V [w, u, ] u ϕ : B −→ r r =: Be (u − f−1(w )) t 7→ wr v 7→ wd−r induces a morphism ϕ∗ : Spec(Be) → Spec(B) which is the composition of a sequence of blowups at the closed point of Spec(B) where t = v = 0. Additionally, it resolves the singularity at Q∞, admits a single closed point Qe∞ lying above Q∞ with uniformizing parameter w, and is an isomorphism outside of the support of the divisor w. Let Ve denote the preimage of V under ϕ∗, and Wf denote the corresponding series of blowups of W at Q∞, by (2.1.10) there is a commutative diagram

∼ Ve = Wf ×W V −−−→ V     y y Wf −−−→ W

Since there is only one point in Ve lying above Q∞, there must be a unique point

P∞ ∈ Wf lying above Q∞. Moreover, since g is ´etale,by Proposition (2.1.8) there is an isomorphism on the completed local rings

Oˆ ∼= Oˆ . Wf ,P∞ V,e Qe∞ Chapter 3. Superelliptic Curves 29

which preserves the order of vanishing of an element in the coordinate ring. We have the following calculations

ord (Y/Z) = −ord (z) = −ord (tx) = −ord (tv/u) = −ord (wd/u)) = −d P∞ P∞ P∞ Qe∞ Qe∞

ordP∞ (X/Z) = ordP∞ (x) − ordP∞ (z) = ord (v/u) − d = ord (wd−r/u)) − d = d − r − d = −r. Qe∞ Qe∞

We now turn our attention to points of the form Pi := [αi : 0 : 1], 1 ≤ i ≤ n. Around r Pi, C satisﬁes the aﬃne equation y = f(x). Shifting coordinates so that Pi = (0, 0), we

r ei can write this equation as y = x fi(x), for some polynomial fi(x) with fi(0) 6= 0. Since

(r, ei) = 1, the same argument as above implies that we introduce a change of coordinates y ur = f (x), v = so that there is an ´etaleneighbourhood of P given by the equation i u i vr = xei . The resolution of singularities given by v 7→ wei , x 7→ wr shows that there is a

unique point Pei ∈ Ce lying above (0, 0). Additionally we have

ord (y) = ord (uwei ) = e Pei Pei i ord (x − α ) = ord (wr) = r. Pei i Pei

The case where d < r is proved similarly.

Proposition 3.1.3. The scheme Ce is a superelliptic curve over V with genus g = (r − 1)(n − 1) . 2 Remark 3.1.4. Note that in the case r = 2 we would have a hyperelliptic curve, and the n − 1 proposition states that its genus is . The fact that this is an integer is ensured by 2 the condition (d, 2) = 1, and the fact that d and n have the same parity.

Proof. It is enough to check the genus on the special ﬁbre Cek. The map

0 1 Ck → Pk \ {∞, [αi : 1]}

sending (x, y) to [x : 1] gives an r-fold cover of the projective line minus n + 1 points. We can then compute

−1 0 χ(Cek) = χ(φ (Ck)) + n + 1 1 = rχ(Pk − {n + 1 points}) + n + 1 = r(2 − n − 1) + n + 1 = 2 − (r − 1)(n − 1). Chapter 3. Superelliptic Curves 30

If g is the genus of Cek, from the Hurwitz genus formula we get 2−2g = 2−(r −1)(n−1), (r − 1)(n − 1) which gives g = . 2

3.1.2 The Zeta Function

Let C be a superelliptic curve over V of genus g, birationally equivalent to the plane

r Qn ei 0 curve deﬁned by an equation y = f(x) = i=1(x − αi) where f has degree d. Let C denote the plane curve minus the points along y = 0. By the Weil conjectures for rigid

cohomology, the zeta function of the special ﬁbre Ck can be written

1 det(1 − TF |Hrig,c(Ck)) Z(Ck,T ) = (1 − T )(1 − qT ) where F is the map induced by Frobenius on cohomology with compact support. Since

Ck is smooth, Poincar´eduality (Equation (2.1)) gives a canonical non-degenerate pairing

1 1 Hrig,c(Ck) × Hrig(Ck)(1) → K

1 −1 where Hrig(Ck)(1) is a K-vector space where the Frobenius action is multiplied by q . It follows that

1 −1 1 det(1 − TF |Hrig,c(Ck/K)) = det(1 − qT F |Hrig(Ck/K))

If V is a K-vector space with an automorphism induced by ρ : C → C, we denote by V l the subspace {w ∈ V |ρw = ζ−lw} and by V − the direct sum V 1 ⊕ · · · ⊕ V r−1. The 1 following two propositions allow us to view HdR(C/V ) as a Frobenius-equivariant lattice 1 0 of Hrig(Ck/K)

Proposition 3.1.5. There is an injective, Frobenius equivariant map

1 1 0 Hrig(Ck/K) → Hrig(Ck/K)

1 0 − whose image is equal to Hrig(Ck/K) .

0 Proof. Let Zk = Ck \ Ck. By Proposition (2.4.4) there is a Frobenius-equivariant exact sequence containing

1 1 0 0 0 → Hrig(Ck/K) → Hrig(Ck/K) → Hrig(Zk/K)(−1) → · · · (3.1) Chapter 3. Superelliptic Curves 31

from which injectivity follows.

0 l Since ρ commutes with the Frobenius and acts as the identity on Zk, we have Hrig(Zk/K) = 1 l 1 0 l 0 for l 6= 0. Therefore Hrig(Ck/K) → HdR(Ck/K) is an isomorphism for 0 < l < r.

Since formation of cohomology commutes with ρ we also have

1 0 ∼ 1 ∼ 1 1 Hrig(Ck/K) = Hrig(Ck/K)/hρi = Hrig(Pk /K) = 0

which completes the proof.

1 1 0 Proposition 3.1.6. The module HdR(C/V ) is a Frobenius-equivariant V -lattice in Hrig(Ck/K) with no subspace which is ﬁxed under ρ.

Proof. From Propositions (2.3.8) and (2.3.12), it follows that there is a Frobenius-equivariant isomorphism 1 ∼ 1 HdR(C/V ) ⊗V K = Hrig(Ck/K). The result follows from Proposition (3.1.5) and [50, Proposition 3.2].

1 0 − 3.1.3 The Vector Space HMW(CK/K)

0 Using the notation of the previous sections, Ck has coordinate ring

1 k[x, y, ] y A = (yr − f(x))

and C0 has coordinate ring 1 V [x, y, ] y A = . (yr − f(x)) An element of the p-adic completion of A, denoted Aˆ, can be represented

X i j z = aijx y 0≤i

The dagger ring A† is the subring of Aˆ consisting of elements represented in the above |j| form, such that there exists a number 0 < c < 1 with |aij|p < c . One can then deﬁne Chapter 3. Superelliptic Curves 32

i the Monsky-Washnitzer cohomology groups HMW (A/K) as the cohomology of the de † † Rham complex over AK := A ⊗V K.

• † We can construct this space explicitly. Let Ω † be the de Rham complex of AK . The AK i K-space HMW (A/K) is then the i-th cohomology group of the chain complex

† d 1 0 −→ AK −→ Ω † −→ 0. AK

r † r−1 0 1 The relation y = f(x) in A leads to the relation ry dy = f (x)dx in Ω † . Writing AK 0 f (x)dx 1 this as dy = r−1 , it follows that any element in Ω † can be written as a sum ry AK

X i j aijx y dx. 0≤i

† We can decompose AK as r−1 † M † AK = Al l=0

† i j where Al only includes the terms aijx y with j = −l (mod r). This leads to a similar decomposition r−1 1 M Ω † = Ωl. (3.2) AK l=0 Since j d(xiyj) = ixi−1yjdx + jyj−1xidy = ixi−1yjdx + f 0(x)xiyj−rdx, r therefore the operator d commutes with the decomposition (3.2), and we have a decomposition of cohomology groups

r−1 i M i l HMW (A/K) = HMW (A/K) . l=0 To view this in another way, the above is the decomposition into invariant subspaces i under the group action of Z/rZ on HMW (A/K) induced by the automorphishm ρ. Each component corresponds to an eigenspace of ρ with eigenvalue ζ−l, where ζ ∈ K is an r − th root of unity.

1 Note that since A† = K[x, ]† is the dagger ring of the coordinate ring of 0 f Chapter 3. Superelliptic Curves 33

1 AK − {zeroes of f}, the cohomology of the l = 0 part is the cohomology of projective space missing n 0 0 1 0 points. In particular, HMW (A/K) = K and HMW (A/K) is a vector space of dimension n.

We will now focus our attention on the case where 0 < l < r. Let Sk = {i : ei ≥ k}.

We will assume the set {αi} is ordered so that S2 = {1, 2, ..., m}. Deﬁne the polynomial h(x) ∈ V [x] to be the greatest common divisor of f(x) and f 0(x). Then

n Y ei−1 h(x) = (x − αi) i=1 e Y = Tk(x) k=2

where we have deﬁned T (x) = Q (x − α ) and e = max{e }. k i∈Sk i i

Remark 3.1.7. In the sections that follow we will regularly use the notation K[x]

i x v(x) = ai(x)u(x) + bi(x)

i−1 with deg(bi(x)) < n. Note that a0 = 0, b0 = v(x), and the leading term of ai(x) is d · x k for i ≥ 1. Thus for any positive k, {ai(x)}i=1 is a basis for K[x]

j a (x) = a (x) + ixi−1 i,j r i j b (x) = b (x). i,j r i

jd + ri Then the leading term of a (x) is xi−1 which is nonzero as (d, r) = 1 and i,j r k d - i. Therefore {ai,j(x)}i=1 is also a basis for K[x]

Consider now the K-vector space homomorphism

K[x] vˆ : K[x] → (u(x))

deﬁned by multiplication by v(x). Since u(x) and v(x) have no common roots, v(x) is not a zero divisor in K[x]/(u(x)). It follows that a polynomial is in the kernel ofv ˆ if and only if ∼ it is divisible by u(x). Thereforev ˆ induces a K-automorphism of K[x]/(u(x)) = K[x]

Using this notation, we can write

j d(xiyj) = ixi−1yjdx + f 0(x)xiyj−rdx r j j = ixi−1yjdx + a (x)yjdx + b (x)h(x)yj−rdx r i r i j j−r = ai,j(x)y dx + bi,j(x)h(x)y dx. (3.3)

3.1.4 Some Useful Order-preserving Functions

Lk Deﬁnition 3.1.8. Given totally ordered sets T1, ..., Tk, deﬁne a total order on i=1 Ti Lk called the lexographical order as follows: For elements (t1, ..., tk), (s1, ..., sk) ∈ i=1 Ti,

(t1, ..., tk) < (s1, ..., sk) if and only if the ﬁrst coordinates ti, si which are diﬀerent, from

the left, satisfy ti < si.

Given the coordinate ring A from the the previous section and any integer t, we can deﬁne a bijective map

t u∞ : Z × {0, 1, ..., d − 1} → (−dt + rZ)

t s −t+kr u∞((k, s)) = −ordP∞ (x y ) = rs + d(−t + kr) = −dt + r(s + kd)

which is order preserving, where Z × {0, 1, ..., d − 1} is given the lexographical order and Chapter 3. Superelliptic Curves 35

−dt + rZ is given the natural order inherited from Z. Suppose A ∈ A is given by

d−1 X X s j A = λs,jx y . j∈Z s=0

Let (j0, s0) = max{(j, s): λs,j 6= 0}. Deﬁne a function o∞ : A → Z by

s0 j0 o∞(A) = −ordP∞ (x y ).

Proposition 3.1.9. Suppose A ∈ Al, given by a sum

d−1 X X s −l+kr A = λs,l+krx y . k∈Z s=0

with j0, s0 deﬁned as above. Then the expansion of A in the completed local ring at

s0 j0 P∞ has the same leading term as the expansion of λs0,j0 x y . In particular, o∞(A) =

−ordP∞ (A).

s0 j0 s0 j0 Proof. It suﬃces to show that ordP∞ (A − λs0,j0 x y ) > ordP∞ (λs0,j0 x y ). Set k0 = s0 j0 s −l+rk (j0 + l)/r. Every term of A − λs0,j0 x y is of the form λs,−l+rkx y with (k, s) < l l (k0, s0). It follows that u∞((k, s)) < u∞((k0, s0)), which gives

s −l+rk l ord(λs,−l+rkx y ) = −u∞((k, s)) l > −u∞((k0, s0))

s0 j0 = ordP∞ (λs0,j0 x y ).

s0 j0 The result follows since ordP∞ (A − λs0,j0 x y ) is bounded by the minimum order of its terms.

Similarly, for i = 1, ..., n, and any integer t, deﬁne

t ui : Z × {0, 1, ..., ei − 1} → (−eit + rZ)

t s −t+rk ui((k, s)) = ordPi ((x − αi) y )

= sr + ei(−t + rk)

= −eit + (s + eik)r Chapter 3. Superelliptic Curves 36

t for k ∈ Z. Then clearly ui is order preserving.

For a polynomial Q(x) ∈ K[x] of degree less than d, we can write the following partial fraction decomposition n ei−1 Q(x) X X Qi,s = . f(x) (x − α )ei−s i=1 s=0 i

for unique constants Qi,s ∈ K. Multiplying through by f(x) and deﬁning fi(x) =

ei f(x)/(x − αi) we get n ei−1 X X s Q(x) = Qi,s(x − αi) fi(x). (3.4) i=1 s=0

Note that if x − αi divides Q(x) with multiplicity t ≤ ei, then the decomposition yields

Qi,s = 0 for s = 0, 1, ..., t − 1.

Suppose A ∈ A. Passing to an extension of K if necessary, by Equation (3.4) we can

ﬁnd constants Ai,s,k such that

n ei X X X s j A = Ai,s,j(x − αi) fi(x)y i=1 j∈Z s=0

Deﬁne j0 := min{j : Ai,s,j 6= 0 for some i, s}, choose i0 such that Ai0,s,j0 6= 0 for some s,

and put s0 = min{s : Ai0,s,j0 6= 0}. Deﬁne

o (A) = ord ((x − α )s0 yj0 ). i0 Pi0 i0

Remark 3.1.10. For any A ∈ A, it is not necessarily true that oi(A) is deﬁned for all i, but if A 6= 0 there exists some i such that it is deﬁned.

Proposition 3.1.11. Fix l, 0 < l < r. Suppose A ∈ Al is given by a sum

n ei−1 X X X s −l+rk A = Ai,s,−l+rk(x − αi) fi(x)y ,

i=1 k≥k0 s=0

j + l where j , s , i are as deﬁned above, and k = 0 . Choose positive integers a, b that sat- 0 0 0 0 r a −b isfy ar −bei0 = 1, so that z := (x−αi0 ) y is a uniformizing parameter at Pi0 . Then the ei −1 bs+aj +1 sr+j e terms of order less than e (r+j ) in the expansions of A and P 0 A f (x) 0 z 0 i0 i0 0 s=s0 i0,s,j0 i0 as series in z in the completed local ring at P are equal. Moreover, o (A) = ord (A). i0 i0 Pi0 Chapter 3. Superelliptic Curves 37

Proof. To prove o (A) = ord (A), it is suﬃcient to prove the inequality i0 Pi0

ord (A − A (x − α )s0 f (x)yj0 ) > ord (A (x − α )s0 f (x)yj0 ), (3.5) Pi0 i0,s0,j0 i0 i0 Pi0 i0,s0,j0 i0 i0 from which it follows

ord (A) = ord (A (x − α )s0 f (x)yj0 ) Pi0 Pi0 i0,s0,j0 i0 i0 = ord (A (x − α )s0 yj0 ) Pi0 i0,s0,j0 i0

= oi(A).

By the minimality conditions used to deﬁne the pair (j0, s0), it follows that each term

s0 j0 s −l+rk in A − Ai0,s0,j0 (x − αi0 ) fi0 (x)y has either the form Ai0,s,−l+rk(x − αi0 ) fi0 (x)y for s j (k, s) > (k0, s0), or the form Ai1,s,j(x − αi1 ) fi1 (x)y for i1 6= i0, j ≥ j0. For the ﬁrst type, one can write

ord (A (x − α )sf (x)y−l+rk) ≥ ord ((x − α )sy−l+rk) Pi0 i0,s,−l+rk i0 i0 Pi0 i0 l = ui0 ((k, s)) l > ui0 ((k0, s0)) = ord (A (x − α )s0 f (x)y−l+rk0 ) Pi0 i0,s0,j0 i0 i0

l where the second inequality comes from the fact that oi0 is order preserving. For the second type, we can compute

ord (A (x − α )sf (x)yj) = ord (A f (x)yj) Pi0 i1,s,j i1 i1 Pi0 i1,s,j i1

≥ rei0 + jei0

≥ (r + j0)ei0

> rs0 + j0ei0 = ord (A (x − α )s0 f (x)y−l+rk0 ) Pi0 i0,s0,j0 i0 i0

which proves Equation (3.5).

For the main part of the proposition, we follow a similar process. It is clear that each term of ei0 −1 X s j0 A − Ai0,s,j0 (x − αi0 ) fi0 (x)y . s=s0

s −l+rk either has the form Ai0,s,l+rk(x − αi0 ) fi0 (x)y where k > k0, or the form Ai1,s,j(x − Chapter 3. Superelliptic Curves 38

s j αi1 ) fi1 (x)y where i1 6= i0, j ≥ j0. We have already seen that the second type has order

greater or equal to ei0 (r + j0) at Pi0 . For the ﬁrst type, we calculate

ord (A (x − α )sf (x)y−l+rk) ≥ ord (y−l+rk) Pi0 i0,s,−l+rk i0 i0 Pi0 l = ui((k, 0)) l = ui((k0 + 1, 0))

= ei0 (r + j0).

It follows that

 ei −1  X0 ord A − A (x − α )sf (x)yj0 ≥ e (r + j ) (3.6) Pi0  i0,s,j0 i0 i0  i0 0 s=s0

e −1 which show that the expansions for A and P i0 A (x − α )sf (x)yj0 in z are equal s=s0 i0,s,j0 i0 i0 e (r+j )−1 up to z i0 0 .

The computations

x − αi0 1−ar br −be br b = (x − α ) y = (x − α ) i0 y = f (x) (3.7) zr i0 i0 i0 y 1+bei0 −aei0 ar −aei0 a e = y (x − αi0 ) = y (x − αi0 ) = fi0 (x) z i0

give

s0 j0 bs0+aj0+1 rs0+ei0 j0 (x − αi0 ) fi0 (x)y = fi0 (x) z ,

yielding the desired result

3.2 Computing a Basis for Cohomology

Proposition 3.2.1. For 0 < l < r the classes [xih(x)dx/yl], 0 ≤ i ≤ n − 2 form a basis 1 l for HMW (A/K) .

The proof of this proposition will be accomplished in several steps. First we will show that these classes generate the de Rham cohomology by giving a reduction algorithm for diﬀerentials with ﬁnite degree in y. Next, we will calculate a bound for the p-adic precision loss incurred by the algorithm, proving that the classes generate the dagger cohomology. Finally we will prove linear independence. Chapter 3. Superelliptic Curves 39

r r−1 0 1 Using the relations y = f(x) and ry dy = f (x)dx, each element in Ωl can be written in the form

1 X Ak(x)dx yl yrk −∞

where Ak(x) ∈ K[x] has degree less than d.

