University of Groningen Twists of Genus Three Curves and Their

University of Groningen

Twists of genus three curves and their Jacobians Meagher, Stephen

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Download date: 28-09-2021 RIJKSUNIVERSITEIT GRONINGEN

Twists of genus three curves and their Jacobians

Proefschrift

ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magniﬁcus, dr. F. Zwarts, in het openbaar te verdedigen op vrijdag 26 september 2008 om 14.45 uur

door

Stephen James Meagher

geboren op 24 september 1980 te Sydney, Australi¨e Promotor Prof. dr. J. Top

Beoordelingscommissie Prof. dr. C-L. Chai Prof. dr. G. B. M. van der Geer Prof. dr. D. Kohel Contents

Introduction 1 Twists of genus three Jacobians 3 1. Introduction 3 2. Definitions and conventions 5 3. Mumford’s theta group, existence of a principal line bundle and χ18 13 4. Existence and properties of the ∆i 20 5. Theorem 1e 25 6. Theorem 1f 27 Twists of genus three curves over finite fields 29 1. Forms and automorphisms 29 2. The points on a form 34 3. The zoo of symmetric quartics 35 4. Points on Fermat 52 5. Points on Klein 54 6. Moduli theoretic interpretation of twists of C and X 54 Bibliography 57 Samenvatting 61 Acknowledgements 63

iii

Introduction

The questions treated in this thesis are motivated by the following simple question: 1 let C be a genus g curve over a ﬁnite ﬁeld Fq. What are the possible values of #C(Fq)?

One fundamental fact is the Hasse-Weil-Serre upper and lower bounds

q +1 g[2√q] #C(F ) q +1+ g[2√q]. − ≤ q ≤ So our question can be rephrased as: which numbers in the Hasse-Weil interval

HW(g, q) := N [q +1 g[2√q], q +1+ g[2√q]] ∩ − can occur as the number of rational points on a genus g curve C over Fq?

This question can be approached from at least two points of view: ﬁx the genus and let q vary or ﬁx q and let the genus of the curve vary. The second perspective has been taken up by many other authors, and we do not consider it here.

The first perspective goes like this: treat genus 0 first, then genus 1, then genus 2, then genus 3, etc. Genus 0 is trivial2 as HW(0, q) = q +1 . Genus 1 was completely understood by the work of Hasse and Deuring:{ the following} numbers can occur subject to the given conditions([58]) N HW(1, q) if p ∤ N q 1 ∈ − − N = q +1 √q if 3 ∤ p 1 and √q Z, ± − ∈ N = q +1 2√q if √q Z, ± ∈ N = q +1 √pq if p =2 or p = 3 and √pq Z, N = q +1± if4 ∤ p 1. ∈ − Genus 2 was treated by Jean-Pierre Serre [48, 49, 50] in the 1980’s. Then we have genus three, which is the first genus where things do not go so smoothly. One question which brings focus to this problem is the following:

Does there exist a constant c so that for every integral prime power q there is a genus three curve C so that

#C(F ) > q +1+3[2√q] c? q − In words this question says: is the Hasse-Weil upperbound good within a constant not depending on the ﬁeld Fq?

1Always assumed to be smooth, projective and connected. 2In fact this shows that any twist of P1 has a rational point, hence a very ample divisor of Fq degree 1 and is therefore isomorphic to P1 . Fq

1 2 INTRODUCTION

Jean-Pierre Serre conjectured that this question has an aﬃrmative3 answer!

Indeed, he more or less sketched a strategy to answer this question, which runs like this: i) For each q construct an indecomposable principally polarised Abelian 3-fold (A, a) over Fq so that

tr(Fr ) > 3[2√q] c. − A − ii) Show that (A, a) as in i) can be chosen to be the Jacobian of a genus three curve C.

Then the Weil trace formula implies that

#C(F )= q +1 tr(Fr ) > q +1+3[2√q] c. q − A − The ﬁrst part of this strategy is considerably easier than the second. For example one can easily choose c = 24 and take A to be (2, 2, 2) isogenous to the third power of an ordinary elliptic curve. If one wants a better constant, this requires some more work. In fact Auer-Top [1] obtain the constant c = 21 and Lauter-Serre [24] obtain the constant c = 3.

The second part is more diﬃcult. However, the Torelli theorem for genus three curves tells us that if (A, a) is not a Jacobian then its [ 1] twist is a Jacobian, and this implies that in any case there is a genus three curve− C so that either

#C(F )= q +1 tr(Fr ) q − A or

#C(Fq)= q + 1 + tr(FrA). We therefore conclude, following Auer-Top [1] and Lauter-Serre [24], that for each q either the Hasse-Weil upperbound is good within a constant or the Hasse-Weil lower bound is good within a constant.

This annoying ambiguity was noticed by Serre and in a letter to Jaap Top [47] he proposed a method for dealing with it. His suggestion is that (A, a) is a Jacobian if and only if the product of its 36 even theta nulls is a square. It is this suggestion which we take up in Chapter 1.

During the last 4 years the author made several stabs at the above conjecture by considering special families of smooth quartics. This philosophy, that exotic or unusual objects should yield interesting results is the motivation of Chapter 2.

In Chapter 2 we study all the smooth quartics with many automorphisms. In particular we establish how to explicitly compute the number of points of a twist of a smooth quartic in terms of a 1-cocycle and the Frobenius of the original untwisted quartic. Highlights of this chapter are treatment of the Klein and Fermat quartics.

3He conjectured the analogous questions for genus g ≤ 8 also have aﬃrmative answers [50] pp18-19, pp71-72. The question itself is probably due to Serre, but we are uncertain about this point. Twists of genus three Jacobians

1. Introduction 1.0.1. Let k be a ﬁeld not of characteristic 2. Let (A, a) be a principally polarised Abelian variety of dimension three over k. 1.0.2. If a is indecomposable and k is algebraically closed there is a smooth genus three curve C and a k isomorphism of principally polarised Abelian varieties1

(Jac(C), λΘ) ∼= (A, a). However, if k admits quadratic extensions, then such a C need not exist (see [45] and [46]). 1.0.3. All of the notions in Theorem 1 will be explained within the ﬁrst 3 sections of this paper, with the exception of parts e) and f) which will be explained in sections 5 and 6 respectively.

Let be the ﬁne moduli scheme of principally polarised Abelian varieties with A3,1,4 a symplectic level 4 structure; it is a smooth scheme over Z[1/2, √ 1]. Let 3′ ,1,4 be the open subscheme corresponding to those principally polarised− AbelianA varieties whose polarisation is indecomposable. We prove

Theorem 1. 1. Let (A/ 3,1,4,a) be the universal principally polarised Abelian scheme with a level 4 structure.A There is a symmetric eﬀective degree 1 polar divisor Θ on A and therefore (Θ) has a a global section s which is unique up OA Θ to multiplication by a unit of Z[1/2, √ 1]. Let χ18 be the product of the 36 even theta nulls on , i.e. − A3,1,4

χ18 := p∗sΘ, even p O where the product runs over the even 2 torsion points p : 3,1,4 A. Then χ18 36 A −→ is unique up to multiplication by an element of Z[1/2]∗ . Moreover, χ18 has the following properties:

a) χ18 is a modular form, i.e. there is an isomorphism 18 p∗ A(Θ) = ωA/ 3,1,4 O ∼ A even p O which is compatible with the classical isomorphism between analytic theta nulls and analytic modular forms (see 3.3.1 and 3.4.4, cf. Moret-Bailly [30]). b) χ18 descends to a unique Katz-Siegel modular form, which will also be denoted by χ18, of level 1 away from characteristic 2, c) Let H be the hyperelliptic locus in 3,1,4. The divisor of χ18, on 3′ ,1,4, is equal to H. A A

1This follows from the properness of Torelli morphism and the irreducibility of the moduli space of Abelian varieties of dimension 3 with an indecomposable principal polarisation(cf. 2.5.4).