3.2.1 The Reduction Process

We will begin by giving a “reduction of poles” procedure on diﬀerentials of the form A(x)yjdx with deg(A(x)) < d, and j = −l + rk with k an integer and 0 ≤ l ≤ r − 1. To ensure that all reduction steps are performed with coeﬃcients in V , we will assume that

αi ∈ K for all i such that ei > 1.

To reduce differentials of the form A(x)yjdx, j > 0, deg(A) < d, one may write A(x) as a linear combination of the ai,j(x)’s, 1 ≤ i ≤ d, and use Equation (3.3) to obtain A(x)yjdx as an exact differential plus a differential of the form B(x)yj−rdx. Repeating this process we can write dx A(x)yjdx = Ae(x) + dν yl for some 0 ≤ l < r, and deg(Ae) < d.

dx To reduce diﬀerentials of the form A(x) , j > 0, we can similarly use Equation (3.3), yj dx but we must ﬁrst reduce to the form B(x)h(x) . One begins with some preliminary yj computations.

−ei For i ∈ S2, recall we set fi(x) = f(x)(x − αi) . By a straightforward calculation,

0 −ei+1 0 f (x)(x − αi) = fi (x)(x − αi) + eifi(x). Chapter 3. Superelliptic Curves 40

For 0 ≤ t ≤ ei − 2, we have

f(x) f(x) 0 dx f(x) dy d e −t−1 j = e −t−1 j − j e −t−1 j+1 (x − αi) i y (x − αi) i y (x − αi) i y 0 r−1 f (x) (ei − t − 1)f(x) dx j f(x)(ry dy) = e −t−1 − e −t j − e −t−1 j+r (x − αi) i (x − αi) i y r (x − αi) i y 0 0 f (x) (ei − t − 1)f(x) dx j f (x)dx = e −t−1 − e −t j − e −t−1 j (x − αi) i (x − αi) i y r (x − αi) i y 0 j f (x) (ei − t − 1)f(x) dx = (1 − ) e −t−1 − e −t j r (x − αi) i (x − αi) i y j = (1 − )(x − α )t(e f (x) + (x − α )f 0(x)) r i i i i i dx − (e − t − 1)(x − α )tf (x) i i i yj je j dx = (t + 1 − i )(x − α )tf (x) + (1 − )(x − α )t+1f 0(x)) r i i r i i yj

Thus dx dx (x − α )tf (x) ≡ c (x − α )t+1f 0(x) (3.8) i i yj i,j,t i i yj (j − r) where ci,j,t = . (r(t + 1) − jei)

Remark 3.2.2. This expression makes sense as the denominator of ci,j,t is nonzero: If

l 6= 0, then r(t + 1) − jei ≡ −lei (mod r) which is nonzero. If l = 0 then j = rk with

k > 0, so r(t + 1) − jei = r(t + 1 − kei) < 0 since t + 1 < ei.

ei−1 0 Note that for all i, h(x) = T2(x) ··· Te(x) divides (x − αi) fi (x), and in general for

1 ≤ tei − 1 we can write

t 0 (x − αi) fi (x) = T2(x)T3(x) ··· Tt+1(x)Ri,t(x)

for some polynomial Ri,t(x).

[1] Let f (x) := f(x)/T2(x). Using a partial fraction decomposition and the fact that [1] deg(A) < d, we can ﬁnd constants Ai,1 and a polynomial Q (x) of degree less than d−m such that [1] m A(x) Q (x) X Ai,1 = + . f(x) f [1](x) (x − α )ei i=1 i Chapter 3. Superelliptic Curves 41

Multiplying both side of the equation by f(x), one sees that

m [1] X A(x) = Q (x)T2(x) + Ai,1fi(x). i=1

By Equation (3.8), we therefore have

m dx dx X dx A(x) ≡ Q[1](x)T (x) + c A (x − α )f 0(x) yj 2 yj i,j,0 i,1 i i yj i=1 m ! X dx = T (x) Q[1](x) + c A R (x) 2 i,j,0 i,1 i,1 yj i=1 dx = T (x)A[1](x) 2 yj

[1] [1] Pm [2] [1] where we have set A (x) := Q (x)+ i=1 ci,j,0Ai,1Ri,1(x). Letting f (x) := f (x)/T3(x), [2] we begin the same process again by ﬁnding a polynomial Q and constants Ai,2 for i ∈ S3 such that [1] [2] A (x) Q (x) X Ai,2 [1] = [2] + e −1 . f (x) f (x) (x − αi) i i∈S3 Multiplying both sides by f(x), one can then write

[1] [2] X T2(x)A (x) = T2(x)T3(x)Q (x) + Ai,2(x − αi)fi(x),

i∈S3

so that

dx dx X dx T (x)A[1](x) ≡ T (x)T (x)Q[2](x) + c A (x − α )2f 0(x) 2 yj 2 3 yj i,j,1 i,2 i i yj i∈S3 ! X dx = T (x)T (x) Q[2](x) + c A R (x) 2 3 i,j,1 i,2 i,2 yj i∈S3 dx = T (x)T (x)A[2](x) 2 3 yj

where we set A[2](x) = Q[2](x) + P c A R (x). i∈S3 i,j,1 i,2 i,2 Continuing in this manner, given A[k](x), in the following manner we can deﬁne a polynomial A[k+1] satisfying

dx dx T (x) ··· T A[k](x) ≡ T (x) ··· T A[k+1](x) . (3.9) 2 k+1 yj 2 k+2 yj Chapter 3. Superelliptic Curves 42

[k+1] [k] Deﬁne f (x) = f (x)/Tk+2(x) and for each i ∈ Sk+2 ﬁnd constants Ai,k+1 and a polynomial Q[k+1](x) such that

[k] [k+1] A (x) Q (x) X Ai,k+1 [k] = [k+1] + (e −k) . (3.10) f (x) f (x) (x − αi) i i∈Sk+2

[k+1] The values Ai,k+1 and the polynomial Q (x) can be computed using

[k] ei−k A (x)(x − αi) Ai,k+1 = (3.11) f [k](x) x=αi [k] [k+1] [k] X f (x) Tk+2(x)Q (x) = A (x) − Ai,k+1 e −k . (x − αi) i i∈Sk+2

After multiplication of Equation (3.10) by f(x) one has

dx dx X dx T (x) ··· T (x)A[k](x) = T (x) ··· T (x)Q[k+1](x) + A (x − α )k−1f (x) 2 k+1 yj 2 k+2 yj i,k+1 i i yj i∈Sk+2 dx X dx ≡ T (x) ··· T (x)Q[k+1](x) + c A (x − α )kf 0(x) 2 k+2 yj i,j,k i,k+1 i i yj i∈Sk+2   X dx = T (x) ··· T (x) Q[k+1](x) + c A R (x) 2 k+2  i,j,k i,k+1 i,k+1  yj i∈Sk+2 and we then deﬁne A[k+1](x) = Q[k+1](x) + P c A R (x). Thus we can i∈Sk+2 i,j,k i,k+1 i,k+1 eventually compute a polynomial B(x) of degree less than n such that

dx dx A(x) ≡ h(x)B(x) . (3.12) yj yj

dx To reduce the diﬀerential h(x)B(x) , write B(x) as a linear combination B(x) = yj Pn−1 i=0 Bibi,r−j(x) so that

n−1 dx X dx h(x)B(x) = B b (x)h(x) yj i i,r−j yj i=0 n−1 ! n−1 X xi X dx = d B − B a (x) . i yj−r i i,r−j yj−r i=0 i=0

dx Repeating this process we can write any form ω = A(x) with j > 0 as an exact yj Chapter 3. Superelliptic Curves 43

xih(x)dx diﬀerential plus a linear combination of the elements { }n−1. By subtracting an yl i=0 appropriate multiple of

r − l f 0(x)dx r − l v(x)h(x)dx d(yr−l) = (r − l)yr−1−ldy = = r yl r yl

xih(x)dx we can then reduce to a linear combination of the elements { }n−2. yl i=0 xidx In order to reduce diﬀerentials to a linear combination of elements { }n−2, we can yl i=0 use the relation

f 0(x) dy = dx ryr−1 v(x) = ydx ru(x) and obtain

xiu(x) dx dy d = (ixi−1u(x) + xiu0(x)) − lxiu(x) yl yl yl+1 l dx = ixi−1u(x) + xiu0(x) − xiv(x) . r yl

dx This equation is valid for i ≥ 0 and has leading term (i + n − dl/r)xi+n−1 6= 0. There- yl dx fore, appropriate multiples can be subtracted from A(x) to obtain a linear combination yl xidx of the elements { }n−2. yl i=0

i l n−2 i l n−2 We have now shown that both sets {[x dx/y ]}i=0 and {[x h(x)dx/y ]}i=0 span the −l 1 0 1 0 ∼ 1 0 ζ -eigenspace of HdR(CK /K). The fact that HdR(CK /K) = HMW(Ck/K) follows either from the comparison isomorphism (2.3.11), or from the lemmas below, which are an adapted version of [12, Lemma 1 and Lemma 2].

3.2.2 Two Lemmas dx Lemma 3.2.3. Suppose ω = A(x) ∈ Ω , where A(x) is a polynomial of degree less yj l blog |ej−r|c than d with coeﬃcients in V . Set e = max{ei} and N = p p . Chapter 3. Superelliptic Curves 44

i) For j > r, then there exist ν ∈ Al, Ae(x) ∈ K[x]

dx ω = Ae(x) + dν (3.13) yl

and NAe(x) has coeﬃcients in V .

ii) For j > 0, there exist ν ∈ Al, Ae(x) ∈ K[x]

dx ω = Ae(x)h(x) + dν (3.14) yl

and NAe(x) has coeﬃcients in V .

Proof. i) Write j = kr + l. By inspection of the reduction algorithm we see that without the conditions of integrality such a ν exists, and that we can write

k R0(x)u(x) X Rt(x) ν = + yj yj−tr t=1

for polynomials Rt such that deg(R0) < d − n, deg(Rk) < n, and deg(Rt) < d for 1 ≤ t ≤ k − 1.

We consider the partial fraction decomposition in the same way as Equation (3.4)

n ei−1 X X s R0(x)u(x) = R0,i,s(x − αi) fi(x). i=1 s=0

For each i, the linear term x − αi divides u(x) with multiplicity 1, and we therefore have

R0,i,0 = 0. We can then write

n ei−1 X X s R0(x)u(x) = R0,i,s(x − αi) fi(x) i=1 s=1

ei−1 X X s = R0,i,s(x − αi) fi(x) (3.15)

i∈S2 s=1

Let Pi denote the closed point in C which lies over the point (αi, 0) in aﬃne space. Then 0 0 Pi is Fqe -rational for some positive integer e. Set V = W (Fqe ) and let K denote its

ﬁeld of fractions. From Proposition (3.1.2) we have ordPi (x − αi) = r and ordPi (y) = ei.

Since (ei, r) = 1, we can ﬁnd integers ai and bi such that air − biei = 1, and therefore Chapter 3. Superelliptic Curves 45

ai bi z(i) := (x − αi) /y is a uniformizing parameter at Pi. By [51, Chapter VIII.5 Theorem 0 0 2], the completion of the local ring of C/V at Pi is V [[z(i)]].

By Proposition (3.1.11), there exists some i such that for the leading terms up to ei(r−j) z(i) the local series expansion for ν at Pi is equal to the expansion of

ei−1 X bis−aij+1 rs−eij R0,i,sfi(x) z(i) . s=1

Since fi(x) is invertible in the local ring at Pi, one can write the expansions

∞ bis−aij+1 X t fi(x) = as,tz(i) (3.16) t=0

0 where as,t ∈ V and as,0 is a unit. We can therefore write the expansion for ν at Pi as

ei−1 ∞ X X rs−eij+t (r−j)ei ν = R0,i,sas,tz(i) + O(z(i) ) s=1 t=0

and the diﬀerential dν as

ei−1 ∞ X X rs−eij+t−1 (r−j)ei−1 dν = (rs − eij + t)R0,i,sas,tz(i) dz(i) + O(z(i) ). (3.17) s=1 t=0

Since ! Ae(x)dx ord ≥ r − 1 − le ≥ r − 1 − (j − r)e , z(i) yl i i the expansion in Equation (3.17) is equal to local expansion of the right side of (3.13). dx In the left side of (3.13), the coeﬃcients of the expansion of A(x) in the completed yj 0 local ring at Pi are elements of V . Chapter 3. Superelliptic Curves 46

In particular we get the following system

0 (r − eij)R0,i,1a1,0 ∈ V (3.18) 0 (2r − eij)(R0,i,1a1,r + R0,i,2a2,0) ∈ V (3.19) 0 (3r − eij)(R0,i,1a1,2r + R0,i,2a2,r + R0,i,3a3,0) ∈ V . . 0 ((ei − 1)r − eij)(R0,i,1a1,(ei−2)r + R0,i,2a2,(ei−3)r + ... + R0,i,ei−1aei−1,0) ∈ V

blog (ej−r)c Recalling that e = max{ei}, N = p p , and using the fact that as,0 is a unit for all

s, by Equation (3.18) we have NR0,i,1 ∈ V . From Equation (3.19) it then follows that

NR0,i,2 ∈ V and continuing in this way we get NR0,i,s ∈ V for 1 ≤ s ≤ ei − 1. We then move the term ei−1 ! X s −j d R0,i,s(x − αi) fi(x)y s=1

to the left-hand side of Equation (3.13) and repeat for other values of i ∈ S2. Eventually

we get that the polynomial NR0(x) has coeﬃcients in V .

We repeat this process, considering next the equation

k ! R0(x)u(x) dx R1(x) X Rt(x) ω − d = Ae(x) + d + (3.20) yj yl yj−r yj−tr t=2

and the decomposition

n ei−1 X X s R1(x) = R1,i,s(x − αi) fi(x). i=1 s=0

Applying Proposition (3.1.11) again, locally at Pi we get

k ei−1 R1(x) X Rt(x) X rs−e (j−r) −(j−2r)e + = R f (x)bis−ai(j−r)+1z i + O(z i ) yj−r yj−tr 1,i,s i (i) (i) t=2 s=0

We follow the same process and for each degree less than −(j − 2r)ei multiply the coefficients of the above expansion by the appropriate power of p so that the corresponding 0 0 coefficients on both sides of Equation (3.20) are in V . We then find that NR1,i,s ∈ V

for all i, s and thus R1(x) ∈ V [x]. Continuing in this way, we see that NRt(x) has integer coeﬃcients for t < k. Chapter 3. Superelliptic Curves 47

For the ﬁnal step we compute the expansion at Pi of the right side of the equation

k−1 ! R0(x)u(x) X Rt(x) dx Rk(x) ω − d + = Ae(x) + d . (3.21) yj yj−tr yl yl t=1

R (x) Using a partial fraction decomposition of k we can write u(x)

n X u(x) R (x) = R k k,i x − α i=1 i

so locally at Pi we have

R (x) k = R λ z−eil + O(z−eil+1) yl k,i i (i) (i)

where f(x)ai u(x) Y e 0 ai+1 0 i λi = x=α = (αi − αi ) . x − αi i i06=i This gives R (x) d k = −e lR λ z−eil−1dz + O(z−eil) yl i k,i i (i) (i) (i) which is equal to the expansion of the expression on the left side of (3.21) since

! Adxe ord ≥ ord (d(x − α )) + ord (y−l) = r − 1 − e l > −e l. Pi yl Pi i Pi i i

0 0 Therefore NRk,i ∈ V for all i, which shows that NRk(x) ∈ V [x] and completes the proof of the ﬁrst part of the proposition.

The proof of the second part follows the exact same procedure as in the ﬁrst, with an added reduction step which gives

k R0(x)u(x) X Rt(x) Rk+1(x)u(x) ν = + + yj yj−tr yl t=1

where Rk+1 is a polynomial of degree less than d − n. Chapter 3. Superelliptic Curves 48

One can write a local expansion at Pi

ei−1 Rk+1(x)u(x) X (r−l)e = R f (x)bis−ail+1zrs−eil + O(z i ) yl k+1,i,s i (i) (i) s=1

ei−1 ∞ X X rs−eil+t (r−l)ei = Rk+1,i,sas,tz(i) + O(z(i) ). (3.22) s=1 t=0

such that as,0 is a unit for all s. Since ! Ae(x)h(x)dx ord ≥ r(e − 1) + r − 1 − e l = (r − l)e − 1, Pi yl i i i

the coeﬃcients of the expansion (3.22) can be compared to the coeﬃcients in the local expansion of k ! R0(x)u(x) X Rt(x) ω − d + , yj yj−tr t=1 0 which gives NRk+1(x) ∈ V by the procedure from the proof of part one.

j Lemma 3.2.4. Let ω = A(x)y dx ∈ Ωl, where 0 < l < r, j > 0, and A(x) is a polynomial 0 with coeﬃcients in V of degree d0 < d . Set N = pblogp(r(d +1)+jd)c.

i) There exists ν ∈ Al, Ae(x) ∈ K[x]

dx ω = Ae(x) + dν (3.23) yl

and NAe(x) has coeﬃcients in V .

ii) There exists ν ∈ Al, Ae(x) ∈ K[x]

dx ω = Ae(x)h(x) + dν (3.24) yl

and NAe(x) has coeﬃcients in V . Proof. i) Writing j = rk − l, the reduction algorithm gives us Equation (3.23) where

d0+1 k−1 d−1 d−n−1 X t j X X t −l+sr X t −l ν = Rk,tx y + Rs,tx y + R0,tx u(x)y . t=0 s=1 t=0 t=0

with Rs,t ∈ V . Consider the completion of the local ring at the closed point at inﬁnity

P∞ ∈ Ce. We have ordP∞ (x) = −r and ordP∞ (y) = −d, so we can choose a local parameter Chapter 3. Superelliptic Curves 49

z = xay−b where a and b are integers such that bd − ar = 1. Recall from section (3.1.4) that the map

l u∞ : Z × {0, ..., d − 1} → −ld + rZ (3.25) t −l+sr (s, t) 7→ −ordP∞ (x y ), (3.26)

l computed as u∞((s, t)) = rt + d(−l + rs) = (t + sd)r − ld, is order preserving. The term d0+1 j which has the largest pole at P∞ is thus Rk,d0+1x y , and we can then write the local expansions

∞ X i ν = aiz l 0 i=−u∞((k,d +1)) ∞ X i dν = (i + 1)ai+1z dz l 0 i=−u∞((k,d +1))−1

with a l 0 = R 0 . Since the degree of A is less than n − 1, the lowest possible −u∞((k,d +1)) k,d +1 e dx order of the expansion of Ae(x) at P∞ is yl

l l −r − 1 − u∞((0, n − 2)) = −1 − u∞((0, n − 1))

thus the coeﬃcients of zi in the expansions of dν and ω are equal for

l i < −1 − u∞((0, n − 1)). (3.27)

Since the expansion for ω is in V [[z]]dz, we have (i+1)ai+1 ∈ V for all i satisfying Equa- 0 blogp(r(d +1)+jd)c l tion (3.27). Letting N = p , it follows that Nai ∈ V if −i > u∞((0, n − 1)). 0 l 0 l In particular, since (k, d + 1) > (0, n − 1), therefore u∞((k, d + 1)) > u∞((0, n − 1)) and

it follows that NRk,d0+1 ∈ V .

We repeat this process using the equation

d0+1 j dx d0+1 j ω − d(Rd0+1,kx y ) = Ae(x) + d(ν − Rd0+1,kx y ) yl

t0 −l+rs0 and performing the same computation with the next highest term Rs0,t0 x y of ν with respect to the lexographical ordering. For any such (s0, t0) we have (s0, t0) > (0, n − 1),

from which it follows that NRs0,t0 ∈ V . Repeating this gives the result for part (i). Chapter 3. Superelliptic Curves 50

To prove the part (ii), we follow the same argument as part (i) with

d0+1 k−1 d−1 X t j X X t −l+rs ν = Rk,tx y + Rs,tx y . t=0 s=1 t=0

dx l and using the fact that the order of Ae(x)h(x) at P∞ is at least −r−1−u ((0, d−2)) = yl ∞ l 0 0 0 −1−u∞((0, d−1)). Since the pairs (s , t ) appearing in ν have only s > 0, we immediately have (s0, t0) > (0, d − 1), and the result follows by the method used in part (i).

To complete the proof of Proposition (3.2.1), it suﬃces to show linear independence of the cohomology classes [xidx/yl]. Proposition 3.2.5. For l, 0 < l < r, the cohomology classes {[xidx/yl]}, 0 ≤ i ≤ n − 2, 1 l form a linearly independent set in the K-vector space HMW(A/K) . Pn−2 i −l Proof. We begin with the ﬁnite case. Let ω = i=0 cix y dx for some ci ∈ K, and suppose there exist constants λi,k ∈ K and integers k1 ≤ k2 such that

k2 d−1 X X i kr−l ω = dν = d(λi,kx y )

k=k1 i=0

l Again consider the order function u∞ from Section (3.1.4). By Proposition (3.1.9), we have

n−2 −l ordP∞ (ω) ≥ ordP∞ (x y ) + ordP∞ (dx) = −r(n − 2) + dl − r − 1 = −r(n − 1) + dl − 1 l = −u∞((0, n − 1)) − 1

We also have

ordP∞ (ν) = −o∞(ν) l = − max{u∞((k, i)) : λi,k 6= 0}, so the order of dν = ω at P∞ is one less than the above expression. Combining these two relations it follows that

l l max{u∞((k, i)) : λi,k 6= 0} ≤ u∞((0, n − 1)) Chapter 3. Superelliptic Curves 51

i.e. λi,k = 0 for (k, i) ≥ (0, n). Therefore k2 ≤ 0, and λi,0 = 0 for i ≥ n.

Write

0 d−1 n 0 ei−1 X X i kr−l X X X s −l+rk ν = λi,kx y = ai,s,k(x − αi) fi(x)y

j=k1 i=0 i=1 k=k1 s=0 for constants ai,s,k ∈ K. Let (k0, s0) = min{(k, s): ai,s,k 6= 0 for some i}. By Proposition (3.1.11) we have, for some i,

l ordPi (ν) = ui((k0, s0)).

We also have l ordPi (dν) = ordPi (ω) ≥ −eil + r − 1 = ui((0, 1)) − 1.

It then follows that (k0, s0) ≥ (0, 1), which implies ai,s,k = 0 for all (k, s) < (0, 1). Hence, one can write

ei−1 X X s −l ν = ai,s,0(x − αi) fi(x)y

i∈S2 s=1 A(x)u(x) = yl for some polynomial A(x) with deg(A(x)u(x)) < d. But deg(u(x)) = n, and from the Pn−1 i initial calculation we have that A(x)u(x) = i=0 λi,0x . It follows that A(x) = 0, so dν = 0.

An immediate consequence is that the sets

xidx xih(x)dx { l }0≤i≤n−2 and { l }0≤i≤n−2 y 0

1 − form K-bases for HMW(A/K) , a vector space of dimension (n − 1)(r − 1).

3.3 The Matrix of Frobenius

Following Gaudry and G¨urel[16], we can compute a representative for the action of the semi-linear p-power Frobenius automorphism

1 1 F : HMW (A/K) → HMW (A/K) Chapter 3. Superelliptic Curves 52

on our selected basis from the previous section. Let σ : V → V denote the Witt vector Frobenius automorphism. We can extend it to polynomials by taking xσ = xp, and to 1 diﬀerentials by taking (dx)σ = pxp−1dx. Setting ∆(x) = (f(x)σ − f(x)p) we can now p extend σ to A† by taking

σ −l/r ∞ 1 p∆ X −l/r pk∆k = y−lp 1 + = yl yrp k ypl+prk k=0

where −l/r (−l/r)(−l/r − 1) ··· (−l/r − k + 1) = ∈ k k! Zp since (r, p) = 1. Finally we have

σ ∞ xih(x)dx X −l/rpk+1xpi+p−1h(x)σ∆k = dx. (3.28) yl k ypl+prk k=0

For each i and l such that 0 ≤ i ≤ n−2 and 0 < l < r, the reduction algorithm is applied to a truncation of the above sum, which yields a (r − 1)(n − 1) × (r − 1)(n − 1) matrix M, an approximation of F with respect to the chosen basis. By semilinearity, the matrix for the q-power Frobenius F a is calculated as M σa−1 M σa−2 M ··· M. Remark 3.3.1. The p-power Frobenius σ on a curve C over k is a morphism C → Cσ, where the defining polynomial of Cσ is the polynomial for C with its coefficients replaced by their images under σ. One is therefore tempted to only consider a basis for cohomology whose coefficients are fixed by σ. This is unnecessary, since the end result will be the same: Suppose A is the matrix of Frobenius with respect to a basis α whose coefficients are stable under σ. Let B be the Frobenius matrix with respect to some other basis β, and suppose Q changes coordinates from α to β. Then B = (Q−1)σAQ, so that

a−1 a−2 a a−1 a−1 a−1 a−2 a−2 Bσ Bσ ··· B = (Q−1)σ Aσ Qσ (Q−1)σ Aσ Qσ ··· (Q−1)σAQ = Q−1Aσa−1 Aσa−2 ··· AQ.

3.4 Working Within a Crystalline Basis

One technical diﬃculty with the computations involved in the previous section is that in the general the basis {xidx/yl} is not a V -stable lattice of the Frobenius operator, i.e. the matrix M computed above may have p-power denominators. The goal of this section is to ﬁnd a lower bound for s such that psM has entries in V . Chapter 3. Superelliptic Curves 53

Let C be a superelliptic curve over V of genus g, birationally equivalent to the plane

r Qn ei curve defined by the equation y = f(x) = i=1(x−αi) , where f has degree d. As usual we will let C0 denote the affine curve minus all points along y = 0, and use subscripts k and K to denote special and generic fibres, respectively. By [50, Thm 2.6 and Proposition 1 1 3.2], the V -module HdR(C/V ) is a Frobenius-stable lattice in HdR(CK /K). 1 We compare a basis for HdR(C/V ) with our chosen basis. The content of the next few pages is largely based on a proposition by Edixhoven [52, Proposition 5.3.1], the details of which can be found in a paper by van den Bogaart [50].

Let D denote the relative eﬀective Cartier divisor of degree 1 on C corresponding to the section above inﬁnity and let s be an integer greater than 1. Consider the V -linear map of sheaves on C ds 1 OC ((s − 1)D) −→ ΩC/V (sD)

1 which is just the restriction of the derivation K (C) → ΩK (C)/K , where K (C) denotes the function ﬁeld of C. Let ds denote the map ds composed with the projection

Ω1 (s ) 1 C/V D ΩC/V (sD) → 1 , ΩC/V (D) and denote the cokernel of ds by Υs. By construction we have an exact sequence of sheaves of OC -modules

Ω1 (s ) ds C/V D OC ((s − 1)D) −→ 1 → Υs → 0. (3.29) ΩC/V (D)

1 1 Note that the support of ΩC/V (sD)/ΩC/V (D) and Υs is contained SuppD. Let P denote the special point of SuppD. By abuse of notation we will denote by Υs both the sheaf as well as its stalk at P .

In some neighbourhood of the support of D, OC (−D) is generated over OC by an element t. From Bourbaki [51, Chapter VIII.5, Theorem 2], the completion of OC,P with respect to its local ring is V [[t]]. Therefore the map ds has the local form

−2 t−sV [[t]]dt M t−(s−1)V [[t]] → ∼= V tidt. t−1 [[t]]dt V i=−s Chapter 3. Superelliptic Curves 54

Since localization and completion are exact, from Equation (3.29) we obtain an exact sequence −2 −(s−1) M i t V [[t]] → V t dt → Υb s → 0 (3.30) i=−s where Υb s denotes the completion of Υs with respect to its local ring. Therefore Υb s is

ﬁnitely generated over V by elements of Υs, so Υb s = Υs. The ﬁrst map in Equation (3.30) is the usual exterior derivative (modulo the ideal (dt/t)), so we can write

−2 M i Υs = (V /(i + 1)V ) t dt i=−s M = (V /pvp(i)V ) ti−1dt. (3.31) −s

Let ϕs denote the map 1 Γ(C, ΩC/V (sD)) → Υs.

blogp(s−1)c blogp(s−1)c 1 The above discussion shows that p Υs = 0, so ker ϕs contains p Γ(C, ΩC/V (sD)).

It turns out that one can relate ker ϕs to a crystalline basis.

Lemma 3.4.1. For s ≥ 2g, there is an isomorphism

∼ 1 ker ϕs/(im ds ∩ ker ϕs) −→ HdR(C/V ) (3.32) which is equivariant for maps induced by automorphisms of C that map D to itself.

Proof. See [50, Lemma 3.10]

Theorem 3.4.2. Let s ≥ 2g. There is a surjective map

− 1 (ker ϕs) → HdR(C/V )

1 Proof. Suppose v ∈ HdR(C/V ). By Lemma (3.4.1), v lifts to an element ve ∈ ker ϕs. Let r Qr−1 i ζ be a primitive r-th root of unity. Diﬀerentiating the identity x − 1 = i=0 (x − ζ ) and multiplying by x gives

r−1 r−1 X Y rxr = (x − ζi)x. l=0 i=0 i6=l Chapter 3. Superelliptic Curves 55

Thus we can write

r−1 r−1 r−1 r−1 r X Y i r−1 X Y i rve = rρ · ve = (ρ − ζ )ρ · ve = (1 + ρ + ··· + ρ ) · ve + (ρ − ζ )ρ · v.e (3.33) l=0 i=0 l=1 i=0 i6=l i6=l

r−1 0 Now (1 + ρ + ··· + ρ ) · ve ∈ (ker ϕs) , so by Proposition (3.1.6) the image of this 1 r−1 term in HdR(C/V ) is zero. Therefore (r − (1 + ρ + ··· + ρ )) · ve is mapped to rv.

Qr−1 i l Each term i=0 (ρ − ζ )ρ · v in (3.33) is killed by by ρ − ζ , and thus belongs to i6=l e l r−1 − − (ker ϕs) . Therefore (r − (1 + ρ + ··· + ρ )) · ve ∈ (ker ϕs) , which proves that (ker ϕs) 1 maps surjectively to rHdR(C/V ). The result follows as r is a unit in V .

− 1 1 0 Corollary 3.4.3. For s ≥ 2g, the images of (ker ϕs) and HdR(C/V ) in HdR(CK /K) are equal.

Proposition 3.4.4. Fix an integer l with 0 < l < r, let s = (d − 1)(r − 1) and as usual let h(x) = (f(x), f 0(x)). Then

1 l i) The V -module Γ(C, ΩC/V (sD)) contains the V -span of the set

xih(x)dxn−2 l , y i=0

and

ii) the image of the restriction map

1 l 0 1 l Γ(C, ΩC/V (sD)) → Γ(C , ΩC/V )

is contained in the V -span of the set

xidxml l y i=0

d(l+r−1) where ml = b r − 2c. xih(x)dx Proof. (i) We consider when an element ω = ∈ Γ(C0, Ω1 )l can be extended yl C/V to all of C, with a pole of order at most s along SuppD.

Let Pi denote the generic point in C lying above (αi, 0) and let P∞ denote the generic point above inﬁnity. Recall from Proposition (3.1.2) that we have ordPi (y) = Chapter 3. Superelliptic Curves 56

xjh(x)dx e , ord (y) = −d, ord (x − α ) = r, and ord (x) = −r. An element ω = i P∞ Pi i P∞ yl 1 can be extended to ΩC/V (sD) if ordPi ω ≥ 0 for each i, and ordP∞ ω ≥ −s.

For 0 ≤ j ≤ n − 2 we have

ordP∞ (ω) = (j + d − n) · ordP∞ (x) + ordP∞ (dx) − lordP∞ (y) = −r(j + d − n) − r − 1 + ld ≥ −r(n − 2 + d − n) − r − 1 + d = −r(d − 2) − r + (d − 1) = −(r − 1)(d − 1) = −s and

j ordPi (ω) = ordPi (x ) + ordPi (h(x)) + ordPi (dx) − l · ordPi (y)

≥ ordPi (h(x)) + r − 1 − lei

= (ei − 1)r + r − 1 − lei

= ei(r − l) − 1 ≥ 0

0 1 which proves part (i). For part (ii), note that an element of Γ(C , ΩC/V ) can be written

X j ω = Aj(x)y dx

L1≤j≤L2

where Aj(x) ∈ V [x] has degree less than d − 1. An order calculation gives

L1 ordPi (ω) ≤ ordPi (AL1 (x)y dx) = ordPi (Aj(x))r + L1ei + r − 1.

Pn Pn Since i=1 ei = d and i=1 ordPi (AL1 (x)) < d, there must be some i such that ei >

ordPi (Aj(x)). In order for ω to extend to this Pi we must have

0 ≤ ordPi (ω)

≤ (ei − 1)r + L1ei + r − 1

= eir + L1ei − 1,

which gives 1 − eir L1 ≥ > −r. ei Chapter 3. Superelliptic Curves 57

1 Now if ω extends to Γ(C, ΩC/V (sD)) we must also have −s ≤ ordP∞ ω, so that

−(d − 1)(r − 1) ≤ −L2d − deg(AL2 (x))r − r − 1.

In particular L2 < r. For l, 0 < l < r, let ωl denote the term in ω whose y-exponent is equal to −l mod r. Then we have

A (x)dx ω = −l + A (x)yr−ldx r−l yl r−l B (x)dx = l yl

For some polynomial Bl(x) ∈ V [x] whose degree satisﬁes

−(d − 1)(r − 1) ≤ ld − deg(Bl(x))r − r − 1,

that is, (d − 1)(r − 1) + ld − r − 1 d(r + l − 1) deg(B (x)) ≤ = − 2. l r r

Corollary 3.4.5. For ﬁxed l, 0 < l < r, let Vl denote the V -span of of the diﬀeren- i x h(x)dx 1 0 l 1 l tials { }0≤i≤n−2 in H (C /K) and let Wl denote the image of H (C/V ) in yl dR K dR 1 0 l HdR(CK /K) . Set e = max{ei}, N = blogp((d − 1)(r − 1) − 1)c, and put

 d(l − 1)  blogp |el − r|c, < 2 N = r . l d(l − 1)  N, ≥ 2 r

Then

Nl −N p Wl ⊂ Vl ⊂ p Wl.

Proof. By Equation (3.31) and Proposition (3.4.4)(i), the diﬀerentials

xih(x)dx pN yl

l −blog ((d−1)(r−1)−1)c belong to (ker ϕ) . Therefore, by Corollary (3.4.3), Vl ⊂ p p Wl. By xidx Proposition (3.4.4)(ii), (ker ϕ)l is contained in the V -span of , for 0 ≤ i ≤ b d(l+r−1) − yl r d(l+r−1) 2c. If A(x) is a polynomial of degree less than b r − 2c with coeﬃcients in V , then Chapter 3. Superelliptic Curves 58

using division with remainder we can write

A(x)dx B (x)dx = 1 + B (x)yr−ldx yl yl 2

d(l − 1) with B1(x),B2(x) ∈ V [x] where B1(x) is a polynomial of degree less than d. If − r d(l − 1) 2 ≥ 0, then B (x) is a polynomial with degree less than or equal to − 2. Other- 2 r wise B2(x) = 0.

By Lemma (3.2.3) (ii) and Lemma (3.2.4) (ii), and using the computation

d(l − 1) r − 2 + 1 + (r − l)d = (d − 1)(r − 1) − 1, r it follows that there exist exact diﬀerentials dν1, dν2, and polynomials Be1(x), Be2(x) ∈

blog ((d−1)(r−1)−1)c B2(x)dx Be2(x)h(x)dx V [x] of degree less than n − 1 such that p p = + dν1 yl yl B1(x)dx Be1(x)h(x)dx and pblogp |el−r|c = + dν . Since e ≤ d and l ≤ r − 1, it follows that yl yl 2 A(x)dx |el − r| ≤ (d − 1)(r − 1) − 1, hence the image of pN lies in V . yl l

Let Mp,Mq denote the (r − 1)(n − 1) × (r − 1)(n − 1) matrices of the p-power and q-power Frobenius maps, respectively,

1 0 − 1 0 − Fp,Fq :HMW (Ck/K) → HMW (Ck/K)

xih(x)dx with respect to the basis , 0 ≤ i ≤ n − 2, 0 < l < r. yl

Let Mp,l denote the (n − 1) × (n − 1) submatrix of Mp deﬁned by restricting the 1 0 l domain of Fp to HMW (Ck/K) .

Theorem 3.4.6. Let N and Nl be as deﬁned in Corollary (3.4.5). Fix 0 < l < r, and

let lp be the unique integer such that 0 < lp < r and lp ≡ lp (mod r). Then

N+Nl i) the matrix p p Mp,l has entries in V

2N ii) the matrix p Mp has entries in V

N 0 iii) if Mf is an approximation of the matrix Mp, correct to precision p for some integer 0 σa−1 σa−2 N ≥ 4N, then the matrix Mf Mf ··· Mf is an approximation of Mq, correct to precision pN 0−4N . Chapter 3. Superelliptic Curves 59

i n−2 0 0 x h(x)dx B 0 B Proof. For each l , 0 < l < r, let l denote the set l0 , and put = y i=0 1 l0 B1 ∪ · · · ∪ Br−1. Additionally, let Hl0 denote a V -bases for HdR(C/V ) and put H = 1 H1 ∪ · · · Hr−1. By Proposition (3.1.6), HdR(C/V ) can be viewed as a Frobenius equiv- 1 0 ariant V -lattice in HMW (Ck/K), so let Ql0 denote the change of basis matrix from Hl0 −1 0 to Bl and let Q denote the change of basis matrix from H to B. Then Mp,l = Qlp AlQl −1 and Mp = QAQ for certain matrices Al,A with entries in V . By Corollary (3.4.5), Qlp

−Nlp −1 −1 −N has entries in p V , and the matrices Ql and Q have entries in p V , so part (i) follows immediately. Using the fact that |el − r| ≤ (d − 1)(r − 1) − 1, so that Nl ≤ N for each l, we have that Q has entries in p−N V , and part (ii) follows.

The matrix Ae = Q−1MQf is an approximation of A, with

0 Ae ≡ A mod pN −2N .

The entries of Ae are therefore in V , and thus

a−1 a−2 a−1 a−2 0 QAeσ Aeσ ··· AeσQ−1 ≡ QAσ Aσ ··· AQ−1 mod pN −4N .