3 4 TWISTS OF GENUS THREE JACOBIANS

d) Let (A/S,a) be a family (smooth and proper) of principally polarised Abelian varieties over a scheme S. Then there is a Zariski open cover Ui of S and local functions ∆i S(Ui) with the following properties: for every fibre (A ,a ) over a closed∈ O point x S with residue field k(x) x x ∈ 2 ∆ (A ,a ) k(x)∗ if and only if (A ,a ) is a Jacobian. i x x ∈ x x The divisor of ∆i(Ax,a) is equal to the hyperelliptic locus. The ∆i are unique up to a square unit of S(Ui). e) In particular, if k C, let τ Odenote the element of the Siegel upper half ⊂ an space of degree 3 corresponding to AC. Let χ18 (τ) denote the product of the 36 analytic even theta nulls at the element τ. Let ξ1, ξ2, ξ3 be a basis for the Kähler differentials of A over k. Let Ω1 denote the 3 3 matrix of 3× 3 3 integrals of the ξi with respect to the homology classes of C /(Z + Z τ) given by the lattice vectors (1, 0, 0), (0, 1, 0), (0, 0, 1) Z3 + Z3τ. Then ∈ an 54 χ18 (τ) (2π) 18 = ∆(A, a) k, (det(Ω1)) ∈ where ∆ is a function as in part d). This does not depend on the embedding k C. Thus if R k and m is a maximal ideal⊂ of R, and (B,b) is a model of (A, a) over⊂R then an 54 χ18 (τ) 2 (2π) 18 mod m (R/m) , (det(Ω1)) ∈ if and only if (B,b) (R/m) is a Jacobian of a genus three curve C over (R/m). Moreover C⊗is hyperelliptic if and only if an 54 χ18(τ) (2π) 18 mod m =0. (det(Ω1)) 2. There is a Katz-Teichmüller modular form Discr, defined for families (smooth and proper) of genus 3 curves, which is compatible with the classical discriminant of a smooth quartic. Moreover, Discr has the property: f) Discr vanishes on the hyperelliptic locus with multiplicity 1 and its square 2 Discr descends to a Katz-Siegel modular form which is equal to χ18 modulo a power of 2.

1.0.4. Remarks. 1) In a letter to Jaap Top from 2003 ([47]), Jean-Pierre Serre suggested that something like Theorem 1 should be true. In fact he more or less conjectured parts e) and f) of Theorem 1. The functions ∆i yield the mysterious square which Serre refers to at the end of the paper [49].

an 2) The vanishing of χ18 on the hyperelliptic locus is a classically known fact for the corresponding Siegel modular form on the Siegel upper half space of degree 3 (see [16]). However these classical facts are strictly complex analytic and do not apply to the scheme 3,1,4 without further techniques. Tsuyumine ([56]) also studied A an the square root of χ18 using automorphic theory and observed that its square root exists on the moduli space of curves; which is roughly half of part f) of Theorem 1. an The ﬁrst author to study χ18 seems to have been Klein (p462 [21]) and he also observed that part f) of Theorem 1 is true. He also remarks upon an analogous result for g = 4.

3) Our reliance upon level 4 structures is the reason we work away from characteristic 2. However, it should be possible to generalise Theorem 1 to characteristic 2 2. DEFINITIONS AND CONVENTIONS 5 by using Mumford’s theta group and so-called Lagrangian decompositions instead of symplectic level 4 structures.

4) Part e) of Theorem 1 is the only way we know to compute the function in part d). It might be possible to do this computation algebraically. However this would necessitate making the isomorphism of part a) explicit.

5) Ritzenthaler and Lachaud [22] have proven a special case of parts e) and f) of Theorem 1 for Jacobians which are (2, 2, 2) isogenous to a product of 3 elliptic curves; they also proved part c) in this case but for the corresponding Siegel modular form. Their proof uses explicit calculations of theta functions along with a result of Howe, Leprevost and Poonen [15] which relates the coeﬃcients of the elliptic curves to a twisting factor of the Jacobian. It is a remarkable example of how much can be achieved with explicit methods. In particular, part e) of Theorem 1 should be thought of as a generalisation of the result of Ritzenthaler and Lachaud. We would not have tried to prove part e) of Theorem 1 had we not been aware of their work.

6) To improve part f) of Theorem 1 to an exact equality it would suﬃce to give the ﬁrst few terms of a Taylor expansion of χ18 and Discr in the neighbourhood of a singular quartic.

1.0.5. The proof of Theorem 1 uses three ingredients: i) the fact that χ18 exists as a global section of a line bundle, and that its locus is the hyperelliptic locus and that is unique up to multiplication by a square unit; ii) the fact that the Torelli morphism factors as a composition of a quotient by an involution and a locally closed immersion; iii) the fact that the classical isomorphism of the line bundle of theta nulls with the line bundle of modular forms is compatible with the algebraic isomorphism of these line bundles.

We prove fact i) by using Mumford’s theory of the theta group [32, 33]; this is essentially a descent argument dressed up in the guise of group theory. Fact ii) is due to Oort and Steenbrink [37]. Fact iii) is due to Moret-Bailly [30].

From fact ii) we deduce the existence of the ∆i as in part d) of Theorem 1. From fact i) we deduce that there is a canonical, up to multiplication by a unit in Z[1/2]∗, map identifying χ18 with the ∆i on an open cover Ui. From fact iii), we deduce that we may identify the ∆i with the analytic χ18 modulo an explicit scaling factor. 1.0.6. This paper is structured in the following way: in the ﬁrst three sections we give the necessary background on moduli spaces of Abelian varieties and curves. We also explain how to construct the eﬀective divisor of Theorem 1 and prove parts a),b) and c) of Theorem 1. In section 4 we study the Torelli morphism and deduce the existence of the functions ∆i of Theorem 1 part d). In section 5 we relate the analytic χ18 to the algebraic ∆i. In section 6 we prove part f) of Theorem 1.

2. Deﬁnitions and conventions 2.1. Basic notation. All rings are commutative with identity. With the exception of this section and section 2.5, all schemes S and rings R considered will be over Z[1/2]. Thus any ring R will necessarily contain 1/2. We will write k for a ﬁeld not of characteristic 2 and we write ks for a separable closure of k.

Let X and T be S schemes, we will write XT for the ﬁbre product 6 TWISTS OF GENUS THREE JACOBIANS

X T. ×S A capital A will usually refer to an Abelian variety or an Abelian scheme over some scheme S which should be understood from the context. Likewise a capital C will refer to a smooth curve of a ﬁxed genus g, either over a ﬁeld k or a scheme S. Such families of Abelian varieties or curves will always be smooth and proper.

A curly letter or will always denote a moduli scheme2. The letter is re- served for moduliA ofM Abelian varieties and the letter for moduli of curves.A M The Jacobian of a curve C will be written as Jac(C) and its theta divisor, when it exists, as Θ. The principal polarisation induced from Θ will be written as λΘ.

The gothic letter t will refer to the Torelli morphism while h3 will denote the Siegel upper half space of degree 3.

We write Gm, µN and (Z/N)S for the group schemes over S = Spec(Z[1/2]) with Hopf algebras

1 Z[1/2,X,X− ],

Z[1/2,X]/(XN 1), −

Z[1/2][Z/NZ] 1 respectively. Let ζN be a primitive Nth root of unity and let S = Spec(Z[ N , ζN ]). For each N we ﬁx throughout this paper an isomorphism

Exp : (Z/N) (µ ) , N S −→ N S i.e., for each N we choose a primitive Nth root of unity. We moreover assume that these isomorphisms are compatible with each other in the obvious way: i.e. if ζm and ζn are the images of 1 under Expm and Expn then ζmζn is the image of 1 under Expnm.

Finally we will write PGL6(R) for the projective linear group of degree 6 with entries in R as a group and not as a group scheme. 2.2. Abelian schemes and families of curves. Let S be a scheme. An Abelian scheme is a smooth proper morphism

π : A S −→ with a section

O : S A −→ so that a) A is a group scheme and b) the geometric ﬁbres of π are connected projective varieties. This necessarily implies that the geometric ﬁbres of π are Abelian varieties and that the group law on A is commutative.

A family of genus g curves is a smooth proper morphism

2 With the exception of section 2.5.6, where we brieﬂy mention the stack Ag,1. And here it is mentioned only out of interest and not need. 2. DEFINITIONS AND CONVENTIONS 7

f : C S −→ whose geometric ﬁbres are connected genus g curves.

2.3. The Picard scheme. 2.3.1. Let π : X S be a morphism of schemes and let T be a S scheme. Consider the absolute Picard−→ functor

Pic(X)(T ) := invertible sheaves L over X / = . { T } ∼T Pullback via π yields a natural transformation of group valued functors

π∗ : Pic(S)(T ) Pic(X)(T ). −→ We deﬁne the relative Picard functor as

Pic(X/S)(T ) := Pic(X)(T )/π∗Pic(S)(T ).

Let Pic´et(X/S) denote the sheaf associated to Pic(X/S) over the ´etale site.

If π : X S is a flat morphism of schemes which is locally projective in the Zariski topology−→ and whose geometric fibres are integral then Theorem 4.8 of Kleiman ([20]) tells us that the functor Picét(X/S) is representable. In particular this theorem applies to the case of π : X S equal to either an Abelian scheme or a family of genus g curves. −→ . τ 2.3.2 We will write Picét(X/S)(T ) for the component of Pic(X/S) containing the identity (cf. [31] p23).

Now assume that π : X S is either an Abelian scheme or a family of genus g curves. −→

In case S is Noetherian Corollary 6.8 and Proposition 7.9 of Mumford ([31] p118) τ tell us that Pic´et(X/S) is an Abelian scheme and Zariski locally projective. We explain how to reduce to the Noetherian case in the following section.

2.3.3. We now explain how to deduce that Pic´et(X/S) is representable for an Abelian scheme or family of curves X/S over a non-Noetherian bases S from the same fact over a Noetherian base. This is required as we deﬁne our moduli spaces for Abelian varieties and curves with respect to families over a non Noetherian base and hence we need to describe what a polarisation of an Abelian scheme is and what the Torelli morphism does over a non-Noetherian base.

If S is non Noetherian we may reduce to the Noetherian case (see 2.3.2.) in the following way: First note, by the sheaf property, that it is necessary and suﬃcient to consider aﬃne S over which X/S is projective.

Then if S = Spec(R) and π : X S is projective, there is a finitely generated R[X , ,X ] ideal I so that −→ 1 · · · n X = Proj(R[X , ,X ]/I). 1 · · · n Let r1,...,rm be the coefficients of some finite set of generators for I as an R module. 8 TWISTS OF GENUS THREE JACOBIANS

3 Let R 1 be the image of Z[1/2] in R and let R0 = R 1[r1, , rm]. We note that R is Noetherian.− Let I = R [X , ,X ] I. Now− put · · · 0 0 0 1 · · · n ∩ X = Proj(R [X , ,X ]/I ). 0 0 1 · · · n 0 Then

R[X , ,X ]/I = R (R [X , ,X ]/I ). 1 · · · n 0 1 · · · n 0 Thus O

τ τ Picét(X/S) = Picét(X0/Spec(R0))S . 2.3.4. When f : C S is a family of curves we will write Jac(C) or Jac(C/S) for Picτ (C/S). When−→π : A S is an Abelian scheme we will write At for Picτ (A/S). ét −→ ét 2.3.5. The definition of Pic(X/S) given in section 2.3.1 makes life difficult in certain applications. When π : X S has a section s : S X, this difficulty can be −→ −→ removed by choosing for each coset of Pic(X)(T )/π∗Pic(T )(T ) a canonical representative.

To be precise, we deﬁne a functor

P (T ) := (L, φ) L Pic(X)(T ) and φ : s∗ L / = X/S { | ∈ T −→OT } ∼T where φ is an isomorphism. An element of PX/S (T ) is called a line bundle, rigidiﬁed or normalized along the section s.

Proposition 2. The functors Pic(X/S) and PX/S are naturally isomorphic. Indeed given L and its coset

L π∗Pic(T ), consider the line bundle O

1 M(L) := L (π∗sT∗ L)− then O

M(L) L π∗Pic(T ) ∈ and there is a canonical isomorphism O

φ : π∗M(L) . L −→OT Moreover, given any other element

L′ L π∗Pic(T ) ∈ we have an isomorphism O

ψ : M(L′) M(L) −→ so that

φ ′ = φ π∗ψ. L L ◦ This yields a natural transformation

Pic(X/S)(T ) P (T ). → X/S 3This unusual deﬁnition is forced on us by our convention that all schemes are over Z[1/2]. 2. DEFINITIONS AND CONVENTIONS 9

This morphism is surjective since if (L, φ) is a pair as above then M(L) is isomorphic to L in a way which is compatible with φ.

When π : X S is an Abelian scheme we will use this alternative definition of Pic(X/S) with−→O as the section s. 2.4. Isogenies, polarisations, the Weil pairing and level structures. 2.4.1. Let A and B be Abelian schemes over S of relative dimension n. An isogeny λ : A B −→ is a morphism of group schemes over S which is surjective. The kernel of λ will be denoted by A[λ]; this is a finite group scheme over S. For a positive integer N we will write [N] : A A −→ for the isogeny which adds a point to itself N times. 2.4.2. A polarisation of an Abelian scheme is an isogeny a : A At −→ which has the following form at each geometric fibre Ap of A: there exists an invertible sheaf Lp on Ap so that

1 a(x)= tx∗ Lp Lp− . We call a polarisation principal if it is an isomorphism.O

2.4.3. It need not be the case that there is an invertible sheaf L over A so that τ s s p∗L Lp mod Pic (Ap/k ) for each geometric point p S(k ). However, given a polarisation≡ a there does exist ([31], p121, Proposition 7.10)∈ an L over A so that λ(L)=2a, where λ(L) is given by the obvious formula

1 λ(L)(x) := tx∗ L L− . We are able to define λ(L) in this way becauseO of our definition of Pic as in 2.3.5 (see [31] p120-121 Def 6.2 Prop 6.10 for another definition). In this case we write H(L) for the kernel of the polarisation λ(L). 2.4.4. The Cartier dual of a finite flat group scheme G is defined as the group functor

S Hom(G(S), G (S)); 7→ m it is represented by a ﬁnite ﬂat group scheme GD (see [38]).

Given a principal polarisation a : A At the canonical isomorphism between At[N] and the Cartier dual of A[N] (see−→ [38]) yields a pairing

e : A[N] A[N] µ . N × −→ N This is called the Weil N pairing; it is symplectic and non-degenerate. 10 TWISTS OF GENUS THREE JACOBIANS

2.4.5. A symplectic level N structure on a principally polarised Abelian scheme (A, a) is an isomorphism of group schemes φ : A[N] (Z/N)g (Z/N)g −→ S × S which takes the Weil N pairing on A[N] to the standard symplectic pairing on (Z/N)g (Z/N)g which is given by the formula S × S ((x, y), (u, v)) Exp (x v y u). 7→ N · − · We will abuse language and also use the phrase ‘level N structure’ to mean a symplectic level N structure.

A level N structure on a family of curves π : C S is a level N structure on its Jacobian variety. −→ 2.5. Moduli schemes, modular forms, and the Torelli morphism. In this section we permit schemes S which are not deﬁned over Z[1/2]. Let N be a positive integer 3 and let ζ be a primitive Nth root of unity. ≥ N

. 1 2.5.1 Consider the functor from the category of schemes over Z[ N , ζN ] to the category of sets given by

S (C, α) C is a family of genus g curves over S and α 7→ { | is a symplectic level N structure on the Jacobian of C / . } ∼=S Here two families of curves f : C S and h : C′ S are considered isomorphic −→ −→ if there is an isomorphism φ : C C′ such that h φ = f. (We note that a more general notion of isomorphism might−→ include automorphisms◦ coming from the base S; if this notion were used, then two elliptic curves over a ﬁeld would be considered isomorphic if their j-invariants were conjugate under a ﬁeld automorphism. How- ever this notion is,as far as we know, not studied anywhere in the literature).