The right side of this equation is the matrix of the q-power Frobenius with respect to the

basis B. We can assume H and B are chosen such that the matrix Q has entries in Qp. Therefore one can write

a−1 a−1 a−2 Mfσ ··· Mf = (QAeσ Q−1)(QAeσ Q−1) ··· (QAQe −1) a−1 a−2 = QAeσ Aeσ ··· AQe −1,

from which part (iii) follows.

3.5 p-Adic Precision Analysis

By the Weil conjectures [4, Appx C], the numerator of the zeta function of Ck can be written

2g 2g Y X i P (T ) = (1 − µiT ) = aiT ∈ Z[T ] i=1 i=0

1/2 where |µi| = q for all i and g is the genus of Ck. Furthermore, we can order the

µi’s such that µg+i = q/µi for i = 1, .., g. Chapter 3. Superelliptic Curves 60

Let Ik denote the subsets of {1, ..., 2g} of length k. For I = (j1, ..., jk) ∈ Ik, write

µI = µj1 ··· µjk . Then for each 1 ≤ i ≤ 2g, we can write

i X ai = (−1) µI

I∈Ii X qg = (−1)i µ{1,...,2g}\I I∈Ii X 1 = (−1)iqg µI I∈I2g−i X 1 = (−1)iqg µ q2g−i I I∈I2g−i i−g = q a2g−i. (3.34)

Therefore the polynomial P (T ) can be obtained by computing each ai for 1 ≤ i ≤ g.

For each ai, we have

2g 2g |a | ≤ qi/2 ≤ qi/2, i i g

2g g/2 which gives |ai| ≤ g q for 1 ≤ i ≤ g.

We also have that

−1 1 P (T ) = det(1 − qT F |HdR(Ck/K)) so if we deﬁne Q(T ) := T 2gP (1/T ), then Q(T ) is the characteristic polynomial of qF −1, whose eigenvalues are therefore the µi’s. It follows that the set {q/µi} = {µi} are the eigenvalues of F , so that Q(T ) is in fact the characteristic polynomial of F .

Each ai is therefore a polynomial with integer coeﬃcients over the entries of the q- power Frobenius matrix, thus our calculations for the matrix of F should be correct with N1 2g g/2 p-adic precision at least p , where N1 := dlogp 2 g q e. In order to ensure that no p- adic denominators arise in our calculation, we can appeal to Theorem (3.4.6), which gives xih(x)dx that the denominators of the p-power Frobenius with respect to the basis { } are yl N2 cleared upon multiplication by p , where N2 := 2blogp((d − 1)(r − 1) − 1)c. To ensure the correct precision of the q-power Frobenius, by the same theorem, we must increase the precision by p2N2 . Therefore we must compute the matrix of pN2 F with accuracy up Chapter 3. Superelliptic Curves 61

to pN1+3N2 .

xih(x)dx By Equation (3.28), the k-th term in the expansion of F (pN2 ) is divisi- yl ble by pk+N2 , and by Lemma (3.2.3) the coeﬃcients of its reduction are divisible by

k+N2−blog (ep(l+r(k−1))−r)c p p where e = max{ei}. Therefore it is suﬃcient to calculate the

ﬁrst N3 terms with

N3 + N2 − blogp(ep(l + r(N3 − 1)) − r)c ≥ N1 + 3N2

and all calculations should be performed in V /pN3+N2 V . If we express Equation (3.28) as y−pl multiplied by a polynomial in τ := y−r, we then are only required to compute this polynomial modulo τ pN3 . Chapter 4

2 Nodal Curves in P

We now present an algorithm for computing the zeta function of a nodal plane curve in two dimensional projective space, by working on its aﬃne complement. We begin with a discussion of hypersurfaces embedded in n-dimensional projective space.

4.1 Cohomology of the Aﬃne Complement of a Hy- persurface in Pn

n In this section the field K can denote any field of characteristic 0. We suppose V ⊂ PK is a hypersurface defined by a homogeneous polynomial f ∈ K[X0, ..., Xn] of degree d, n and set U = PK \ V . Then U is a smooth, affine variety, with coordinate ring g A = { : s ≥ 0, g ∈ K[X , ..., X ] is homogeneous of degree ds}. f s 0 n

i Since U is aﬃne of dimension n, its de Rham cohomology HdR(U) vanishes for i > n, and can be calculated as the cohomology of the de Rham complex

d 1 d n 0 → A −→ ΩA −→· · · ΩA → 0.

k n+1 The elements of ΩA can be represented by a rational diﬀerential form on AK with poles along V , that is, 1 X ω = A (X)dX , f(X)s J J where we have used the shorthand notation X = (X0,X1, ..., Xn), J = (j1, ..., jk), dXJ = dXj1 ∧ · · · dXjk and the sum runs over k-tuples 0 ≤ j1 < ··· < jk ≤ n. By the deﬁnition of the exterior derivative d, if one sets the degree of dXi to be 1, then d preserves degrees,

62 Chapter 4. Nodal Curves in P2 63

so the AJ (X)’s can be taken to be homogeneous polynomials with deg AJ = ds − k. To k give a criterium that is suﬃcient to know when ω comes from ΩA, it is convenient to deﬁne the Euler contraction operator ∆.

n+1 Deﬁnition 4.1.1. Let ∆ denote the map on rational forms over AK , sending k-forms to rational (k − 1)-forms, and satisfying the following properties:

1 X 1 X i) ∆ A (X)dX = A (X)∆(dX ) B(X) J J B(X) J J k X i−1 ii) ∆(dXJ ) = (−1) Xji dXj1 ∧ · · · ∧ dXdji ∧ · · · dXjk i=1

k One can then show [35, Proposition 2.2] that ω comes from ΩA if and only if ∆(ω) = 0. Additionally, ∆ is exact [35, Lemma 2.8], from which it then follows that any element k ω ∈ ΩA can be written

1 X ω = A (X)∆(dX ) f(X)s J J k+1 1 X X i−1 = AJ (X) (−1) Xj dXj ∧ · · · ∧ dXdj ∧ · · · dXj f(X)s i 1 i k i=1 where the ﬁrst sum runs over subsets J of size k + 1, and AJ (X) is a homogeneous n polynomial of degree sd − k − 1. In particular, all elements of ΩA have the form

G ω = Ω f s where G is a homogeneous polynomial of degree sd − n − 1, and

n X i Ω = (−1) XidX0 ∧ · · · ∧ dXdi ∧ · · · ∧ dXn. i=0

n 0 To calculate exact diﬀerentials in ΩA, consider the subvariety U ⊂ U obtained by remov- 0 ing points along the hyperplane X0 = 0. Then the coordinate ring of U can be written

in aﬃne coordinates xi := Xi/X0, i = 1, ..., n. Using the notation x = (x1, ..., xn) and n−1 f(1, x) = f(1, x1, ..., xn), one can write an element ω ∈ ΩU 0 as

n 1 X ω = Ai(x)dx1 ∧ · · · ∧ dxci ∧ · · · ∧ dxn f(1, x)s i=1 Chapter 4. Nodal Curves in P2 64

for arbitrary polynomials Ai(x), so that

n X ∂ Ai(x) dω = dxi ∧ dx1 ∧ · · · ∧ dxci ∧ · · · ∧ dxn ∂x f(1, x)s i=1 i n X ∂ Ai(x) = (−1)i−1 dx ∧ · · · ∧ dx . ∂x f(1, x)s 1 n i=1 i

By direct computation one has

1 dx1 ∧ · · · ∧ dxn = 2n (X0dX1 − X1dX0) ∧ · · · ∧ (X0dXn − XndX0) X0 1 = n dX1 ∧ · · · ∧ dXn X0 n 1 X − X dX ∧ · · · ∧ dX ∧ dX ∧ dX ∧ · · · ∧ dX Xn+1 i 1 i−1 0 i+1 n 0 i=1 1 = n+1 Ω X0 so that

n X ∂ Ai(x) 1 dω = (−1)i−1 Ω ∂x f(1, x)s Xn+1 i=1 i 0 n X ∂ Ai(X1/X0, ..., Xn/X0) 1 = (−1)i−1X Ω 0 ∂X f(1,X /X , ..., X /X )s Xn+1 i=1 i 1 0 n 0 0 n sd−n X ∂ X Ai(X1/X0, ..., Xn/X0) = (−1)i−1 0 Ω ∂X f(X)s i=1 i

n Since Ai(x) can be chosen arbitrarily, it follows that exact diﬀerentials in ΩA are combi- ∂ nations of elements of the form (G/f s)Ω for G a homogeneous polynomial of degree ∂Xi sd − n, and thus the n-th de Rham cohomology group of of U can be written

G n SpanK s Ω: G is homogeneous of degree sd − n − 1, s ≥ 1 ΩA f n−1 = s . dΩA ∂(G/f ) SpanK Ω : 0 ≤ i ≤ n, s ≥ 1,G is homogeneous of degree sd − n ∂Xi

The following theorem of Dimca gives two important pieces of information regarding representatives for cohomology classes in the complement of a hypersurface.

Theorem 4.1.2. Let S be the singular locus of V . Then Chapter 4. Nodal Curves in P2 65

n (i) any cohomology class in HdR(U) can be represented by a meromorphic diﬀerential form of the type G(X) ω = Ω f(X)n for some polynomial G(X); 1 (ii) for i < n − dim S, any exact form ω = P A (X)dX ∈ Ωi can be f(X)s |J|=i J J A written   1 X ω = d B (X)dX f(X)s−1 J J  |J|=i−1

for some polynomials BJ (X); 1 (iii) there is a positive constant k = k(V ) such that any exact form ω = P A (X)dX f(X)s |J|=i J J can be written   1 X ω = d B (X)dX f(X)s+(i+1)k J J  |J|=i−1

for some polynomials BJ (X).

Proof. See [36], Theorem B, as well as the paragraph following Theorem A.

Dimca also oﬀers the following conjecture that the constant k(V ) depends only on the dimension of the ambient space. 1 Conjecture 4.1.3. Any exact form ω = P A (X)dX can be written f(X)s |J|=i J J

  1 X ω = d B (X)dX f(X)s+n J J  |J|=i−1 for some polynomials BJ (X).

Assuming the validity of the conjecture, we obtain an algorithm to compute a basis n for HdR(U) and a “reduction of poles” procedure. We simply need to compute a basis for the quotient of certain ﬁnitely generated K-vector spaces. Namely, set

G Z := Span : G is homogeneous of degree sd − n − 1 s K f s ∂(G/f s) B := Span : W ∈ {X,Y,Z},G is homogeneous of degree sd − n . s K ∂W Chapter 4. Nodal Curves in P2 66

Then by 4.1.2, we have for s ≥ n,

n ∼ Zs HdR(U) = . Zs ∩ Bs+n

This procedure for computing cohomology is not so explicit, however. In the nonsingular case, one use Gröebnerbasis techniques from the partial derivatives of f to write any differential as the sum of an exact differential and a differential with lower pole order. In the singular case this does not work, since for any fixed pole order there will necessarily be exact differentials which cannot be written as a polynomial combination of the partial derivatives of f.

Though the constant k(V ) seems diﬃcult to ﬁnd for a general variety, in the case of a 2 reduced curve embedded in PK one can take k(V ) = 1 [36, Example 3]. In addition, from the following proposition we see that if the curve is irreducible, then the only “interesting” 2 de Rham cohomology group of the complement is HdR(U).

2 Proposition 4.1.4. Suppose C is a reduced curve in PK of degree d with isolated sin- 2 gularities. Let Σ denote the singular points of C, U = PK \ C, and let m = #Σ. Then 0 1 HdR(U) = K, and HdR(U) has dimension r − 1 over K, where r is the number of irreducible components of C. If Σ consists of ordinary multiple points with multiplicities 2 Pm 2 k1, ..., km, then HdR(U) has dimension (d−1)(d−2)+r−1− i=1(ki −1) . In particular, 2 if C is an irreducible nodal curve, then dimK HdR(U) = (d − 1)(d − 2) − m.

Proof. Since cohomological dimension is preserved by ﬂat base extension, we may assume ∗ 2 ∗ that K is algebraically closed. Let P = PK \ Σ and C = C \ Σ. The Gysin sequence (Theorem (2.4.1)) for the pair (P∗,C∗) gives a long exact sequence

k ∗ k k−1 ∗ k+1 ∗ · · · → HdR(P ) → HdR(U) → HdR (C ) → HdR (P ) → · · · (4.1)

0 ∗ ∼ 0 which immediately shows there is an isomorphism HdR(P ) = HdR(U). By aﬃneness of 3 k ∗ k U,HdR(U) = 0, and using the fact that HdR(P ) → HdR(U) is trivial for k = 1 or 2 (see [53, Ch. 6, Ex. 3.9 (ii)]) gives us short exact sequences

1 0 ∗ 2 ∗ 0 → HdR(U) → HdR(C ) → HdR(P ) → 0 2 1 ∗ 3 ∗ 0 → HdR(U) → HdR(C ) → HdR(P ) → 0. Chapter 4. Nodal Curves in P2 67

2 The Gysin sequence for the pair (PK , Σ) gives an exact sequence

k 2 k ∗ k−3 k+1 2 · · · → HdR(PK ) → HdR(P ) → HdR (Σ) → HdR (PK ) → · · · (4.2)

k ∗ ∼ k 2 which shows that HdR(P ) = HdR(PK ) for 0 ≤ k ≤ 2. Additionally, using the fact that 3 2 4 ∗ HdR(PK ) = HdR(P ) = 0, we obtain a short exact sequence

3 ∗ 0 4 2 0 → HdR(P ) → HdR(Σ) → HdR(PK ) → 0.

Assembling this information, have

0 0 2 dimK HdR(U) = dimK HdR(PK ) = 1, 1 0 ∗ dimK HdR(U) = dim HdR(C ) − 1, 2 1 ∗ 3 ∗ dimK HdR(U) = dim HdR(C ) − dim HdR(P ) 1 ∗ 0 4 2 = dim HdR(C ) − dim HdR(Σ) + dim HdR(PK ) 1 ∗ = dim HdR(C ) − m + 1.

1 Therefore the dimension of HdR(U) is equal to one less than the number of connected components of C \ Σ, which is one less than the number of irreducible components of C. 1 ∗ It remains to calculate the dimension of HdR(C ). Denote by Ce → C the nonsingular model of C. If P ∈ Σ is ordinary with multiplicity k, then there are k points in Ce lying (d − 1)(d − 2) ki(ki − 1) above Σ. Recalling that the genus g of Ce is − P , we have 2 2

0 ∗ 1 ∗ ∗ dimK HdR(C ) − dimK HdR(C ) = χ(C ) m X = χ(Ce \{ ki points}) i=1 m X = 2 − 2g − ki i=1 m X = 2 − (d − 1)(d − 2) + ki(ki − 2). i=1

0 ∗ 1 ∗ Pm Since dimK HdR(C ) = r, therefore dim HdR(C ) = (d−1)(d−2)+r −2− i=1 ki(ki −2), so ﬁnally we obtain

m 2 X dimK HdR(U) = (d − 1)(d − 2) + r − 2 − ki(ki − 2) − (m − 1) i=1 Chapter 4. Nodal Curves in P2 68

which gives the result.

4.2 Basic Properties of Nodal Curves

2 We now restrict our attention to the study of nodal curves in P . Let Ck be a reduced, k irreducible, plane curve over the ﬁeld = Fq whose singular points over the algebraic closure k are nodes. That is, the singularities of Ck are of multiplicity 2 and have distinct tangent directions. Suppose Ck is deﬁned by an irreducible homogeneous polynomial

f ∈ k[X,Y,Z] of degree d. If Σk is the singular locus of Ck and m = #Σk, then we know

(e.g. [37, 8.3, Proposition 5]) that the genus of the nonsingular model of Ck (i.e. the

geometric genus of Ck) is (d − 1)(d − 2) g = − m. 2

Computing the zeta function of Ck is clearly equivalent to computing the zeta function 2 of its complement Uk in Pk . Since Uk is smooth and aﬃne, its zeta function Z(Uk; T ) can i be recovered from the induced action of Frobenius on HMW(Uk/K).

4.2.1 Computing a Lift

Suppose f ∈ V [X,Y,Z] is a homogeneous polynomial with reduction modulo p equal to f. Then f deﬁnes a scheme C → Spec(V ) whose ﬁbre over the special point of Spec(V )

can be identiﬁed with Ck. Let CK denote the ﬁbre over the generic point of V .

For a point P ∈ CK , there exist elements X0,Y0,Z0 ∈ V such that P can be repre-

sented as the triple (X0,Y0,Z0), and at least one of X0,Y0,Z0 is a unit. One can deﬁne a map

ρC : CK → Ck

P 7→ P = (X0, Y 0, Z0)

sending P to the point (X0 mod p, Y0 mod p, Z0 mod p).

−1 Proposition 4.2.1. If P ∈ Ck is non-singular, then ρC (P ) contains only non-singular −1 points. If P is a node, then ρC (P ) contains at most one node.

Proof. Without loss of generality we can assume Z0 = 1. Deﬁne f 0(X,Y ) := f(X,Y, 1) −1 and f0(X,Y ) := f(X,Y, 1). Suppose that P is non-singular and Q ∈ ρC (P ). On the Chapter 4. Nodal Curves in P2 69

2 aﬃne subscheme of PK deﬁned by Z = 1, let J(Q) denote the Jacobian matrix evaluated Q, that is df df J(Q) = 0 (Q), 0 (Q) . dX dY df df This matrix, modulo p, is equal to [ 0 (P ), 0 (P )] which has rank 1 since P is non- dX dY singular. Therefore J(Q) has rank 1, so Q is non-singular in CK .

−1 Suppose now that P is a node and Q = (X0,Y0, 1) ∈ ρC (P ) is singular. After a

change of variables, we can assume P = (0, 0, 1), and f0 = XY + h, where each term of h ∈ V [X,Y ] either has degree greater than 2 or is divisible by p. The Hessian matrix of f0 at Q is  2 2  d h d h " # (Q) 1 + (Q) a 1 + a  dX2 dXdY  = 1 2  d2h d2h  1 + (Q) (Q) 1 + a3 a4 dXdY dY 2

where ai ∈ pV for each i. Therefore the determinant of the Hessian is in 1 + pV , and therefore a unit, hence Q is a node. By a version of Hensel’s lemma [54, Chapter III.4.5, 0 0 0 2 Corollary 2], it follows that there exists a unique pair Q = (X0,Y0 ) ∈ V such that df df Q0 ≡ (0, 0) mod p, and 0 (Q0) = 0 (Q0) = 0. Since the pair (X ,Y ) satisfy the same dX dY 0 0 0 0 properties, we must have that X0 = X0 and Y0 = Y0 . Therefore Q is unique.

Corollary 4.2.2. CK is a K-curve whose singularities consist of at most m nodes cor-

responding to a subset of nodes in Ck.

Definition 4.2.3. Suppose f ∈ k[X,Y,Z] is a homogeneous polynomial of degree d, 2 defining a nodal curve Ck ⊂ Pk . Suppose f ∈ V [X,Y,Z] is a homogeneous polynomial of degree d whose coefficients module pV are equal to the coefficients of f. Let C denote the

corresponding V -scheme, and CK the ﬁbre over the generic point of Spec(V ). We will say f is an equisingular lift of f if for each node P ∈ Ck there a unique node P ∈ ρC (P ). We will call f a ﬁnite equisingular lift if f is an equisingular lift of f, and additionally

the coeﬃcients of f are in Vﬁn.