When N 3 this functor is representable by a smooth irreducible projective scheme ≥ g,N over Z[1/N, ζN ] ([31] p143 Corollory 7.14, [42] p134 Theorem 10.10, p142 RemarkM 2, p104 Theorem 8.11; for smoothness see [13] p103 Lemma 3.35; for irreducibility see [7]).

Likewise for N 3 the functor ≥

S (A, a, α) A an Abelian scheme of relative dimension g over S 7→ { | with a principal polarisation a and a symplectic level N structure α / , } ∼=S 1 is represented by an irreducible smooth projective scheme g,1,N over Z[ N , ζN ] ([31] p139 Theorem 7.9; for smoothness see [40] p242 TheoremA 2.33; for irreducibility see p xi [5] and [10]). 2.5.2. One consequence of the representability of the above functors is the existence of universal families of curves and Abelian varieties.

Thus there is a family of genus g curves

π : C −→Mg,N with a level N structure

Z g Z g α : Jac(C)[N] ( /N) g,N ( /N) g,N . −→ M × M 2. DEFINITIONS AND CONVENTIONS 11

Moreover given any family of genus g curves

f : D S −→ with a level N structure

β : Jac(D)[N] (Z/N)g (Z/N)g , −→ S × S there is a unique morphism h : S so that f = h∗π and β = h∗α up to −→ Mg,N canonical isomorphism. We call the pair (π : C g,N , α) the universal genus g curve with level N structure. −→M

Similarly, there is a universal principally polarised Abelian scheme of relative dimension g with a level N structure. 2.5.3. Let C/S be a family of genus g curves. The Jacobian has a canonical polarisation (see Prop 6.9 [31] p118)

t λΘ : Jac(C) Jac(C) . −→ 1 Moreover if C/S has a section s : S C then λΘ− is obtained by applying the Picard functor to the morphism −→ −

j : C(T ) Jac(C)(T ) : t t s , −→ 7→ − T here T is an S scheme and sT is the base change of the section s to T . This follows from the proof of Proposition 7.9 in [31]. 2.5.4. For N 3, we therefore have a morphism of ﬁne moduli schemes ≥ t : : C (Jac(C), λ ), Mg,N −→Ag,1,N 7→ Θ which we call the Torelli morphism; it is a proper morphism of degree 2 and is ramiﬁed along the hyperelliptic locus.

The moduli spaces g,N and g,N are irreducible. Moreover the locus of decomposably polarised AbelianM schemesA is closed. Then since the Torelli morphism is proper4 onto its image, for g = 2 and 3, we have the Torelli theorem: Let (A, a) be an indecomposably principally polarised Abelian variety of dimension g = 2 or 3 over an algebraically closed ﬁeld k. Then there is a smooth projective irreducible genus g curve C over k such that

(Jac(C), λΘ) ∼= (A, a). Our Theorem 1 is really just a reﬁned version of the Torelli theorem for genus 3. There is a more precise version of this theorem for non-algebraically closed ﬁelds and even for Abelian schemes over a Noetherian base (see [45] and [46]).

2.5.5. Let π : A S be an Abelian scheme and let ΩA/S be the sheaf of relative K¨ahler diﬀerentials−→ on A. We write

ωA/S := det(π ΩA/S), ∗ for the determinant of the Hodge bundle π ΩA/S. A Siegel modular form of weight ρ is deﬁned to be a global section of ∗

ρ ωA/S.

4This can be proven by using irreducibility of the moduli of principally polarised Abelian threefolds, the existence of the Neron model, and semi-stable reduction of curves. See also Oort- Ueno [39] for a proof which does not use irreducibility of the moduli of Abelian varieties. 12 TWISTS OF GENUS THREE JACOBIANS

2.5.6. The definition of the previous paragraph is not “functorial” in that it is defined relative to a fixed Abelian scheme A/S. To remedy this Katz has suggested the following definition of a modular form5.

Let g and N be ﬁxed positive integers, and let S be a given base scheme. A Katz- Siegel modular form of weight ρ and level N, relative to S, consists of the following data: (D) For every S scheme T and for every principally polarised Abelian scheme π : A T of relative dimension g with a level N structure α, there is given a −→ ρ global section sA/T of ωA/T subject to the condition: (C) Given a principally polarised Abelian scheme A′/T ′ with a level N structure α′, a morphism f : T ′ T and an isomorphism h : A′ A ′ , compatible with the −→ −→ T level structures αT ′ and α′, we have ρ det(dh) (f ∗sA/T )= sA′/T ′ . We note that if N = 1 then the condition on level structures is empty.

Thus to give a Katz-Siegel modular form is the same as giving a compatible system of modular forms indexed by Abelian schemes.

This definition can be made completely in terms of stacks. To do this first define a category ωρ fibred over the stack whose objects are pairs (s, (A/S,a)), A/ g,1 g,1 A A ρ where (A/S,a) is a principally polarised Abelian scheme and s is a section of ωA/S.

A section from g,1 to ωA/ g,1 is equivalent to a Siegel-Katz modular form of weight ρ. A A

2.5.7. Let f : C S be a family of genus g curves. Let ΩC/S be the sheaf of relative Kähler differentials.−→ A Teichmüller modular form of weight ρ is a section of the bundle

ρ ρ ω := det(f ΩC/S) . C/S ∗ A Katz-Teichmüller modular form tC/S is defined in exact analogy with the definition of a Katz-Siegel modular form. 2.5.8. Let f : C S be as in the preceding paragraph, and consider the Jacobian −→ Jac(C)/S and its sheaf of relative Kähler differentials ΩJac(C)/S. The Kodaira- Spencer class, gives a canonical isomorphism between

ωC/S and the bundle

ωJac(C)/S deﬁned in section 2.5.5. To be more precise, we need a relative version of Serre duality for smooth proper family of curves over an arbitrary base S. The standard reference for Grothendieck duality is Hartshorne’s Residues and Duality [12], and a deduction of the case we need can be found in [25] p243 Remark 4.20. Then we can construct the isomorphism above in the following way:

5At least in the case of an Abelian scheme of relative dimension1[17]. It seems Mumford predicted this definition by defining what a line bundle on a stack is in [35] p64.). Our definition GL excludes twisting π∗ΩA/S by a arbitrary representation of g , since we do not need the general definition. 3. MUMFORD’S THETA GROUP, EXISTENCE OF A PRINCIPAL LINE BUNDLE AND χ1813

A Kodaira-Spencer deformation argument shows that the pullback tangent bundle 1 of Jac(C) via the zero section O of Jac(C)/S is isomorphic with R f C . Serre ∗O duality then yields an isomorphism between π ΩJac(C)/S and f ΩC/S. Taking de- terminants then yields the desired isomorphism.∗ ∗

Thus for a family of Jacobians, a Siegel modular form (on the family of Jacobians) deﬁnes a Teichm¨uller modular form (on the family of curves) and vice versa.

2.5.9. We say that a Katz-Teichm¨uller modular form tC/S of genus g descends to a Katz-Siegel modular form sA/S of dimension g if for every family of genus g curves C/S we have

tC/S = sJac(C)/S, using the identiﬁcation of ωC/S and ωJac(C)/S explained in section 2.5.8.

3. Mumford’s theta group, existence of a principal line bundle and χ18 3.1. Mumford’s theta group as a descent gadget. 3.1.1. Let A and B be Abelian schemes over S, let λ : A B be an isogeny and let L be a line bundle over A. −→

We are interested to know when there exists a line bundle M over B so that λ∗M ∼= L and to classify all such M. When such an M exists we say that L descends.

Consider the Cartesian product

p2 A B A / A × p1 λ A / B. λ

We will write p12, p13 and p23 for the three distinct projections

p : A A A A A. ij ×B ×B −→ ×B The following conditions turn out to be necessary for L to descend

1) there is an isomorphism φ : p∗L p∗L, 1 −→ 2 2) up to natural isomorphism we have, p∗ φ = p∗ φ p∗ φ. 13 23 ◦ 12 We call a pair (L, φ : p1∗L p2∗L), where φ is an isomorphism satisfying condition 2), a line bundle with descent−→ datum with respect to λ. Line bundles with descent datum form a category in an obvious way. Pullback via λ gives a functor from the category of ’line bundles over B’ to the category of ‘line bundles over A with descent datum with respect to λ’.

A special case of Grothendieck’s descent theorem states ([4] p134 Theorem 4) that the functor λ∗ induces an equivalence of categories (in fact this holds for any faith- fuly ﬂat quasi compact morphism λ). In other words conditions 1) and 2) are necessary and suﬃcient for L to descend. 3.1.2. Let K be the kernel of λ. Consider the isomorphism

i : A K A A : (x, y) (x, x + y). × −→ ×B 7→ 14 TWISTS OF GENUS THREE JACOBIANS

We deﬁne two morphisms m : A K A and p : A K A by × −→ × −→ m(x, y)= x + y and

p(x, y)= x.

Then i gives an isomorphism of (A K,p,m) with the ﬁbre product (A B A, p1,p2) of (λ : A B, λ : A B). × × −→ −→ Similarly the isomorphisms i and

j : A K K A A A : (x,y,z) (x, x + y, x + y + z), × × −→ ×B ×B 7→ give an isomorphism of 1 1 1 (A K K,i− p j(a,b,c),i− p j(a,b,c),i− p j(a,b,c)) × × ◦ 12 ◦ ◦ 13 ◦ ◦ 23 ◦ with the threefold ﬁbre product (A A A, p ,p ,p ). ×B ×B 12 13 23 In other words i and j identify p12, p13 and p23 with following morphisms

1 i− p j(a,b,c) = (a,b), ◦ 12 ◦

1 i− p j(a,b,c) = (a,b + c), ◦ 13 ◦

1 i− p j(a,b,c) = (a + b,c). ◦ 23 ◦ The conditions for L to descend via λ can then be restated as:

1) there is an isomorphism φ : p∗L m∗L, 2) up to natural isomorphism we have,−→ 1 1 1 (i− p j)∗φ = (i− p j)∗φ (i− p j)∗φ; ◦ 13 ◦ ◦ 23 ◦ ◦ ◦ 12 ◦ or equivalently, for every S scheme T : the group scheme K acts on L, i.e. we have

1) for each b K(T ) there is an isomorphism φ : L t∗L, ∈ b −→ b 2) up to natural isomorphism for each b,c K(T ) we have φ = (t∗φ ) φ . ∈ c+b c b ◦ c Thus L descends to B if there is an action of K on L. The Mumford theta group G(L), which we deﬁne in 3.1.3, gives a way of encoding all possible actions of such a K on L. 3.1.3. Now assume that L is relatively ample. Consider the isogeny 1 λ(L)(x) := tx∗ L L− , where x A(T ) for some S scheme T . LetO H(L) be the kernel of λ(L). If x H(L∈)(T ) then there is an isomorphism ∈

φ : L t∗ L . T −→ x T We write G(L)(T ) for the set of all pairs (x, φ : LT tx∗ LT ) with x H(L)(T ) and φ an isomorphism as above. Then G(L)(T ) is endowed−→ with the structure∈ of a group (x, φ) (y, ψ) := (x + y, (t∗φ) ψ). · y ◦ There is a natural projection

G(L)(T ) H(L)(T ) → 3. MUMFORD’S THETA GROUP, EXISTENCE OF A PRINCIPAL LINE BUNDLE AND χ1815 and the kernel of this map is naturally identiﬁed with Gm(T ), which lies in the centre of G(L)(T ). Hence we have a short exact sequence

1 / Gm(T ) / G(L)(T ) / H(L)(T ) / 1. The functor T G(L)(T ) is represented by the scheme 7→ G H(L). m × However the group structure of G(L)(T ) is not that of the product G (T ) H(L)(T ). m × Thus as a group scheme G(L) is a non-trivial extension of H(L) by Gm.

We call G(L) the Mumford theta group of L (cf. [32]). 3.1.4. Given a subgroup scheme K H(L) an action of K on L is equivalent to ⊂ giving a subgroup scheme K˜ of G(L) which is isomorphic to K under the canonical projection from G(L) to H(L).

Thus Grothendieck’s descent theorem tells us: for a ﬁxed subgroup scheme K H(L), ⊂ let π be the isogeny from A to A/K, then there is a bijection between lifts K˜ of K G(L) / H(L) O O

= K˜ ∼ / K and equivalence classes of pairs (M, φ) where M is a line bundle over A/S and φ : L π∗M is an isomorphism. −→ 3.1.5. The commutator pairing

e (T ) : H(L)(T ) H(L)(T ) G (T ) L × −→ m is deﬁned in the following way: let x, y H(L)(T ) and let gx and gy be lifts of x and y respectively in G(L)(T ). We then∈ deﬁne

1 1 eL(T )(x, y) := gxgygx− gy− , which is independent of the choice of lifts gx and gy as Gm(T ) is in the centre of G(L)(T ).

From the definition of multiplication on G(L)(T ) we see that this pairing is equal to 2 the Weil N pairing when H(L)= A[N] (e.g. if N = m and L = [m]∗M for some principal and ample line bundle M). Moreover we note that for a subgroup scheme K of H(L) to lift to G(L) as in section 3.1.4 it is necessary that eL(T ) vanishes on K(T ) K(T ) for all S schemes T . This is because a subgroup scheme K of H(L) is commutative× and thus any lift of it to G(L) must also be commutative and therefore have trivial commutator. 3.2. Existence of an effective symmetric polar divisor for level 4. 3.2.1. Let π : A S be an Abelian scheme with a principal polarisation −→ a : A At, −→ and a level 4 structure (see section 2.4.5 for the definition)

φ : A[4] (Z/4)g (Z/4)g . −→ S × S 16 TWISTS OF GENUS THREE JACOBIANS

Following 2.4.3 there is an invertible sheaf L over A so that λ(L)=2a.

The invertible sheaf M := L [ 1]∗L satisﬁes [ 1]∗M = M and − − ∼ N λ(M)=4a. Thus H(M) = A[4] and the Mumford theta group of G(M) ﬁts into an exact sequence

1 / Gm / G(M) / A[4] / 0 . We first show that a lift of A[2] A[4] to G(M) exists. As we will see, this is ⊂ possible as M = L [ 1]∗L. − . 3.2.2 To lift A[2] toNG(M) we proceed as follows. Let x1, , x2g be generators of A[2](S). For each x we want to find a lift g G(M)(S) so· · · that i i ∈ 2 gi = 1 and gigj = gjgi. Then the subgroup of G(M) generated by the gi is a lift of A[2]. The commutativity condition is satisfied for any lifts gi and gj of xi and xj as

1 1 4 gigj gi− gj− = e4(xi, xj ) = (e4(yi,yj )) =1 where 2yi = xi and 2yj = xj (see 3.1.5). In order to obtain a lift gi of xi so that 2 2 2 hi g = 1 we construct a lift hi so that h = λ with λi Gm(S). We then put gi = . i i i ∈ λi

We construct such an hi in the following way. Since ker(λ(L)) = A[2] we have H(L)= A[2]. Therefore we may lift x to γ G(L). Note that γ has the form i i ∈ i (xi, φi) for some isomorphism φi : L tx∗i L. Now deﬁne an element γi′ of G([ 1]∗L) by the formula −→ − (x , [ 1]∗φ ). i − i Consider the following functors from the category of line bundles on A to itself:

F (L) := t∗(L [ 1]∗L) x − and O

G(L) := (t∗ L) (t∗ [ 1]∗L). x x − Then F and G are naturally isomorphic, andO we will write ξL for a natural isomorphism from F to G. We also note that

t∗ F (φ)= φ [ 1]∗φ and G(φ)= t∗ φ t∗ [ 1]∗φ. x − x x − 2 2 Then γ = γ′ = λi for someOλi Gm(S) and O i i ∈ 1 hi = (xi, ξL− tx∗i F (φi)) 2 ◦2 is a lift of xi to G(M)(S) such that hi = λi . 3.2.3. Let L be the line bundle on A/A[2] corresponding to the lift described in section 3.2.2, so that [2]∗L ∼= M. Consider the following element of At(S):

1 D := [ 1]∗L L− . − O 3. MUMFORD’S THETA GROUP, EXISTENCE OF A PRINCIPAL LINE BUNDLE AND χ1817

2 1 t Then D⊗ = [2]∗D = [ 1]∗M M − = . Let D′ A [4](S) be such that ∼ − ∼ OA ∈ N 2D′ = D, and put

L′ = L D′. 0 Note that L and L′ yield the same polarisationO on A as D′ Pic (A/S)(S). ∈ Now [ 1]∗L′ = L′ and H(L′) = 0. Thus L′ yields a principal polarisation of A and − ∼ L′ is symmetric in the sense that [ 1]∗L′ = L′. − ∼

The sheaf π L′ may have no global section; and indeed if L′ is rigidiﬁed along the ∗ zero section O, it follows from Theorem 5.1 of [10] that π L′ is anti-ample. So in ∗ this case π L′ can not have a global section. ∗

However, we can arrange for a twist of L′ to have a global section. Put

1 L := L′ π∗π L′− . ∗ Then we have a canonical isomorphismOψ : π L S thus L is eﬀective and has a 1 ∗ −→O canonical global section s := ψ− (1). Let Θ be the divisor of s; then

L = (Θ). ∼ OA Thus Θ is an relatively ample Cartier divisor which is principal and symmetric.

We have therefore proven Lemma 3. Let (A/S,a) be a principally polarised Abelian variety with a level 4 structure. Then A/S has a symmetric effective polar divisor Θ and Θ is unique up to translation by a 2-torsion point. 3.3. Symmetric polar divisors, theta nulls and modular forms. Let (A/S,a) be a principally polarised Abelian scheme of relative dimension g with a level 4 structure and let Θ be an effective symmetric relatively ample Cartier divisor on A/S such that λ( A(Θ)) = a. Such a Θ exists by Lemma 3 and is unique up to translation by a 2-torsionO point. We fix Θ throughout the sequel, and we will see that our choice of Θ does not matter. 3.3.1. Following Moret-Bailly ([30]) we define the sheaf of the zeroth theta null

TH := O∗ (Θ). g OA Now we set

1 M = (Θ) π∗O∗ (Θ)− . OA OA Then Theorem 5.1 of [10] yields O

8 4 π M = det(π ΩA/S)− . ∗ ∼ ∗ Moreover

1 π M = (O∗ A(Θ))− . ∗ ∼ O Thus

8 4 THg = det(π ΩA/S) . ∼ ∗ 18 TWISTS OF GENUS THREE JACOBIANS

3.4. Even 2-torsion points. 3.4.1. Let p A[2](S) be a 2-torsion point; let x be some geometric point of A ∈ in the image of p. Let I = A( Θ) be the sheaf of ideals corresponding to the symmetric eﬀective line bundleO −(Θ). OA Let be the local ring of A at x and let m be the maximal ideal of and OA,x x OA,x let Ix be the stalk of I at x. We say that p is even if

max n N I mn { ∈ | x ⊂ x} is even for every such x. In words this says that a section p is even if and only if it intersects the theta divisor of each geometric fibre at a point of even multiplicity of the theta divisor. This agrees with the classical notion. 3.4.2. It is not a priori clear from the definition in the preceding paragraph that even p exist. We therefore offer an alternative characterisation which proves their existence.

Let t : A A be the translation by p morphism. The line bundle t∗ (Θ) is p −→ pOA eﬀective with a unique non-zero global section (up to multiplication by S∗ (S)). Moreover we have an isomorphism O

t∗ (Θ) = [ 1]∗t∗ (Θ) pOA ∼ − pOA which induces an involution ǫ µ (S) on t∗ (Θ) and therefore on t∗I. p ∈ 2 pOA p

We claim that p is even if and only if ǫp = 1 on each connected component of S.

Indeed, let x be a geometric point of A in the image of p and let f be an element of A,x which generates (tp∗I)x. Without loss we can assume that S is connected. ThenO

ǫ f = f [ 1]. p ◦ − Thus f mn with n even if and only if ǫ = 1. ∈ x p The existence of even p then follows from the fact that each geometric ﬁbre of A/S has even 2-torsion points (see [32], where ǫp is called e : p304 Property iv) of e , and p307 Proposition 2), and such points must be the pullback∗ of a global 2-torsion∗ point if A/S has a level 2 structure.

3.4.3. Let Θπ(x) be the divisor of Aπ(x) given by restricting Θ to Aπ(x). Then p is even (cf. 3.4.1) precisely if Θπ(x) contains x with even multiplicity.

If the relative dimension g of A/S is three and the polarisation a of A/S is indecomposable, then for generic π(x) we may identify Θπ(x) with the theta divisor of the Jacobian of a smooth genus three curve over the residue ﬁeld of π(x).

Let u be a half canonical divisor of C, i.e. 2u = KC; such a u is a called a theta characteristic. By the theorem of Riemann-Kempf ([19]) the multiplicity of x in 0 Θπ(x) is equal to h (x + u). By Mumford’s results on theta characteristics ([36]) g 1 g we know there are exactly 2 − (2 +1) = 36 such x.

3.4.4. We keep the notation of the previous paragraph. The function 3. MUMFORD’S THETA GROUP, EXISTENCE OF A PRINCIPAL LINE BUNDLE AND χ1819

1 if h0(x + u) is not even, x − 0 7→ (1 if h (x + u) is even. is quadratic for the Weil 2 pairing and has Arf invariant 1. Choosing coordinates ¯ for Ax[4](k(x)), we therefore compute that

x = O. even x X Thus by the theorem of the cube ([10] p1) we have a canonical isomorphism

36 t∗ (Θ) = (Θ) . pOA ∼ OA even p O Since we work under the hypothesis that g = 3 and π : A S has a level 4 structure it is possible to improve upon Theorem 5.1 of [10] and−→ prove that

36 18 TH3 = det(π ΩA/S) . ∼ ∗ It suﬃces to prove this over C and this follows the fact that over the degree 3 Siegel 36 an upper half space h3 the bundle TH3 has a global section χ18 which transforms as a modular form (cf. [16]). And therefore

18 36 36 det(π ΩA/S) = THg = O∗(L ) = O∗( tp∗L) = p∗L. ∗ ∼ ∼ ∼ ∼ even p even p O O 3.5. Deﬁnition of χ18, descent to level 1 and vanishing on the hyperelliptic locus.

3.5.1. Let π : A 3,1,4 be the universal principally polarised Abelian scheme with a level 4 structure.−→ A Let Θ be an eﬀective symmetric line bundle as in Lemma 3.

The Satake embedding embeds 3,1,4 into projective space, and the projective closure of this image has a boundaryA of codimension 2. Thus through any two closed points x, y of there is a complete curve Z so that x, y Z. A3,1,4 ⊂ A3,1,4 ∈ Therefore the ring global sections of 3,1,4 is equal to OA Z[1/2, √ 1]. − Let s be a global section of (Θ). The section s is unique up to multiplication Θ OA by a unit of Z[1/2, √ 1]. At each geometric ﬁbre A of A/S the section s cuts out − x a symmetric polar divisor Θx (note x is now a geometric point of S and not A in constrast to section 3.4) of Ax. Such polar divisors Θx are unique up to translation by a 2-torsion point.