Proposition 4.2.4. Let f ∈ k[X,Y,Z] be a homogeneous irreducible polynomial deﬁning

a nodal curve Ck. Then there exists an equisingular lift of f.

Proof. Suppose P1, ..., Pm are the nodes of Ck. From the paragraph after Proposition 1.5 of a paper by Deligne and Mumford [55], it is shown that there exists a scheme C , proper ∼ and ﬂat over Spec(V [[t1, ..., tm]]), such that C ×Spec(k) = Ck, and in the completed local Chapter 4. Nodal Curves in P2 70

rings one has ∼ V [[x, y, t1, ..., tm]] ObPi,C = . (xy − ti)

An equisingular lift of f is the deﬁning polynomial of the ﬁbre of C at t1 = t2 = ··· = tm = 0.

We can compute an equisingular lift of a polynomial f up to precision pN as follows.

Let Md be the set of monomials in X,Y,Z of degree d. For each monomial M ∈ Md define a parameter c , and put F = P c M. Let f be any lift of f to [X,Y,Z], M M∈Md M V and let P1, ..., Pm be lifts of the nodal points of f. One can then consider the system of N−1 equations in the variables {cM }M∈Md and with coefficients in the ring V /p V , defined by

∂F 1 ∂f (P ) = (P ) (4.3) ∂X i p ∂X i ∂F 1 ∂f (P ) = (P ) ∂Y i p ∂Y i ∂F 1 ∂f (P ) = (P ) ∂Z i p ∂Z i

where one interprets c/p to mean an element b ∈ V /pN−1V such that pb = c. Any solution of this system yields an approximation fe := f − pF of an equisingular lift of f.

In general, a ﬁnite equisingular lift will not exist, however, one can construct such a lift explicitly provided that the number of singularities of f is suﬃciently small.

Proposition 4.2.5. For p 6= 2, suppose f ∈ k[X,Y,Z] is homogeneous of degree d with

(p, d) = 1, deﬁning a reduced curve Ck with m nodes, rational over k, and possessing no d + 1 other singularities. If m ≤ , then there exists a ﬁnite equisingular lift of f. 2 Proof. We will prove the proposition by constructing an equisingular lift of f explicitly.

Let f be any lift of f to a homogeneous polynomial of degree d with coeﬃcients in Vﬁn. 2 Let P i = (Xi, Y i, Zi) ∈ Pk , i = 1, ..., m denote the distinct singular points of the curve

Ck, and for each i, lift P i to a Vﬁn-triple Pi = (Xi,Yi,Zi). Since the partial derivatives Chapter 4. Nodal Curves in P2 71

of f vanish modulo p at each Pi, we can deﬁne elements ai, bi, ci, ∈ Vﬁn by

1 df a = (X ,Y ,Z ) i p dX i i i 1 df b = (X ,Y ,Z ) i p dY i i i 1 df c = (X ,Y ,Z ) i p dZ i i i

dF Suppose F ∈ Vﬁn[X,Y,Z] is homogeneous of degree d, and satisﬁes (Xi,Yi,Zi) = dX dF dF a , (X ,Y ,Z ) = b , (X ,Y ,Z ) = c for all i. Then the polynomial f − pF is a i dY i i i i dZ i i i i lift of f, whose partial derivatives are zero at each Pi. By Corollary (4.2.2), f − pF is a rational equisingular lift of f. The method is to construct the polynomial F using a Lagrange interpolation-type approach.

To ﬁnd a suitable polynomial F , it suﬃces to construct, for each point Pi, homoge-

neous polynomials Fi,X ,Fi,Y , and Fi,Z of degree d with coeﬃcients in Vﬁn, satisfying the following conditions: ∂F ∂F ∂F 1. i,X (P ) = i,Y (P ) = i,Z (P ) = 1 ∂X i ∂Y i ∂Z i ∂F 2. i,W (P ) = 0 for all other choices W ∈ {X,Y,Z}, j ∈ {1, ..., m}. ∂W j Assuming this is possible, one could write

m X F := aiFi,X + biFi,Y + ciFi,Z i=1 satisfying the conditions for F . Therefore the task is to construct suitable polynomials

Fi,X ,Fi,Y ,Fi,Z . Without loss of generality we can assume i = 1 and P1 = (X1,Y1, 1).

We ﬁrst will show that one can reduce to the case P1 = (0, 0, 1). Consider the linear transformation of the projective plane

X 7→ X − X0Z

Y 7→ Y − Y0Z Z 7→ Z,

under which the points P1, ..., Pm get mapped to some points Q1, ..., Qm, with Q1 =

(0, 0, 1). Assuming that one could construct polynomials G1,X ,G1,Y ,G1,Z satisfying the Chapter 4. Nodal Curves in P2 72

above properties for Q1, ..., Qm, it follows that for any elements a, b, c ∈ V , the polynomial

Ga,b,c(X,Y,Z) := aG1,X +bG1,Y +cG1,Z satisﬁes ∂Ga,b,c/∂X(Q1) = a, ∂Ga,b,c/∂Y (Q1) = b,

∂Ga,b,c/∂Z(Q1) = c, and ∂Ga,b,c/∂W (Qj) = 0 for all W ∈ {X,Y,Z} and j 6= i. Now deﬁne

Fa,b,c(X,Y,Z) := Ga,b,c(X + X0Z,Y + Y0Z,Z)

and note that for j = 1, ..., m we have

∂F ∂G a,b,c (P ) = a,b,c (Q ) ∂X j ∂X j ∂F ∂G a,b,c (P ) = a,b,c (Q ) ∂Y j ∂Y j ∂F ∂G ∂G ∂G a,b,c (P ) = X a,b,c (Q ) + Y a,b,c (Q ) + a,b,c (Q ). ∂Z j 0 ∂X j 0 ∂Y j ∂Z j

The partial derivatives of Fa,b,c at Pj therefore vanish for all j 6= 1, and one also has

      ∂Fa,b,c/∂X(P1) 1 0 0 a        ∂Fa,b,c/∂Y (P1)  =  0 1 0  b  . ∂Fa,b,c/∂Z(P1) X0 Y0 1 c

Since the 3 × 3 matrix above is invertible, there exist choices for a, b, c such that Fa,b,c

satisﬁes the criteria for each of F1,X ,F1,Y , and F1,Z at the points P1, ..., Pm.

It remains to show that, under the conditions of the proposition, it is possible to construct F1,X ,F1,Y , and F1,Z when P1 = (0, 0, 1), so for the remainder of the proof we

will ﬁx P1 = (0, 0, 1).

Claim For each j 6= 1, there exists a homogeneous polynomial Gj ∈ Vﬁn[X,Y,Z] of dG dG dG degree 2, with G (P ) = 1, and j (P ) = j (P ) = j (P ) = G (P ) = 0. j 1 dX j dY j dZ j j j 2 2 2 Proof of Claim. Write Gj = Z + aX + bY + cZX + dZY + eXY . Since Gj is

homogeneous, the vanishing of its partial derivatives at Pj will immediately give Gj(Pj) = 0, so we may ignore this condition. The conditions of the claim give a systems of equations

over Vﬁn  a        2Xj 0 Zj 0 Yj  b  0    0 2Yj 0 Zj Xj  c  =  0          0 0 Xj Yj 0  d  −2Zj e Chapter 4. Nodal Curves in P2 73

If either Xj or Yj is a unit modulo p, then the matrix on the left has rank 3 modulo p

(since p 6= 2), so there exists a solution in Vﬁn. Otherwise, Pj ≡ P1 mod p, which is impossible since P j and P 1 are distinct points, proving the claim.

With Gj given from the claim for each j 6= 1, deﬁne a homogeneous polynomial of degree d − 1 m d−2m+1 Y G = Z Gj. j=2 Then the polynomial

1 ∂G ∂G 1 F := Z − (P )X − (P )Y G = Zd + ... 1,Z d ∂X 1 ∂Y 1 d

satisﬁes the required conditions, since clearly its partial derivatives vanish at Pj for each j 6= 1, and

∂F 1,Z (P ) = 1 ∂Z 1 ∂F 1 ∂G ∂G 1,Z (P ) = − (P )G(P ) + (P ) = 0 ∂X 1 d ∂X 1 1 ∂X 1 ∂F 1 ∂G ∂G 1,Z (P ) = − (P )G(P ) + (P ) = 0. ∂Y 1 d ∂Y 1 1 ∂Y 1

Deﬁning F1,X and F1,Y to be XG and YG, respectively, it is immediate that these polynomials also satisfy the required criteria.

If f is an equisingular lift of f, then f defines a relative reduced normal crossings 2 2 divisor C ⊂ PV . Let U denote the complement of C in PV , so that the special fibre of U can be identified with Uk. Then from Theorem (2.3.11), there is an isomorphism

i ∼ i Hrig(Uk) = HdR(UK ) which is natural in the sense that the de Rham cohomology inherits a Frobenius automorphism. Chapter 4. Nodal Curves in P2 74

4.2.2 The Zeta Function of a Nodal Curve

k 2 To calculate the zeta function of a -curve Ck in Pk , it suﬃces to calculate the zeta function of its aﬃne complement Uk, since

2 1 Z(Uk/k; T )Z(Ck/k; T ) = Z( ; T ) = . Pk (1 − T )(1 − qT )(1 − q2T )

Let A denote the coordinate ring of Uk, and let Na(A) denote the number of Fqa - homomorphisms A → Fqa . Then #Uk/Fqa , the number of Fqa -rational points on Uk is equal to Na(A). The Lefschetz trace formula reads [56, Theorem 4.1]

2 X i 2a −a i Na(A) = (−1) Tr(q F∗ |HMW(A/K)) i=0 2a 2a −a 1 2a −a 2 = q − Tr(q F∗ |HMW(A/K)) + Tr(q F∗ |HMW(A/K)).

We therefore have

∞ ! X T l Z(Uk/k; T ) = exp #Uk/ l Fq l l=1 ∞ ! X T a = exp (q2a − Tr(q2aF −a|H1 (A/K)) + Tr(q2aF −a|H2 (A/K))) . ∗ MW ∗ MW a a=1

By Lemma (1.0.3) we can write

∞ ! X T a exp Tr(q2aF −a|Hi (A/K)) = det(1 − q2aF −1T |Hi (A/K))−1 ∗ MW a ∗ MW a=1

and obtain P1(T ) k k Z(U / ; T ) = 2 (1 − q T )P2(T )

2 −1 i where Pi(T ) := det(1 − q F∗ T |HMW(A/K)).

Suppose now that Ck is a reduced, irreducible, nodal curve of degree d with m singu-

larities. We then have by Proposition (4.1.4) that P1 = 1 and therefore we can write

P (T ) Z(Ck/k; T ) = (1 − T )(1 − qT )

2 −1 2 where P (T ) = det(1 − q F∗ T |HMW(A/K)) ∈ 1 + Z[T ] is a polynomial of degree (d − Chapter 4. Nodal Curves in P2 75

1)(d − 2) − m.

(d − 1)(d − 2) Now the normalization Cek is a smooth, projective curve over k of genus g = − 2 m, and its zeta function has the form

Pe(t) Z(Ck/k; t) = (1 − t)(1 − qt)

2 2g where Pe(T ) ∈ Z[T ] is a polynomial of the form c0 + c1T + c2T + ··· + c2gT whose roots −1/2 α1, ..., α2g satisfy αjαg+j = 1/q for j = 1, ..., g, each αj has complex absolute value q , g−i and c2g−i = q ci (see Section (3.5)).

Proposition 4.2.6. P (T ) = Pe(T )h(T ) where h(T ) ∈ Z[T ] is a polynomial of degree m of the form k Y ri h(t) = (1 + λiT ) i=1

where r1, ..., rk are positive integers and λi ∈ {1, −1}.

Proof. Suppose Ck/Fq has a node at a point Q, and r is the smallest integer such that Q is defined over Fqr . Then Q has r conjugates lying in Ck/Fqr by applying the q- power Frobenius map to the coordinates of Q. Suppose Ck has no other singularities. There is a Zariski-open set around Q such that C satisfies a polynomial equation 2 2 Ax + Bxy + Cy + g(x, y) with coefficients in Fqr , and where Q corresponds to the point 2 (0, 0). Let δ ∈ F q be a square root of B − 4AC.

case 1. δ ∈ Fqr In this case, the tangent directions at Q (and all its conjugates) are deﬁned over Fqr , and each conjugate blows up to two points in Cfk/Fqr . We then have ( #Ck/Fqs + r, r | s #Cfk/Fqs = #Ck/Fqs , r - s

Hence,

∞ rk ! Z(Cfk/k; T ) X T = exp r Z(Ck/k; t) rk k=1 1 = 1 − T r Chapter 4. Nodal Curves in P2 76

case 2. δ∈ / Fqr In this case there is no point lying above any of the conjugates in Cek/Fqr . Since δ satisﬁes a quadratic equation over Fqr , it follows that δ ∈ Fq2r , and we have  #Ck/ s + r, 2r | s  Fq #Cfk/Fqs = #Ck/Fqs − r, r | s, 2r - s  #Ck/Fqs , r - s

It then follows that

∞ rk ! Z(Cfk/k; T ) X T = exp r (−1)k Z(Ck/k; T ) rk k=1 1 = 1 + T r

The case of other nodes deﬁned over other extensions of Fq follows immediately.

In light of the proof of Proposition (4.2.6), finding the polynomial h(T ) is computa- tionally easy. Then, in order to calculate P˜(T ), one needs only to calculate the first g + 1 coefficients of P (T )h(T ) via the following algorithm. Put

2g+m P (T ) = 1 + a1T + ··· + a2g+mT m h(T ) = 1 + b1T + ··· + bmT .

˜ 2g If we deﬁne bi = 0 for i > m, then the coeﬃcients c1, ..., cg of P (T ) = 1+c1T +···+c2gT can be computed from the linear system

c1 = a1 − b1

c2 = a2 − c1b1 − b2 . .

cg = ag − cg−1b1 − cg−1b2 − · · · − bk.

˜ g−i One can then calculate the other coeﬃcients of P (T ) through the equality c2g−i = q ci

from the functional equation for C˜k. Chapter 4. Nodal Curves in P2 77 4.3 A Crystalline Lattice of the Aﬃne Complement

2 Suppose that C is a relative reduced normal crossings divisor on PV . That is, étale-locally 2 on PV , C can be identified with an open subset of the coordinate axes. Then the fibres CK and Ck are nodal curves lying in the projective plane. Let Σ ⊂ C denote the set of singular V -valued points of C, and let m = #Σ. Let ρ : Ce → C be the normalization of C, so that Ce is a smooth V -curve with two points lying above each point in Σ. We 2 2 2 2 will also define U = PV \ C. Let i : C → PV , i2 :Σ → PV and j : U → PV denote the inclusion maps, and put i1 = i ◦ ρ.

2 2 • Associated to the pairs (PV ,C) and (PK ,CK ) are the log-de-Rham complexes Ω( 2 ,C) PV • and Ω 2 with a natural isomorphism of hypercohomology (PK ,CK )

i 2 ∼ i 2 HdR((PV ,C)/V ) ⊗V K = HdR((PK ,CK )/K)

• • induced by the map Ω( 2 ,C) ⊗V K → Ω( 2 ,C ). PV PK K

• • From [18, Corollary 2.2.6] the inclusion Ω 2 → j∗ΩU induces an isomorphism (PK ,CK ) K of K-vector spaces i 2 i HdR((PK ,CK )/K) → HdR(UK ),

i 2 i therefore we can view HdR((PV ,C)/V ) as a V -lattice inside HdR(UK ).

i From Theorem (2.3.11), HdR(UK ) inherits a Frobenius automorphism. From Propo- i 2 sition (2.3.12) combined with Theorem (2.3.10) it follows that HdR((PV ,C)/V ) is a Frobenius-equivariant lattice.

• We now let (X,Z) be any smooth pair. For a positive integer k, deﬁne Ω(X,Z)(kZ) = • Ω(X,Z) ⊗OX OX (kZ), where OX (kZ) is the sheaf on X of meromorphic functions with poles i 2 of order k along Z. Our goal now is to use Proposition (4.3.2) to compute HdR((PV ,C)/V ) using diﬀerential forms in Ω• (kC). To aid in the proof of Proposition (4.3.2) we (P2,C) introduce a convenient lemma.

Lemma 4.3.1. Let R be a ring, n a nonnegative integer, and let M be an R-module. Let M =: M n ⊃ M n−1 ⊃ · · · M 0 ⊃ M −1 = {0} be a decreasing ﬁltration of sub-R-modules, and for 0 ≤ i ≤ n let M (i) := M i/M i−1 be the i-th part of the graded module. Suppose (i) ai ∈ R is a element such that aiM = 0. Then a0a1 ··· anM = 0.

(n) Proof. Let x be an element of M, and let x be its image in M . Then anx = 0, Chapter 4. Nodal Curves in P2 78

n−1 n−2 so anx ∈ M . Similarly, an−1anx ∈ M , and continuing in this manner one has

a0a1 ··· anx = 0. Proposition 4.3.2. Let (X,Z) be a smooth pair of relative dimension n over an aﬃne scheme S = Spec(R), where R is a ring with characteristic 0. Suppose j ≤ n is a positive i integer, and let k be a positive integer such that the sheaves Ω(X,Z)(kZ) are acyclic for all 0 ≤ i ≤ j. Then i) if j = n, there is a map

Γ(X, Ωn (kZ)) n (X,Z)/S HdR((X,Z)/S) → n−1 d(Γ(X, Ω(X,Z)/S(kZ)))

whose kernel is killed upon multiplication by lcm(1, ..., k)2n−1 and whose cokernel is killed upon multiplication by lcm(1, ..., k)2n.

ii) if j < n, there is a map

ker(d : Γ(X, Ωj (kZ)) → Γ(X, Ωj+1 (kZ))) j (X,Z)/S (X,Z)/S HdR((X,Z)/S) → j−1 dΓ(X, Ω(X,Z)/S(kZ))

whose kernel is killed upon multiplication by lcm(1, ..., k)2j−1 and whose cokernel is killed upon multiplication by lcm(1, ..., k)2j+1.

n Proof. From Remark (2.2.14), since S is aﬃne one can compute HdR((X,Z)/S) as the n • • hypercohomology H (X, Ω(X,Z)/S) of the complex of sheaves of OX -modules Ω(X,Z)/S. Deﬁne a complex of sheaves Q• such that the sequence

• • • 0 → Ω(X,Z)/S → Ω(X,Z)/S(kZ) → Q → 0 (4.4)

is exact. This induces a long exact sequence of homology sheaves

i • φi i • i • · · · → H (Ω(X,Z)/S) −→ H (Ω(X,Z)/S(kZ)) → H (Q ) i+1 • φi+1 i+1 • → H (Ω(X,Z)/S) −−→ H (Ω(X,Z)/S(kZ)) → · · ·

from which we obtain the short exact sequence

i • 0 → cokφi → H (Q ) → ker φi+1 → 0.