We therefore deﬁne

χ18 = p∗sΘ. even p A[2](S) O∈ 18 36 It is a global section of det(π ΩA/S) ∼= THg and thus a level 4 Katz-Siegel modular form of weight 18. We will∗ see in section 3.5.3 that it vanishes with multiplicity 1 on the locus of hyperelliptic Jacobians, it seems it also vanishes on the locus of Abelian three folds with a decomposable polarisation but we will not need this. 20 TWISTS OF GENUS THREE JACOBIANS

It is natural to ask how χ18 depends on the choice of Θ and sΘ. For ﬁxed Θ, we 36 see that χ18 is unique up to multiplication by an element of Z[1/2]∗ .

It is in fact independent of our choice of Θ. Given another choice Θ′, satisfying the hypothesis of Lemma 3, there is a unique 2-torsion point p such that

p∗Θ=Θ′.

In this case there is a section sΘ′ ′ such that

tp∗sΘ = sΘ′ ′ . 3.5.2. Let π : A S be any principally polarised Abelian scheme over a base S in which 2 is invertible.−→ Without loss, we may assume that S is connected.

There exists an Galois morphism f : S′ S so that A ′ /S′ has a level 4 structure −→ S and so that the covering group is PSp6(Z/4) (for example take S′ = A[4]S). Using the universal property of , we obtain a weight 18 modular form χ on A ′ /S′. A3,1,4 18 S We now show that χ18 descends to A/S.

The group PSp6(Z/4) is an extension of PSp6(Z/2) by an Abelian group H of ex- ponent 2. Now H acts trivially on 2-torsion points, and thus χ18 descends to a level 2 Katz-Siegel modular form. Moreover, PSp6(Z/2) is simple and non-commutative, and therefore has no non-trivial 1 dimensional representations. Thus χ18 descends as a level 2 modular form to a level 1 Katz-Siegel modular form.

In this way we obtain χ18 as a Katz-Siegel modular form of weight 18 and level 1 over Z[1/2].

3.5.3. We now show that χ18 vanishes on the hyperelliptic locus with multiplicity 1.

Let A/S be as in the preceding paragraph. Let (Ax,ax) be a ﬁbre of a closed point x S with indecomposable polarisation ax. Let k(x) be the residue ﬁeld of x and ∈ s let Θx be a symmetric polar divisor of ax over the separable closure k(x) of k(x).

We can assume that there is a curve C over k(x) such that s s (Jac(C), λ) k(x) ∼= (Ax,ax) k(x) and that Θx is a theta divisor ofO Jac(C). O

Then Θx contains an even p if and only if C is hyperelliptic and in this case p is s 0 unique. This is because Θx = a + b u Jac(C)(k ) with 2u = KC and h (u) even, and so we have that h0(p{+ u) is− even}⊂ and p + u = a + b for some a,b C(ks). Hence there is a morphism f : C P1 with polar divisor a + b. Thus f∈is 2:1 from C to P1. Such an f is unique−→ up to an automorphism of P1 and hence p is also unique.. Moreover, it is easy to show that the multiplicity of such an p must be exactly 2, i.e. p is even.

Thus χ18 vanishes with multiplicity 1 on the locus of hyperelliptic Jacobians; it seems it also vanishes on the locus of decomposable principally polarised Abelian varieties, but we will not need this.

4. Existence and properties of the ∆i 4.1. Properties of the Torelli morphism. 4. EXISTENCE AND PROPERTIES OF THE ∆i 21

4.1.1. Let N be an integer 3 and let be the ﬁne moduli space of smooth ≥ Mg,N genus g curves with a level N structure. Let g,1,N be the ﬁne moduli space of principally polarised Abelian varieties of dimensionA g with a level N structure.

Let t : be the Torelli morphism. Mg,N −→Ag,1,N

A point x g,N (S) corresponds to a pair (C,β) where C is a family of genus three curves∈ over M S and β is a level N structure for its Jacobian variety.

If C is non-hyperelliptic (i.e. a dense set of its ﬁbres are non-hyperelliptic) then the automorphism scheme of Jac(C)/S is isomorphic with Aut(C/S) [ 1] (see [46] and [45]). Otherwise, if C is hyperelliptic (i.e. all its geometric× h ﬁbres− i are 6 hyperelliptic curves ), we have Aut(C/S) ∼= Aut(Jac(C)/S) and the hyperelliptic involution is taken to the automorphism [ 1]. − Thus unless C is hyperelliptic, there is no automorphism of C which takes β to β. We therefore have a non-trivial involution −

τ : Mg,N −→Mg,N given on S valued points by the formula

τ(C,β) = (C, β). − Moreover, the ﬁxed locus of τ is precisely the hyperelliptic locus.

Let V be the geometric quotient of under τ and let g,N Mg,N ι : V Mg,N −→ g,N be the quotient morphism. Since the [ 1] automorphism of the Jacobian takes β to β we see that the Torelli morphism− factors through ι. We therefore have a morphism−

i : V ; g,N −→Ag,1,N and the image of i obviously avoids the locus of Abelian g folds with a decomposable polarisation. By the Torelli theorem for g = 3, every indecomposably principally polarised Abelian threefold occurs in the image of i. Thus V = ′ , where 3,N A3,1,N ′ is the locus of indecomposably principally polarised Abelian threefolds. A3,1,N Oort and Steenbrink ([37]) have shown that i is a locally closed immersion. We will therefore identify V with a locally closed subscheme of . g,N Ag,1,N 4.1.2. The moduli spaces g,N and g,1,N are smooth over Z[1/N, ζN ], which means they are locally regular.M A

Let U be an aﬃne cover of so that U = Spec(R ) with R regular. { N,i} Mg,N N,i N,i N,i Assume further that τ(UN,i)= UN,i.

Let Rτ be the ring of invariants of R under the action of the group τ . N,i N,i h i

6This deﬁnition is equivalent to C having an involution whose quotient is a family of P1’s over S (see [26]). 22 TWISTS OF GENUS THREE JACOBIANS

Since RN,i is locally regular, one can choose (after a possible refinement of the cover U ) the R to be free of rank 2 as an Rτ module. This implies7 that there { N,i} N,i N,i is a function ∆ Rτ , unique up to a square unit of Rτ , so that N,i ∈ N,i N,i τ RN,i = RN,i[ ∆N,i] and moreover τ( ∆N,i)= ∆N,i. This lastp condition implies that ∆N,i vanishes with multiplicity 1 on− the fixed locus of τ. Hence ∆ Rτ vanishes with p p N,i ∈ N,i p multiplicity 1 on the image of the fixed locus of τ in Vg,N .

But the ﬁxed locus of τ is precisely the hyperelliptic locus. Thus the ∆N,i cut out the same divisor as χ18 (see 3.5.3).

Proposition 4. Let k be a ﬁeld not of characteristic 2. Let (A, a, α) 3,1,N (k) be a principally polarised Abelian threefold deﬁned over k with a level N∈ Astructure α. Suppose that (A, a, α) UN,i as above for g = 3 and that the polarisation a is indecomposable. Then ∈ ∆ (A, a) k2 N,i ∈ if and only if (A, a) ∼= (Jac(C), λΘ) for some genus three curve C over k. Proof: There are two cases to consider: i) (A, a) is a Jacobian and ii) (A, a) is the [ 1] twist of a non-hyperelliptic Jacobian. In the second case there is a unique non-− hyperelliptic curve C over k and a unique quadratic extension k2 of k (cf. [45], [46], Theorem 3.2) so that

(A, a) k2 ∼=k2 (Jac(C), λΘ) k2. In case i) we have that ∆ O(C, α) k and O N,i ∈ 2 2 ∆N,i(A, a)p = ( ∆N,i) (Jac(C), λΘ) = ( ∆N,i(C, α)) , 2 and so ∆N,i(A, a) k . p p ∈

In case ii) (Jac(C), λΘ) is the [ 1] twist of (A, a) over k2. Consider the obvious Galois representation −