By Theorem (2.2.20), the sheaves cokφi and ker φi+1 are killed by lcm(1, ..., k). Note n+1 ´ that since Ω(X,Z)/S = 0 we therefore have ker φn+1 = 0. Etale-locally on X one can write Chapter 4. Nodal Curves in P2 79

OX/S = R[x1, ..., xn], and OX/S(kZ) ⊂ R[x1, ..., xn, 1/x1, ..., 1/xn], so that for a section

s ∈ OX/S(kZ), ds = 0 implies s ∈ R since R has characteristic 0. It follows that the 0 • 0 • map φ0 : H (Ω(X,Z)/S) → H (Ω(X,Z)/S(kZ)) is locally equal to the identity R → R, and therefore cokφ0 = 0. From the short exact sequence above, cokφi and ker φi+1 together form a graded ﬁltration of Hi(Q•), and it follows from Lemma (4.3.1) that Hi(Q•) is killed by lcm(1, ..., k)2 for 1 ≤ i ≤ n − 1, and by lcm(1, ..., k) for i = 0 or n.

From [57, Remark 2.1.6 (i)], there exists a spectral sequence from sheaf homology converging to hypercohomology with

a,b a b • a+b • E2 = H (X, H (Q )) ⇒ H (Q ).

a,b a,b We then have that E2 , and consequently E∞ , is killed by lcm(1, ..., k) if b is equal 2 a,b to 0 or n, and by lcm(1, ..., k) for any other b. For a + b = j, the modules E∞ are graded pieces of a ﬁltration of the j-th hypercohomology group. Using Lemma (4.3.1) and taking into account cases where b is 0 or n, we have that Hn(Q•) is killed by lcm(1, ..., k)2(n−1)+2 = lcm(1, ..., k)2n, and Hi(Q•) is killed by lcm(1, ..., k)2i+1 for 0 ≤ i < n.

From Equation (4.4) one obtains a long exact sequence of hypercohomology

i−1 • i • ψi i • i • · · · → H (Q ) → H (Ω(X,Z)/S) −→ H (Ω(X,Z)/S(kZ)) → H (Q ) → · · ·

i • i−1 • It follows that cok ψi is a submodule of H (Q ) and H (Q ) maps surjectively onto 2i−1 ker ψi. Putting everything together we have ker ψ0 = 0, ker ψi is killed by lcm(1, ..., k) 2i+1 for 1 ≤ i ≤ n, cok ψi is killed by lcm(1, ..., k) for 0 ≤ j < n, and cok ψn is killed by lcm(1, ..., k)2n.

• One now can compute the hypercohomology of Ω(X,Z)/S(kZ) from an diﬀerent spectral sequence (see [57, Remark 2.1.6 (ii)]) in which

a,b b a a+b • E1 = H (X, Ω(X,Z)/S(kZ)) ⇒ H (Ω(X,Z)/S(kZ)).

a a,b By assumption Ω(X,Z)/S(kZ) is acyclic for all a ≤ j, so the terms E1 for a + b ≤ j are all zero except for when b = 0. It follows that

a−r,b+r−1 a,b a+r,b−r+1 Er → Er → Er Chapter 4. Nodal Curves in P2 80

are all zero maps for r ≥ 2 and a + b ≤ j, therefore the terms below the j-th diagonal

of the spectral sequence degenerate at E2. Since there is only one nonzero term in each diagonal, it computes the entire hypercohomology group (rather than a graded part of it). We then have

j,0 j+1,0 j • j,0 ker(E1 → E1 ) H (Ω(X,Z)/S(kZ)) = E2 = j−1,0 j,0 im(E1 → E1 ) j j+1 ker(d : Γ(X, Ω(X,Z)/S) → Γ(X, Ω(X,Z)/S)) = j−1 . dΓ(X, Ω(X,Z)/S)

n+1,0 n,0 This proves the cases where j < n. For j = n, we have E1 = 0, so ker(E1 → n+1,0 n,0 n E1 ) = E1 = Γ(X, Ω(X,Z)/S) which completes the proof.

By Serre vanishing, there exists some integer k satisfying the condition of Proposition (4.3.2). The next several pages will be devoted to ﬁnding such an integer.

Let e and k be integers with k ≥ 0. For a sheaf of OP2 -modules F, we denote the k k ∗ 2 2 Serre twist of F by F(e) := F ⊗O 2 O (e). We will also set Ω (e) = Ω ⊗O i1(O (e)) P P Ce Ce Ce P and Ωk (0) = Ωk . We have the following two useful lemmas. Ce Ce Lemma 4.3.3.

i 2 i) If e ≥ −2 and i > 0, then H ( , O 2 (e)) = 0. PV PV

i 2 k i 2 k ii) If i, e, k > 0, then H (PV , Ω 2 (e)) = 0. If i, e, k ≥ 0, then H (PV , Ω 2 (e)) is a free PV PV V -module.

Proof. The cases where i > 0 are a standard calculation, see for instance [4, Theorem k III.5.1] for part (i) and [18, Proposition 3.1.2] for part (ii). Let F = Ω 2 (e), k = 0, 1 or 2. PV i 2 Then H (PV , F) is a finitely generated module over the local ring V , so to show it is a free module is equivalent to showing it is flat over V . One can use Grothendieck’s criterion i 2 for cohomological flatness (EGA III.7.8.4) which gives that H (PV , F) is flat if and only i 2 if the dimension of H (Ps, Fs) is independent of s ∈ Spec(V ). By (EGA III.7.9.4), the

Euler characteristic of Fs is independent of s. For e > 0, all cohomology groups vanish 0 2 except for the zeroth, so the Euler characteristic is simply dim H (Ps, Fs), which proves 0 2 0 2 H (Ps, Fs) is ﬂat. For e = 0, from [18, Proposition 3.1.2] we get that H (PV , F) is free of rank 1 if k = 0 and vanishes for k > 0.

Lemma 4.3.4. Chapter 4. Nodal Curves in P2 81

i) If i > 0 and e > d − 3, then Hi(C, O (e)) = 0. For all i ≥ 0, if e > d − 3 or e = 0 e Ce then Hi(C, O (e)) is free. e Ce ii) If i, e > 0, then Hi(C, Ω1 (e)) = 0. If i, e ≥ 0, then Hi(C, Ω1 (e)) is free. e Ce e Ce Proof. Since Ce is one dimensional, we only need to treat the cases i = 0 and i = 1. By the projection formulas, [4, II, Exercise 5.1 and III, Exercise 8.2], H1(C, O (e)) ∼ e Ce = H1( 2 , i (O (e))) ∼ H1( 2 , (i O )(e)). The map i] : O → ρ O is injective with PV 1∗ Ce = PV 1∗ Ce C ∗ Ce cokernel supported on Σ which we will denote MΣ. Thus there is an exact sequence of sheaves 0 → i O (e) → i O (e) → M → 0 ∗ C 1∗ Ce Σ which gives rise to a long exact sequence in cohomology, part of which includes

H1( 2 , i O (e)) → H1( 2 , i O (e)) → H1( 2 ,M ) = 0. PV ∗ C PV 1∗ Ce PV Σ

Therefore, H1( 2 , i O (e)) = 0 is enough to ensure that H1(C, O (e)) vanishes as well. PV ∗ C e Ce ∼ Using the fact that IC = O 2 (−C) = O 2 (−d) we have an exact sequence PV PV

0 → O 2 (e − d) → O 2 (e) → i∗OC (e) → 0. PV PV

1 2 From the induced long exact sequence in cohomology, it follows that H (PV , i∗OC (e)) = 0 1 2 2 2 if H ( , O 2 (e)) = H ( , O 2 (e − d)) = 0, which is satisﬁed for e ≥ d − 2 by Lemma PV PV PV PV (4.3.3). This proves the ﬁrst statement of part (i).

Regarding the ﬁrst statement of part (ii), Serre duality gives an isomorphism

H1(C, Ω1 (e)) ∼= H0(C, Ω1 (e)−1 ⊗ Ω1 )∗ e Ce e Ce Ce 0 ∗ ∗ = H (C,e i1OP2 (−e))

∗ which vanishes if e > 0 since i1OP2 (1)) is an ample invertible sheaf.

Let F = O (e) or Ω1 (e). Just as in the proof of Lemma (4.3.3), Hi(C, F) is a free Ce Ce e i V -module if dim H (Ces, Fs) is a constant function of s ∈ Spec(V ). As before, the Euler 0 characteristic of Fs is independent of s, and equal to dim H (Ces, Fs) for e ≥ d − 2 if F = O (e) and for e > 0 if F = Ωk (e). Therefore H0(C, F) is free in these cases by Ce Ce e Grothendieck’s criterion.

It remains to show that Hi(C, O ) and Hi(C, Ω1 ) are free. This result follows again e Ce e Ce Chapter 4. Nodal Curves in P2 82

0 1 from Grothendieck’s criterion, that the Euler characteristic dim H (Ces, Fs)−dim H (Ces, Fs) for these sheaves is independent of s ∈ Spec( ), and the fact that dim H0(C , O ) = 1 V es Ces 0 1 (d − 1)(d − 2) and dim H (Ces, Ω ) = g = − m. Ces 2 Theorem 4.3.5.

i 2 1 i) If i > 0 and e > max{0, d − 3}, then H (PV , Ω( 2 ,C)(e)) = 0. If e > max{0, d − 3} PV 0 2 1 or e = 0 then H (PV , Ω( 2 ,C)(e)) is a free V -module. PV i 2 2 0 2 2 ii) If i, e > 0, then H (PV , Ω( 2 ,C)(e)) = 0. If e ≥ 0, then H (PV , Ω( 2 ,C)(e)) is free. PV PV • Proof. Let W = W1(Ω( 2 ,C)). By remark (2.2.17), there exist short exact sequences of PV 2 PV -modules

• • •−1 0 → Ω 2 → W → i1∗Ω → 0 (4.5) PV Ce • • •−2 0 → W → Ω( 2 ,C) → i2∗ΩΣ → 0. (4.6) PV

For an integer e, we can tensor both sequences with O 2 (e) and take the corresponding PV long exact sequence in cohomology, obtaining two sequences

i 2 k i 2 k i 2 k−2 · · · → H ( , (e)) → H ( , Ω 2 (e)) → H ( , i Ω (e)) → · · · (4.7) P W P (P ,C) P 2∗ Σ i 2 k i 2 k i 2 k−1 · · · → H ( , Ω 2 (e)) → H ( , W (e)) → H ( , i1∗Ω (e)) → · · · (4.8) P P P P Ce

k−2 The sheaf i2∗ΩΣ (e) is supported only on Σ, therefore it is acyclic and its zeroth cohomology group is free. Considering sequence (4.7), we have that H2( 2, Ωk (e)) P (P2,C) is isomorphic to H2( 2, k(e)), and H1( 2, Ωk (e)) is isomorphic to a quotient of P W P (P2,C) 1 H (P2, W k(e)).

Suppose i, e > 0. From Lemma (4.3.3) we have Hi( 2, Ωk (e)) = 0, which implies P P2 i 2 k ∼ i 2 k−1 ∼ i k−1 H ( , W (e)) = H ( , i1∗Ω (e)) = H (C,e Ω (e)) from the exact sequence (4.8) and P P Ce Ce the projection formula [4, Ch. III, Ex. 8.2]. By Lemma (4.3.4), this is zero if k = 2, and also for k = 1 provided e > d − 3.

It remains to verify the freeness conditions hold for i = 0. For the cases involving e 6= 0, it follows as in previous proofs using Grothendieck’s criterion for cohomological ﬂatness, and the fact that the Euler characteristic is equal to the dimension of the zeroth cohomology group, which is the same for both ﬁbres over Spec(V ). For e = 0, sequence (4.8) begins 0 2 k 0 2 k 0 k−1 0 → H ( , Ω 2 ) → H ( , W ) → H (C,e Ω ) → · · · P P P Ce Chapter 4. Nodal Curves in P2 83

0 2 k 0 k−1 By Lemma (4.3.3) and Lemma (4.3.4), H ( , Ω 2 ) and H (C,e Ω ) are free V -modules, P P Ce 0 0 which proves that H (P2, W k) is free. For if x ∈ H (P2, W k) is a torsion element, then its 0 k−1 0 2 k image in H (C,e Ω ) is zero. By exactness, x lives in the free submodule H ( , Ω 2 ), so Ce P P x = 0. By exactness of the sequence

0 2 k 0 2 k 0 k−2 0 → H ( , ) → H ( , Ω 2 ) → H (Σ, Ω (e)) → · · · P W P (P ,C) Σ

the same reasoning shows that H0( 2, Ωk ) is free. The last statement of the theorem P (P2,C) ∼ follows from the fact that OP2 (k0C) = OP2 (k0d). 2 Corollary 4.3.6. For the pair (PV ,C), the conditions of Proposition (4.3.2) are satisﬁed when k ≥ 1. Proof. This is immediate from the acyclicity results of Lemma (4.3.3) part (i) and The-

orem (4.3.5) parts (i) and (ii), and the fact that O 2 (C) = O 2 (d). PV PV 2 2 2 Proposition 4.3.7. Let H denote the image of HdR((P ,C)/V ) in HdR(UK /K), and 2 M let Vs denote the V -span in H (UK /K) of elements of the form Ω where M is a dR f s monomial of degree sd − 3. Then H = V2, and for s > 2,

4blogp(s−1)c p Vs ⊂ H ⊂ Vs

• • Proof. By deﬁnition, the complex Ω( 2 ,C) is the subcomplex of j∗ΩU consisting of ele- PV ments ω of pole order 1 along C such that dω also has pole order 1 along C. More precisely, k k k+1 Ω( 2 ,C) is the subsheaf of Ω 2 (C) consisting of diﬀerentials ω such that dω ∈ Ω 2 (2C) PV PV PV lies in the image of the natural inclusion

k+1 k+1 Ω 2 (C) → Ω 2 (2C). PV PV

2 2 2 Since dω = 0 for any ω ∈ Ω 2 (C), we therefore have Ω( 2 ,C) = Ω 2 (C), and similarly PV PV PV 2 2 Ω( 2 ,C)(sC) = Ω 2 ((s + 1)C). PV PV

For s ≥ 2, by Corollary (4.3.6) there is a map

2 2 Γ(PV , Ω 2 (sC)) 2 2 PV HdR((PV ,C)/V ) → 2 1 (4.9) d(Γ(PV , Ω( 2 ,C)((s − 1)C))) PV

whose kernel and cokernel are killed by lcm(1, ..., s − 1)4 = up4blogp(s−1)c for some unit u ∈ V . For s = 2 this map is therefore an isomorphism, so it follows immediately that 4blog (s−1)c H = V2 ⊂ Vs, and that the V -module Vs/H is killed by p p , giving the result. Chapter 4. Nodal Curves in P2 84

2 2 For any integer s ≥ 1 denote the sub-V -module of Γ(PV , Ω 2 ((s + 1)C)) consisting PV exact diﬀerentials derived from pole order s by Bs, that is,

s 2 1 d(g/f ) Bs := d Γ(PV , Ω 2 (sC)) = spanV { Ω: w ∈ {X,Y,Z}, g ∈ V [X,Y,Z]sd−2}. PV dw

The following is a version of the previous proposition which lends itself better to computations.

2 2 Proposition 4.3.8. Let B ⊂ Γ(PV , Ω 2 (2C)) be a set of diﬀerentials such that the image PV 2 of B in HdR(UK /K) is a V -basis for H. Let s ≥ 2 be an integer, and let r be an integer r 2 2 such that p HdR((PV ,C)/V ) is free.

2 2 Then for any ω ∈ Γ(PV , Ω 2 (sC)), there exists some ω˜ ∈ spanV (B) and dν ∈ Bs + PV −r p B2 such that p4blogp(s−1)cω =ω ˜ + dν.

Proof. From Corollary (4.3.6) there is a map

2 2 2 2 Γ(PV , Ω 2 (2C)) Γ( , Ω (sC)) PV PV P2 2 1 −→ 2 1 d(Γ(PV , Ω( 2 ,C)(C))) d(Γ(PV , Ω( 2 ,C)((s − 1)C))) PV PV

4blogp(s−1)c 2 1 whose cokernel is killed by p . From the fact that d(Γ(PV , Ω( 2 ,C)((s − 1)C))) ⊂ PV Bs, one can write p4blogp(s−1)cω = ω0 + dν0

0 2 2 0 for some ω ∈ Γ(PV , Ω 2 (2C)) and dν ∈ Bs. The kernel of the map PV

2 2 Γ(PV , Ω 2 (2C)) PV id⊗1K 2 1 −−−→ H d(Γ(PV , Ω( 2 ,C)(C))) PV is the submodule of torsion elements, therefore one can write

r 0 r 00 p ω = p ωe + dν

00 2 1 with ω ∈ spanV (B) and dν ∈ dΓ(PV , Ω( 2 ,C)(C)) ⊂ B2. e PV If (d, p) = 1, then elements of dΓ( 2, Ω1 ((s−1)C)) = dΓ( 2, Ω1 (sC))∩Γ( 2, Ω2 (sC)) P (P2,C) P P2 P P2 have a convenient interpretation in terms of syzygies of Jacobian ideal (∂f/∂X, ∂f/∂Y, ∂f/∂Z), Chapter 4. Nodal Curves in P2 85

that is, nontrivial homogeneous polynomial triplets (C1,C2,C3) such that

∂f ∂f ∂f ∂f C + C + C + C = 0. 1 ∂X 1 ∂X 2 ∂Y 3 ∂Z

For a polynomial P ∈ V [X,Y,Z], we will denote its partial derivatives with respect to X,Y , and Z by P ,P , and P . Any element ω ∈ dΓ( 2, Ω1 (sC)) can be represented X Y Z P P2 by homogeneous polynomials A1,A2, and A3 of degree sd − 2 by the formula

s s s ω = ((A1/f )X + (A2/f )Y + (A3/f )Z )Ω A + A + A A f + A f + A f = 1X 2Y 3Z − s 1 X 2 Y 3 Z Ω. f s f s+1

Therefore ω belongs to Γ( 2, Ω2 (sC)) if and only if A f + A f + A f = fB for P P2 1 X 2 Y 3 Z some homogeneous polynomial B of degree sd − 3 (or B = 0). Supposing this is the 1 case, by writing f = (Xf + Y f + Zf ), one sees there is a syzygy (C ,C ,C ) = d X Y Z 1 2 3 (A1 − XB/d, A2 − Y B/d, A3 − ZB/d) of degree sd − 2 in the partial derivatives of f such s s s that ω = ((C1/f )X + (C2/f )Y + (C3/f )Z )Ω.

Conversely, given a triple of homogeneous polynomials (C1,C2,C3) of degree sd − 2 satisfying C f + C f + C f = 0, one can deﬁne a diﬀerential ω ∈ dΓ( 2, Ω1 ((s − 1 X 2 Y 3 Z P (P2,C) 1)C)) by

s s s ω := ((C1/f )X + (C2/f )Y + (C3/f )Z )Ω C + C + C = 1X 2Y 3Z Ω. f s

This gives a one-to-one correspondence between syzygies of the Jacobian ideal of degree sd − 2 over K and elements of dΓ( 2, Ω1 ((s − 1)C)). One can in general filter the P (P2,C) • complex j∗ΩU by pole order along C, and show that, after a slight modification, the filtered complex is isomorphic to the de Rham-Koszul complex of f, whose associated

spectral sequence can be identiﬁed with the Koszul complex of the elements (fX , fY , fZ ) (see, for example, [58]).

4.4 The Matrix of Frobenius

2 Let Ck be a nodal curve with m singular points in Pk . Suppose Ck has coordinate ring A and is deﬁned by an equation f = 0 where f is a polynomial of degree d. Let f denote an equisingular lift of f with coeﬃcients in the ring V , and let C denote the corresponding Chapter 4. Nodal Curves in P2 86

2 V -scheme. Letting U = PV \ C, the coordinate ring of U is given by g A = { : s ≥ 0, g ∈ V [X,Y,Z] is homogeneous of degree ds} f s

† † † Let A denote its dagger ring, and let AK := A ⊗V K. A left inverse of the q-power • Frobenius can be deﬁned on Ω † , following [28]. AK

Deﬁne the K-linear operator ψ on K[X,Y,Z]

( a1 a2 a3 X q Y q Z q , if ai ≡ 0 (mod q) for all i ψ(Xa1 Y a2 Za3 ) = 0, otherwise

f(Xq,Y q,Zq) − f q Set ∆ := . For a positive integer s > 0, let r, l be the unique p † integers with l > 0, 0 ≤ r ≤ q − 1, such that s + r = ql. We then extend ψ to AK by the formula

g g ψ( ) = ψ( ) f s f ql−r = ψ(gf r(f σq − p∆)−l) ∞ ! X −l(−l − 1) ··· (−l − k + 1) = ψ gf r (−p∆)kf(Xq,Y q,Zq)−l−k k! k=0 ∞ ! X l + k − 1 = ψ gf r pk∆kf(Xq,Y q,Zq)−l−k k k=0 ∞ X l + k − 1 ψ(gf r∆k) = pk k f k+l k=0

dw 1 dw The operator ψ extends naturally to diﬀerentials by setting ψ( ) = for w ∈ w q w † {X,Y,Z}. In particular, for an element g ∈ AK ,

Ω 1 Ω ψ(gΩ) = ψ(gXY Z)ψ = ψ(gXY Z) , XYZ q2 XYZ which is well-deﬁned in the de Rham complex since XYZ divides ψ(AXY Z). By deﬁni- tion, ψ is a left inverse of the q-power Frobenius map, and since the Frobenius induces i an automorphism F∗ on the cohomology groups HMW(A/K), it must be that ψ induces −1 2 −1 F∗ . The matrix of q F∗ can then computed as follows. Chapter 4. Nodal Curves in P2 87

b Let B = {β } denote a basis as in Proposition (4.3.8), and write β = i Ω with i i f 2 bi ∈ V [X,Y,Z]2d−3. For each βi we can write

b XYZ Ω q2ψ(β ) = q2ψ( i )ψ( ) i f 2 XYZ f q−2b XYZ Ω = q2ψ( i ) f q q2XYZ ∞ q−2 k X ψ(f biXYZ∆ ) = pk Ω XY Zf k+1 k=0 ∞ X k = p βi,k (4.10) k=0

ψ(f q−2b XYZ∆k) where we have set β = i Ω. i,k XY Zf k+1

For the purpose of computational efficiency, it is convenient to define an operator ψp, † a left inverse of the p-power Frobenius on AK , rather than the q-power Frobenius. As a1 a2 a3 above, one defines ψp first on K[X,Y,Z] to be the semi-linear map that sends X Y Z

a1/p a2/p a3/p to X Y Z if a1, a2 and a3 are divisible by p and is zero otherwise, and satisﬁes −1 ψp(ab) = σp (a)ψp(b) for a ∈ K, b ∈ K[X,Y,Z], and where σ : K → K denotes the † p-power Frobenius. Analogous to the above deﬁnitions, we then extend ψp to AK , and

then to diﬀerentials, replacing q by p where appropriate. One can then compute ψ(βi) a as ψp (βi), which avoids having to work with polynomials whose degree is a multiple of q.

For each βi,k, one performs a reduction of poles to write it as a linear combination of

elements of B plus an exact diﬀerential. More precisely, let Wk denote the V -subspace of Γ( 2, Ω2 ((k + 2)C)) spanned by the β ’s via the inclusion P P i

2 2 2 2 Γ( , Ω 2 (2C)) → Γ( , Ω 2 ((k + 2)C)) P P P P b b f k i Ω 7→ i Ω f 2 f k+2

By Proposition (4.3.8), if k − 4blogp kc ≥ 0 we have

k k−4blogp kc ˜ p βi,k = p βi,k + dν

˜ k−r−4blogp kc for some βi,k ∈ Wk and dν ∈ p Bk+1. We then have the following proposition. Chapter 4. Nodal Curves in P2 88

2 Proposition 4.4.1. If p ≥ 5, then for each i, q ψ(βi) is cohomologically equivalent to P 0 a V -linear combination ai,jβj. Moreover, let N ≥ 1 be an integer, and N be the 0 smallest integer such that k − 4blogp kc ≥ N for all k ≥ N . Then if the sum of the 0 2 ﬁrst N terms of q ψ(βi) in the expansion (4.10) is equivalent to a V -linear combination P a˜i,jβj, then N a˜i,j ≡ ai,j mod p .

Proof. The ﬁrst assertion follows immediately from the previous paragraph, and the fact that n − 4blogp nc ≥ 0 for all n ≥ 1 when p ≥ 5. The second follows from the fact that 0 k k−4blogp kc ˜ N for k ≥ N , the k + 1-th term p βi,k reduces to p β ≡ 0 mod p .

k To calculate the reduction of p βi,k one can use linear algebra. For instance, one may compute a basis {γi} for the vector space Bk+1, and view βi,k as an element of

0 k−4blogp kc ˜ Wk + Bk+1 with basis B = {βi} ∪ {γi}. One then computes p βi,k as the ﬁrst (d − 1)(d − 2) − m coordinates with respect to B0.

4.5 p-Adic Precision Analysis

The numerator of the zeta function of Cek is a polynomial

2g 2g Y X i Pe(T ) = (1 − µiT ) = ciT ∈ Z[T ] i=1 i=0 (d − 1)(d − 2) where g = − m, and |µ | = q1/2. As before we have c = qg−ic , and 2 i 2g−i i

2g 2g |c | ≤ qi/2 ≤ qi/2, i i g

If P (t) is the numerator of the zeta function of Ck, by Proposition (4.2.6), we have m P (t) = Pe(t)h(t), where h(t) = b0 +b1t+···+bmt ∈ Z[t] is symmetric or antisymmetric, m of degree m, and has roots with complex absolute value 1. In particular |bi| ≤ i .

To compute P (t) from Section (4.2.2) it suffices to compute its first g + 1 coefficients. Writing 2g+m P (t) = a0 + a1t + ··· a2g+mt ,

we have, for k ≤ g, Chapter 4. Nodal Curves in P2 89

X |ak| = a˜ibj i+j=k m X m ≤ |c | j k−j j=0 m X m2g ≤ q(k−j)/2 j g j=0 m 2g X m ≤ qg/2 q−j/2 g j j=0 2g = qg/2(q−1/2 + 1)m g 2g = q(g−m)/2(q1/2 + 1)m g

so the polynomial Pe(t) can be calculated in Z/pN1 Z, where

2g N = dlog 2 q(g−m)/2(q1/2 + 1)m e. 1 p g

b Let B = { i Ω} denote a basis for H2 (U /K)) as in Proposition (4.4.1), and let M f 2 dR K 2 −1 denote the matrix of q F∗ with respect to B. Then by the statement of the proposition, M has entries in V , and can be calculated with p-adic precision pN−4blogp Nc by reducing 2 the ﬁrst N terms in the series expansions for each q ψ(βi). Therefore, choosing N2 such that

N2 − 4blogp N2c ≥ N1, one may compute a matrix M˜ satisfying

M˜ ≡ M mod pN1 ,

by reducing the ﬁrst N2 terms in the expansion of each basis element, and performing all operations in Z/pN2 Z. One then derives the numerator of the zeta function from the lowest g + 1 coeﬃcients of the polynomial det(MT˜ − I). Chapter 5

Algorithms and Complexity Estimates

In this chapter we outline both algorithms, and give asymptotic bounds for the complexity k of each, with respect to both the genus g of the curve and the index a = [ : Fp]. In both algorithms we assume that p is ﬁxed and that for two objects of bit size N, multiplication can be performed in time O(N 1+ε). Both algorithms also make use of the

p power Frobenius operator σp on V , which can be computed as follows. Write V = Zp[θ], and let P (x) be the minimal polynomial of θ. Then σ(θ) is a root of P (x), and can be calculated up to the required precision by Newton interpolation

X ← X − P 0(X)/P (X)

p initialized at θ (see [12]). After this value has been stored, the action of σp can be calculated for any element using O(a) ring operations.

5.1 Superelliptic Curve

5.1.1 Algorithm

a k k k For p an odd prime, set q = p , = Fq, and V = W ( ). Let f(x) ∈ [x] be a polynomial of degree d that has n distinct roots, m of which are k-rational multiple roots, such that the multiplicity of any root is not divisible by p. Denote by e the greatest multiplicity of any root of f. Let r > e be a prime diﬀerent from p, and set g = (n − 1)(r − 1)/2. The following algorithm outlines the procedure for computing the zeta function of the k-curve deﬁned by the equation yr = f(x).

90 Chapter 5. Algorithms and Complexity Estimates 91

Step 1. Using the procedure of section (3.1.1), lift f to a polynomial f ∈ Vﬁn[x] such that yr = f(x) deﬁnes a superelliptic curve over V .

Step 2. Letting e denote the greatest multiplicity of any linear factor of f, compute integers 2g g/2 N1 = dlogp(2 g q )e, N2 = 2blogp((d−1)(r−1)−1)c, and N3 the smallest integer

such that N3 − blogp(ep(l + r(N3 − 1)) − r)c ≥ N1 + 2N2. xih(x) N3+N2 N2 Step 3. Performing computations in V /p V , compute the ﬁrst N3 terms of p F yl for 0 < l < r, 0 ≤ i ≤ n − 2. One eﬀective way to do this, following [16] is to set 1 τ = 1/yr, ∆ = (f(x)σ − f(x)p), and S = 1 + p∆τ p, so that p

xih(x) dx F = τ (pl div r)pxpi+p−1h(x)σS−l/r . yl ypl

One can use Newton’s method to quickly compute X = S−1/r as a root of the 1 function F (X) = 1 − SXr : set X0 = 1 for the initial value, and let

1 X = X − F (X )/F 0(X ) = ((r + 1)X − SXr+1). i+1 i i i r i i

After each iteration one should use the relation f = τ −1 to rewrite the resulting expression so that the degree in x is less than d, and reduce modulo τ pN3 .

[k] ei−k Step 4. For i ∈ S2, k = 0, 1, ..., ei − 2, precompute the polynomials f (x)/(x − αi) and

Ri,k+1(x). For i, k = 0, ..., d − n − 1 compute the polynomials bi(x), ai+1(x) from Pd−n−1 k section (3.1.3) and constants λi,k such that i=0 λi,kbi(x) = x . Use the stored data and the processes of section (3.2) to reduce the degree in τ of the polynomials xih(x) found in Step 3, until one can write it as a linear combination of diﬀerentials . yl 1 Let M 0 denote the resulting matrix, and let M = M 0 (with entries viewed in pN2 1 N1+2N2 Vﬁn up to precision p . pN2

Step 5. Compute the norm matrix |M| = M σa−1 M σa−2 ··· M, and det(|M| − T ). Com-

1 N1 1 N1 pute integers in [− 2 p , 2 p ] congruent to the integers of the resulting polynomial modulo pN1 . Chapter 5. Algorithms and Complexity Estimates 92

5.1.2 Complexity Analysis

Assume p is ﬁxed. Assume also that multiplication of two objects of bit size N can be performed in time O(N 1+ε). Let R = V /pN3+N2 V . Then any element of R can be stored 2+ε 1+ε 1+ε 1+ε using a(N3 + N2) = O(a g ) bits. Note that N3 = O(a g ), and d ≤ nr = O(g). Step 1 and 2 can clearly be performed in O(1) time and space. For Step 3, one can compute S in O(a) ring operations. The expression we need to represent S−j/r is

a polynomial over R of degree less than d in x and of degree less than pN3 in τ, and 2+ε 1+ε 3+ε 3+ε thus requires O(dpN3a g ) = O(a g ) bits of storage. Only O(1) operations are required to compute S−1/r, therefore the computation of S−j/r for j = 1, ..., r − 1 is O(ra3+εg3+ε) in runtime and space. The precomputations for Step 4 can be done in comparable time to the rest of the step, and requires storing O(d − n) polynomials of degree less than d over R, as well as another (d − n)2 elements of R for a total of O(d(d − n)a2+εg1+ε) bits. One uses the reduction process of section (3.2) for each of the 2g polynomials found in the previous

step. For j = pN3, pN3 − 1, ..., 1, we repeat the following process on the polynomial associated with τ j, which is of degree less than d over the ring R. From Equation (3.11), [k+1] [k] one computes A (x) from A (x) using O(|Sk+2|d) ring operations to find the values [k+1] Ai,k+1 for i ∈ Sk+2 as well finding the quotient Q (x) using polynomial long division. Repeating this for k = 0, ..., e − 2 gives a total of O((d − n)d) ring operations until one 2 obtains a polynomial B(x) of degree less than d−n. Using the λi,k’s, another O((d−n) ) ring operations are required to compute B(x) as linear combination of the polynomials 3+ε 5+ε bi,r−j(x), i = 0, ..., d − n − 1. In total, this step is O(a g ) in complexity. The final step in computing the norm matrix |M| and its characteristic polynomial adds a time cost of O(a3+εg4+ε) (see the resource analysis of Kedlaya [12]). Overall, the complexity of this algorithm is O(a3+εg5+ε).

5.2 Nodal Plane Curve

5.2.1 Algorithm

The following is an algorithm to compute the zeta function of a projective, irreducible, k k planar -curve C of degree d that has m nodes over = Fq and no other singularities. 2 (d−1)(d−2) Suppose that C is deﬁned in Pk by an equation f = 0, let g = 2 − m be the genus, and let Uk denote its complement.

2g (g−m)/2 1/2 m Step 1. Let N1 = dlogp 2 g q (q + 1) e, and let N2 be the smallest integer such

that k − 4blogp kc ≥ N1 for all k ≥ N2. Lift f to a homogeneous polynomial Chapter 5. Algorithms and Complexity Estimates 93

N2 f ∈ Vﬁn[X,Y,Z] of degree d, such that f ≡ fe mod p , where fe is an equisingular d + 1 lift of f deﬁning a V -scheme C. Note that if m ≤ , we can take f = fe. 2

g 2 2 2 Step 2. Calculate a basis B = {βi}i=1 for the image of Hcrys((Pk , C)/V ) in HMW(Uk/K) by taking the V -span of diﬀerentials of the form AΩ/f 2 modulo the subgroup generated by diﬀerentials of the form ∂(B/f 2)/∂W Ω. Here A and B represent monomial of degrees 2d − 3 and 2d − 2, respectively, and W is X, Y , or Z.

N2 2 Step 3. Performing calculations in Z/p Z, evaluate the ﬁrst N2 terms of q ψ(βi) from (4.10).

Step 4. Using exact diﬀerentials with pole order one degree greater, reduce the terms found

in Step 3 to a linear combination of the βi’s and write these as the i-th rows of a matrix M.

Step 5. Compute det(I−TM) modulo pN1 , and calculate the numerator of the zeta function from the ﬁrst g + 1 coeﬃcients.

5.2.2 Complexity Analysis

We will assume that the parameters p and m are fixed, and that multiplication of two objects of bit size N can be performed in time O(N 1+ε). In the process of Gaussian elimination for a matrix of dimension M by N, a column with a leading 1 and zeros beneath is formed using O(NM) field operations. Therefore, since its rank is less or equal to L = min(M,N), the number of field operations required to reduce the matrix to row echelon form is O(NML). Our algorithm is implemented using matrices over the ring R = V /pN2 V . For an element b ∈ R, we allow b/p to mean any element such that when multiplied by p is equal to b. Similarly, one can perform the row operation of dividing by p when all entries of the row are divisible by p. 2+ε 1+ε Elements of the ring R can be stored using aN2 = O(a g ) bits. As in the algorithm for superelliptic curves, the p-power Frobenius σ can be calculated in R with O(a) operations by precomputation of σ(θ). Similarly, one can precompute σ−1(θ) by Newton interpolation initialized at θpa−1 , and from then on compute the action of σ−1 on R in O(a) ring operations. The space of homogeneous polynomials of degree D in variables X,Y,Z has dimension D+2 2 = (D + 1)(D + 2)/2. From Equation (4.3), Step 1 therefore requires finding a particular solution to a linear system of 3m equations in (d + 1)(d + 2)/2 variables. This Chapter 5. Algorithms and Complexity Estimates 94 can be accomplished by row reducing a matrix of dimension 3m by (d + 1)(d + 2)/2 + 1 to echelon form and solving the system, which requires O(m2d2) = O(g) ring operations. This step is therefore O(a2+εg2+ε) in time and O(a2+εg2+ε) in space. 2d−1 For Step 2, one generates the 2 -dimensional vector space of all differentials with 3d−1 pole order two as a subspace of the 2 -dimensional vector space of differentials with pole order three. Then take βi’s to be representatives of this space modulo exact differ- 2d entials, which is a subspace generated by 3 2 elements. One may perform this step by 3d−1 2d−1 2d 6 3 row reducing a 2 by 2 + 3 2 matrix, which uses O(d ) = O(g ) operations in R and thus requires time O(a2+εg4+ε).

For Step 3, one must first calculate the numerator of each βi,k from (4.10) for k = 2 2 a 0, ..., N2 − 1, i = 1, .., 2g + m. One computes q ψ(βi) = (p ψp) (βi) by iteration, in each step evaluating all the terms up to pole order N2. The numerator of a differential with pole order k is a homogeneous polynomial of degree at most N2d − 3 over the ring, 2 3 2 4+ε 4+ε therefore the storage requirement for each q ψ(βi) is O(aN2 d ) = O(a g ). In the 2 first iteration, one evaluates the first N2 terms of p ψp(βi). In each subsequent iteration, one evaluates N2 − k terms of the expansion of ψp on a differential with pole order k, for k = 1, ..., N2. This requires applying ψp to (N2 − 1)(N2 − 2)/2 polynomials of 2 2 degree at most N2dp (a space of size O(N2 d ) over the ring). Each polynomial in the argument of ψp can be computed in O(1) operations, and since ψp is a composition of −1 4 2 selecting coordinates and applying σ , this requires O(aN2 d ) ring operations. Adding 2 the polynomials of corresponding degree together requires O(N2 ) operations on a space 2 2 4+ε 5+ε 7+ε 7+ε of size O(N2 d ) over the ring. In total, this step is O(a g ) in space and O(a g ) in time. We can speed this up by roughly a factor of a at a cost of O(a3+εg2+ε) storage k by precomputing the values of σp at the coefficients of f for k = 1, ..., a − 1 and storing these in a file. In Step 4, the elements stored in Step 3 are then embedded into the space of differ-

N2d−1 entials with pole order N2, which has dimension 2 . The space of exact diﬀerentials N2d which sits inside here is generated by 3 2 elements. One then must solve a matrix N2d−1 (N2−1)d−1 N2d 6 6 system of size 2 by 2 + 3 2 , which requires O(N2 d ) ring operations. This limiting step is O(a7+εg10+ε) in time and O(a5+εg8+ε) in space. The ﬁnal step in computing the characteristic polynomial requires O(g3) ring operations for a time cost of O(a2+εg4+ε). Chapter 6

Experiments

In this section we give the results of several experiments conducted using somewhat coarse implementations of each algorithm. The programming language used is MAGMA V2.19-7 and the machine used is the Sphere computer at the University of Toronto. This has a SunFire X4200 quad-core processor with 16 GB of RAM. In general the memory usage for these particular implementations is quite high due to their being somewhat crudely written, as well as the fact that they rely on certain intrinsic functions of the programming language for solving large systems of equations which trade time for memory consumption.

6.1 Examples of Superelliptic Curves

We begin with an example of a genus 5 curve deﬁned over F510 by the equation

y3 = x2(x5 + x4 + x + t)

10 5 4 3 2 where t generates F510 over F5 and has minimal polynomial z +3z +3z +2z +4z +z+2.

In this case we set N1 = 29, N2 = 2, and N3 = 37. The highest working precision we will need is 539, and the zeta function should be calculated modulo 529. After 3 minutes and 18 seconds, and using 80MB of space, the algorithm revealed the ﬁrst six coeﬃcients of the numerator of the zeta function of the normalized curve to be 1−1253T +10171416T 2+ 10359663716T 3 + 177276031807004T 4 − 154385140679896875T 5, and one can then compute the size of its Jacobian as N = 88806455475258350626437233651431509.

As a second example, we compute the zeta function of a genus 8 curve over F320 with

95 Chapter 6. Experiments 96 planar equation y5 = x4(x − 1)4(x − t)4(x − t2)4

20 13 11 10 9 where F320 generated over F3 by t with minimal polynomial z + 2z + z + z + z + 8 5 4 3 z + 2z + 2z + 2z + z + 2. Here N1 = 67, N2 = 6 and N3 = 86. The program outputted 1 − 222102T + 26390120031T 2 − 2538440313760890T 3 + 216995406474678950790T 4 − 15512966168826218109491232T 5 +960365390614818936440153417604T 6 giving a total of

N = 1797010425151142564400620717847005400323356502183342964163 points on the Jacobian. The running time was 22 minutes and the memory usage for this example was approximately 1 GB.

3 2 4 Consider now a curve defined over F7 with affine equation given by y = x (x −x−1). This does not define a superelliptic curve by Definition (3.1.1), since (d, r) = 3. We apply 11 the same algorithm with N1 = 5, N2 = 2, N3 = 12, and compute with precision 7 the reduction of the first 12 terms of the Frobenius multiplied by 72. At the final step we divide by 49, and take the result modulo 75. The processing time was 1.82 seconds with 33 MB of memory usage, and the program outputted the polynomial

(1 − T )2(1 + T + 7T 2)(1 + 2T − 3T 2 + 14T 3 + 49T 4).

This result can be interpreted as follows. The normal model is a curve C of genus 3, with three points at inﬁnity which are permuted by the automorphism ρ :(x, y) 7→ (x, ζy), where ζ is a primitive cube root of unity. If we let Σ denote the points at inﬁnity, then the Gysin sequence (Proposition (2.4.4)) gives

1 1 0 − 0 − 0 → Hrig(C/K) → Hrig(C /K) → Hrig(Σ/K) → · · · where C0 denotes the aﬃne curve minus the points along y = 0. In this case (cf. Propo- 0 3 sition (3.1.5)) Hrig(Σ/K) = K[w]/(w − 1) is non-trivial, with ρ given as the K-linear map ρ(w) = ζw. This decomposes into eigenspaces K ⊕ Kw ⊕ Kw2, with the Frobenius 0 − 2 2 acting trivially on each, and Hrig(Σ/K) = Kw˙ ⊕ Kw˙ . We conclude that (1 − T ) is 0 − the characteristic polynomial of the Frobenius acting on Hrig(Σ/K) , and that the zeta function of Ce is (1 + T + 7T 2)(1 + 2T − 3T 2 + 14T 3 + 49T 4) . (1 − T )(1 − 7T ) Chapter 6. Experiments 97

6.2 Examples of Nodal Plane Curves

It should be mentioned that the author suspects from two observations that the algorithm for nodal curves could be improved. Firstly, the precision bound k − 4bkc ≥ N in Proposition (4.4.1) in practice seems more than necessary in ensuring correct precision of the result, and for the implementations below we instead use k − 2bkc ≥ N. This in fact is not an arbitrary choice, since it is claimed [41] that this bound is suﬃcient, however this author hesitated on one detail of the proof. One also suspects (as in the case with superelliptic curves) that there could exist a more explicit reduction method for diﬀerentials. Nevertheless, we demonstrate that this algorithm correctly computes the zeta function of two interesting cases of nodal curves.

For the first example we use is a random genus 5 quintic defined over F11 with a single node at (0, 0, 1) and defining polynomial

−2X5 + 2X4Y + 2X3Y 2 + X3YZ − 5X3Z2 − 5X2Y 3 − 2X2Y 2Z + X2YZ2 +2X2Z3 + 5XY 4 + 5XY 3Z − 5XY 2Z2 − Y 5 − Y 4Z − 5Y 3Z2 − 4Y 2Z3.

The nodal point is at (0,0,1) and the discriminant of the quadratic terms on the aﬃne

plane Z = 1 is 32 ≡ −1 mod 11 which is not a square in F11. We conclude that the numerator of the zeta function of the nodal curve is the degree 10 polynomial coming from the normalization, multiplied by the linear factor 1 + T . For this curve we can take

N1 = N2 = 6, so that calculation are performed on the ﬁrst 6 terms of the Frobenius expansion with maximal precision 116, and the ﬁnal matrix of Frobenius is calculated also with precision 116. The program outputted the polynomial

(1+T )(1−T −4T 2 +3T 3 +66T 4 −104T 5 +726T 6 +363T 7 −5324T 8 −14641T 9 +161051T 10)

which is the correct numerator of the zeta function. The running time for this example was 44 seconds and the memory usage was 321 MB.

For a second example we apply the algorithm to a curve considered by Lauder [38].

It is the genus 4 sextic with 6 nodes over F7 deﬁned by the equation

X6 − X5Y − 2X5Z + 2X4Y 2 + 7/2X4Z2 + X3Y 3 − 4X3Y 2X − 3X3Z3 +1/2X2Y 4 + 5X2Y 2Z2 − X2YZ3 + 7/2X2Z4 − 2XY 4Z − 4XY 2Z3 −2XZ5 + Y 6 + 3/2Y 4Z2 + Y 3Z3 + 3/2Y 2Z4 + Z6 = 0. Chapter 6. Experiments 98

The precision requirements from Chapter 5 are N1 = 6, N2 = 8, and therefore we are supposed to calculate the first 8 terms of the Frobenius series with maximal precision 78 and final precision 76. We can perhaps do slightly better by investigating the factor of the zeta function coming from the singular points on this curve. Two of the singularities, 2 s1 and s2, occur at Z = Y = 1, and X satisfies X + 5X + 2 = 0. Since the discriminant

of this polynomial is 25 − 8 ≡ 3 mod 7 is not a square in F7 we must pass to F72 to 2 ﬁnd roots. Letting t generate F72 over F7 with minimal polynomial x − x + 3 = 0 so 2 14 that we may designate s1 = [t : 1 : 1] and s2 = [t : 1 : 1]. Then locally around s1, the polynomial has the form

t27X2 + t13XY + t3Y 2 + higher order terms

and the discriminant of the quadratic terms is

δ = t26 − 4t30 = 5

2 which is a square in F72 . Therefore we can deduce that 1 − T is a factor of the zeta function of the curve. We can perform a similar computation to ﬁnd that the remaining

four singular points are conjugates defined over F74 , with the discriminant of the local 4 quadratic term not equal to a square in F74 . We deduce from this the factor 1 + T , and conclude that the zeta function of the curve has the form (1 − T 2 + T 4 − T 6)Q(T ), where Q(T ) ∈ Z[T ] is a polynomial of degree 8 whose roots have absolute value 7−1/2. As in 8 i/2 Section (3.5), the i-th degree coefficient of Q(T ) is bounded by i 7 , and it follows that the largest coefficient of (1 − T 2 + T 4 − T 6)Q(T ) for degree less than or equal to four is at most 3627. Hence the polynomial can be calculated with precision 7N1 where

N1 = dlog7(2 · 3627)e = 5, and one can then take N2 = 5.

Using these precision bounds, after 53 seconds and 400 MB of memory the program outputted the polynomial

(1 − T 2 + T 4 − T 6)(1 + 2T + 11T 2 + 40T 3 + 72T 4 + 280T 5 + 539T 6 + 686T 7 + 2401T 8)

which is the numerator of the zeta function of the curve.

For a third example we consider a random genus 5 curve over F72 deﬁned by the Chapter 6. Experiments 99 polynomial

t26X5 − X4Y + t27X4Z + t11X3Y 2 − 3X3YZ + t18X3Z2 + t10X2Y 3 + t36X2Y 2Z + t25X2YZ2 + t39X2Z3 + t35XY 4 + t33XY 3Z + t5XY 2Z2 + t26XYZ3 + t42Y 5 + t36Y 4Z + t31Y 3Z2 + t45Y 2Z3

2 where the minimal polynomial for t over F7 is x − x + 3. The only singular point is at [0, 0, 1], and the discriminant of the quadratic terms at Z = 1 is

δ = t52 − 4t39+45 = t28 which is a square in F72 , and hence h(T ) = 1−T . The parameters of the algorithm require the ﬁrst 11 terms of the Frobenius expansion to be computed with working precision 711 and ﬁnal precision 79. The program outputted the numerator of the zeta function of the curve as

(1 − T )(1 + 13T + 151T 2 + 1179T 3 + 8021T 4 + 59805T 5 +393029T 6 + 2830779T 7 + 17764999T 8 + 74942413T 9 + 282475249T 10).

For this example the program ran for 170 minutes and consumed approximately 8 GB of memory. Bibliography

[1] C. F. Gauss, Disquisitiones Arithmeticae. 1801. [2] B. Dwork, “On the rationality of the zeta function of an algebraic variety,” Amer- ican Journal of Mathematics, vol. 82, no. 3, pp. 631 –648, 1960. [3] P. Deligne, “La conjecture de Weil. I,” Publications Mathématiquesde l’Institut des Hautes Etudes´ Scientifiques, vol. 43, no. 1, pp. 273–307, 1974. [4] R. Hartshorne, Algebraic Geometry. New York: Springer-Verlag, 1977. [5] R. Schoof, “Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p,” Mathematics of Computation, vol. 44, no. 170, pp. 483–494, 1985. [6] J. Pila, “Frobenius Maps of Abelian Varieties and Finding Roots of Unity in Finite Fields,” Mathematics of Computation, vol. 55, no. 192, pp. 745–763, 1990. [7] L. M. Adelman and M.-D. Huang, “Counting Points on Curves and Abelian Vari- eties over Finite Fields,” J. of Symbolic Computation, vol. 32, pp. 171–189, 2001. [8] M.-D. Huang and D. Ierardi, “Counting Points on Curves over Finite Fields,” J. of Symbolic Computation, vol. 25, pp. 1–21, 1998. [9] P. Gaudry and R. Harley, “Counting Points on Hyperelliptic Curves over Finite Fields,” in Algorithmic Number Theory, ser. Lecture Notes in Computer Science, vol. 1838, Springer Berlin Heidelberg, 2000, pp. 313–332. [10] T. Satoh, “The canonical lift of an ordinary elliptic curve over a finite field and its point counting,” J. of Ramanujan Math. Soc., vol. 15, no. 4, pp. 247–270, 2000. [11] P. Monsky and G. Washnitzer, “Formal Cohomology I,” Annals of Math., vol. 88, pp. 181–217, 1968. [12] K. S. Kedlaya, “Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology,” Journal of Ramanunan Mathematical Society, vol. 16, pp. 323–338, 2001.

100 BIBLIOGRAPHY 101

[13] P. Gaudry and N. G¨urel,“Counting points in medium characteristic using Kedlaya’s algorithm.,” Experimental Mathematics, vol. 12, no. 4, pp. 395–402, 2003. [14] D. Harvey, “Kedlayas algorithm in larger characteristic,” 2007. arXiv: math/0610973. [15] J. Denef and F. Vercauteren, “An Extension of Kedlaya’s Algorithm to Hyperel- liptic Curves in Characteristic 2,” Journal of Cryptology, vol. 19, no. 1, pp. 1–25, 2006. [16] P. Gaudry and N. G¨urel,“An Extension of Kedlayas Point-Counting Algorithm to Superelliptic Curves,” in Advances in Cryptology ASIACRYPT 2001, ser. Lecture Notes in Computer Science, C. Boyd, Ed., vol. 2248, Springer Berlin Heidelberg, 2001, pp. 480–494.

[17] J. Denef and F. Vercauteren, “Counting points on Ca,b curves using MonskyWash- nitzer cohomology,” Finite Fields and Their Applications, vol. 12, no. 1, pp. 78 –102, 2006. [18] T. G. Abbott, K. S. Kedlaya, and D. Roe, “Bounding picard numbers of surfaces using p-adic cohomology,” Séminaires et Congrés, vol. 21, pp. 125–159, 2009. [19] A. G. B. Lauder, “Deformation theory and the computation of zeta functions,” Proc. London Math. Soc, vol. 3, pp. 565–602, 2004. [20] ——, “Counting solutions to equations in many variables over finite fields,” Foun- dations of Computational Mathematics, vol. 4, no. 3, pp. 221–267, 2004. [21] R. Gerkmann, “Relative rigid cohomology and point counting on families of elliptic curves,” J. Ramanujan Math. Soc., vol. 23, no. 1, pp. 1–31, 2008. [22] ——, “Relative Rigid Cohomology and Deformation of Hypersurfaces,” Interna- tional Mathematics Research Papers, 2007. [23] P. Berthelot, “Cohomologie rigide et cohomologie rigide àsupports propres, Première partie,” Prépublication, Universitéde Rennes, 1996. [24] ——, “Finitude et puretécohomologique en cohomologie rigide,” Invent. Math., vol. 128, pp. 329–377, 1997. [25] A. G. B. Lauder, “A recursive method for computing the zeta functions of varieties,” LMS J. Comput. Math., vol. 9, pp. 222–269, 2006. [26] ——, “Ranks of elliptic curves over function fields,” LMS J. Comput. Math., vol. 11, pp. 172–212, 2008. [27] G. M. Walker, “Computing zeta functions of varieties via fibration,” 2009. BIBLIOGRAPHY 102

[28] R. Kloosterman, “Point counting on singular hypersurfaces,” in Algorithmic Num- ber Theory, 8th International Symposium, ANTS-VIII Proceedings, A. van der Pooten and A. Stein, Eds., vol. 5011, Banff, Canada: Springer Verlag, 2008, pp. 327– 341. [29] P. Deligne, “Théoriede Hodge: II,” Publications Mathématiquesde l’IHES´ , vol. 40, pp. 5–57, 1971. [30] K. Koyama, “Fast RSA-type schemes based on singular cubic curves y2 + axy ≡ x3 mod n,” in Eurocrypt ’95, ser. Lec. Notes. in Comp. Sci. 1995, pp. 329–340. [31] S. Arita, S. Miura, and T. Sekiguchi, “An addition algorithm on the Jacobian varieties of curves,” J. Ramanujan Math. Soc., vol. 19, no. 4, pp. 235–251, 2004. [32] I. Déchène,“Arithmetic of Generalized Jacobians,” in Algorithmic Number Theory, ser. Lecture Notes in Computer Science, vol. 4076, Springer Berlin Heidelberg, 2006, pp. 421–435. [33] S. D. Galbraith and B. A. Smith, “Discrete logarithms in generalized Jacobians,” 2006. arXiv: math/0610073. [34] D. R. Kohel, “Constructive and destructive facets of torus-based cryptography,” preprint, 2004. [35] P. A. Griffiths, “On the periods of certain rational integrals: I,” Annals of Mathe- matics, vol. 90, pp. 496–541, 1969. [36] A. Dimca, “On the de Rham cohomology of a hypersurface complement,” American Journal of Mathematics, vol. 113, no. 4, pp. 763–771, Aug., 1991. [37] W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry, 2008 Edition, ser. Math. Lec. Note Series. W. A. Benjamin Inc, 1969. [38] A. G. B. Lauder, “Degenerations and limit Frobenius structures in rigid cohomology,” London Math. Soc. J. Comp. Math., 2011. [39] N. Jacobson, Basic Algebra: II, ser. Basic Algebra. W.H. Freeman & Company, 1989. [40] Q. Liu, Algebraic Geometry and Arithmetic Curves. Oxford University Press, 2002. [41] G. M. Walker, Explicit crystalline lattices in rigid cohomology, 2011. arXiv: 1110. 4049. [42] R. Elkik, “Solutions d’équationsàcoefficients dans un anneau hensélien,” Ann. Sci. Ecole´ Norm. Sup., vol. 6, pp. 553–603, 1973-74. BIBLIOGRAPHY 103

[43] A. Shiho, “Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology,” J. Math. Sci. Univ. Tokyo, vol. 9, pp. 1–163, 2002. [44] 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, 1994, pp. 11–50. [45] K. Kato, “Logarithmic structures of Fontaine-Illusie,” in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), John Hopkins Univ. Press, 1989, pp. 191–224. [46] R. Hartshorne, “On the de Rham cohomology of algebraic varieties,” Publications mathématiquesde l’I.H.E.S.´ , vol. 45, pp. 5–99, 1975. [47] F. Baldassarri, M. Cailotto, and L. Fiorot, “Poincaréduality for algebraic de Rham cohomology,” manuscripta mathematica, vol. 114, no. 1, pp. 61–116, 2004. [48] P. Berthelot, “Géométrierigide et cohomologie des variétésalgébriquesde car- actéristique p,” Mém.de la Soc. Math. de France, 2nd ser., vol. 23, pp. 7–32, 1986. [49] N. Tsuzuki, “On the Gysin isomorphism of rigid cohomology,” Hiroshima Math. J., vol. 29, pp. 479–527, 1999. [50] T. van den Bogaart, About the choice of basis in Kedlaya’s algorithm, 2008. arXiv: 0809.1243. [51] N. Bourbaki, Elémentsde´ mathématique:Algèbre commutative, VIII and IX. Springer, 2006. [52] B. Edixhoven, “Point counting after Kedlaya,” Notes from the EIDMA-Stieltjes graduate course in Leiden, 2003. [Online]. Available: www.math.leidenuniv.nl/ ~edix/talks/2003_09_22-kedlaya-counting.pdf. [53] A. Dimca, Singularities and Topology of Hypersurfaces, ser. Universitext (1979). Springer-Verlag, 1992. [54] N. Bourbaki, Commutative Algebra, ser. Elements of Mathematics I-VII. Paris: Hermann, 1972. [55] P. Deligne and D. Mumford, “The irreducibility of the space of curves of a given genus,” Publ. Math., Inst. Hautes Etud.´ Sci., vol. 36, pp. 75–109, 1969. [56] M. van der Put, “The cohomology of Monsky and Washnitzer, introductions aux cohomologies p-adiques,” Mém.Soc. Math. France, vol. 23, pp. 33–59, 1986. [57] A. Dimca, Sheaves in Topology. Berlin: Springer, 2004. BIBLIOGRAPHY 104

[58] A. Dimca and G. Sticlaru, “Koszul complexes and pole order ﬁltrations,” 2011. arXiv: 1108.3976.