ρ : Gal(k /k) Aut(A[N](k )). 2 −→ 2 This is trivial as A has k rational N torsion. Let σ be the generator of Gal(k2/k). The Galois representation

ρtw : Gal(k /k) Aut(Jac(C)[N](k )) 2 −→ 2 is then given by the formula

ρtw(σ) = [ 1]ρ(σ) = [ 1]. − − Thus Jac(C) does not have k rational N torsion but Jac(C) k2 has k2 rational N torsion. N tw 6 Let α be the symplectic isomorphism of Jac(C)[N](k2) with (Z/N) given by composing the isomorphism of Jac(C)k2 with Ak2 with the level N structure

7 Here the author is indebted to Marius van der Put for pointing out to him that such a ∆N,i existed and that p∆N,i and ∆N,i vanished with the stated multiplicities on the hyperelliptic locus of Mg,N and Vg,N respectively. 4. EXISTENCE AND PROPERTIES OF THE ∆i 23

α : A[N](k ) (Z/N)6. 2 −→ Then the ﬁbre of the Torelli morphism above (A, a) is given by the Gal(k2/k) orbit

(C, αtw), (C, [ 1] αtw) . { − ◦ } And therefore

[ ∆ ( (C, αtw), (C, [ 1] αtw) )] k k, N,i { − ◦ } ∈ 2 − hence p

∆ (A, a) = [ ∆ ( (C, αtw), (C, [ 1] αtw) )]2 k k2. N,i N,i { − ◦ } ∈ − This completes the proofp of Proposition 4.

4.1.3. Assume now that over the cover UN,i, the line bundle ′ (H) is trivial. OA3,1,N Then we have isomorphisms

′ τ φi : (H) RN,i, OA3,1,N −→ such that

φi(χ18) UN,i UN,j = φj (χ18) UN,i UN,j . | ∩ | ∩ Such a system of compatible isomorphisms (φi) is unique up to multiplication by an element of Γ( 3,1,N , ∗ ′ ) = Z[1/2N, ζN ]∗ (the codimension of decomposably A OA3,1,N polarised Abelian threefolds is 2, and hence the global units are constants, cf. 3.5.1).

Consider the pullback t∗χ18 of χ18 via the torelli morphism t.

1 We can extract a square root of t∗χ over C, and hence over Z[ , √ 1], thus 18 2N −

t∗χ18 3,N (H). ∈ OM Assume moreover that over thep cover WN,i = Spec(RN,i) the bundle 3,N (H) is trivial. OM

Let

ψ : (H) R , i O −→ N,i be a trivialisation such that

ψi( t∗χ18) WN,i WN,j = ψj ( t∗χ18) WN,i WN,j . | ∩ | ∩ Since the ∆N,i are onlyp deﬁned up to a squarep unit, we have

ψi( t∗χ18) = ( ∆N,i).

Therefore we may choose φi as abovep so thatp

φi(χ18) = ∆N,i. We have therefore proven 24 TWISTS OF GENUS THREE JACOBIANS

Proposition . 5 Given any system of isomorphisms (φi′ ) such that

φi′ (χ18) UN,i UN,j = φj′ (χ18) UN,i UN,j , | ∩ | ∩ 1 there is a constant c Z[ , ζ ]∗, depending only on the system (φ′ ), so that ∈ 2N N i φi′ (χ18)= c∆N,i.

Moreover, there exists a system of isomorphisms (φi) such that

φi(χ18) UN,i UN,j = φj (χ18) UN,i UN,j , | ∩ | ∩ and φi(χ18) = ∆N,i.

4.2. Construction of ∆i. 4.2.1. Let A/S be an Abelian scheme of relative dimension 3 with a principal polarisation a. We now show the existence of the ∆i as in part d) of Theorem 1.

Let fN : S′ S be a Galois cover such that AS′ has a level N structure (cf. 3.5.2). −→ 1 Now let XN,i be a Zariski cover of S such that UN,i := f − (XN,i) is aﬃne with a function ∆ coming from the ﬁne moduli space as in 4.1.2. N,i A3,1,N

We ﬁrst note that since χ18 is a Katz-Siegel modular form, it follows from Propo- sition 5, that ∆N,i descends to a function δN,i on XN,i with locus equal to the hyperelliptic locus.

Moreover, given distinct N and M we have, by Proposition 5, that

δN,i = γδM,i, 2 for some unit γ Z[1/2MN,ζMN ]∗ . Since Z[1/2MN,ζMN ]∗ is generated by 1/2M, 1/2N, ζ and∈ ζ , we may choose ∆ on X X so that M N i N,i ∩ M,i

fM∗ ∆i = ∆M,i and fN∗ ∆i = ∆N,i. This choice can be made independently of M and N so that we have, up to multi- 2 plication by an element of Z[1/2]∗ , a unique choice of ∆i with the property that that there is a trivialisation of (H) sending χ to ∆ . We have a version of OS 18 i Proposition 4 for the ∆i

Proposition 6. Let x be a closed point of Ui S with residue field k. Let (Ax,ax) be the fibre of A over x. Then ⊂ ∆ (A, x) k2 i ∈ if and only if there is a curve C over k such that (Jac(C), λΘ) ∼= (Ax,ax). Proof: From the refined Torelli theorem, we know that there is a curve C over k and an element D k such that ∈ √ √ (Jac(C), λΘ) k( D) ∼= (Ax,ax) k( D).

We want to show that k(√D)=Ok( ∆i(Ax,ax)). FromO our construction we know 2 that if D k∗ , then for any two M,N ∈ p

∆i(Ax,ax) k(A[M]) k(A[N]), 2 ∈ ∩ and so ∆ (A ,a ) k∗ . i x x ∈ p

Now suppose k(√D) is not contained in k(A[4]), but k( ∆i(Ax,ax)) k(A[4]). Now A is the [ 1] twist of Jac(C), and so if ρ is the representation of Gal(k⊂(A[4])/k) − p 5. THEOREM 1E 25

s s on A[4]ks (k ) then ρ is the Galois representation of Gal(k(A[4])/k) on Jac(C)[4]ks (k ), which is a contradiction.− So

∆ (A ,a ) / k(A[4]) i x x ∈ Consequently, p

k(A[4], √D)= k(A[4], ∆i(Ax,ax)) and thus k(√D)= k( ∆i(Ax,ax)). p

4.2.2. We have a versionp of Proposition 5 for the ∆i: Proposition . 7 Given any system of isomorphisms (φi′ ) such that

φi′ (χ18) Ui Uj = φj′ (χ18) Ui Uj , | ∩ | ∩ there is a constant c Z[1/2], depending only on the system (φ′ ), so that ∈ i φi′ (χ18)= c∆i.

Moreover, there exists a system of isomorphisms (φi) such that

φi(χ18) Ui Uj = φj (χ18) Ui Uj , | ∩ | ∩ and φi(χ18) = ∆i.

5. Theorem 1e

an 5.1. Deﬁnition of the analytic χ18.

5.1.1. Let h3 be the Siegel upper half space of degree 3: that is the complex do- main of 3 3 symmetric complex matrices with positive deﬁnite imaginary part. Let πan : A×an h be the universal complex analytic Abelian threefold over h . −→ 3 3 3 Analytic uniformisation provides a family f : C h3 h3 of complex vector spaces, and a family h :Λ h of lattices of rank× 6 whose−→ ﬁbre above a point τ is −→ 3 1 3 3 1 3 h− (τ) := Z + τZ f − (τ)= C τ ⊂ ×{ } so that

Aan = (C3 h )/Λ ∼ × 3 as analytic manifolds above h3.

Let z = (z (τ),z (τ),z (τ)) denote the standard coordinates of C3 τ C3 h . 1 2 3 ×{ }⊂ × 3 3 The Riemann theta function is the holomorphic function on C h3 deﬁned by the formula ×

π√ 1ntτn+2π√ 1n z ϑ(z; τ) := e − − · . n Z3 ∈ X an an We have a canonical level 2 structure on π : A h3 given by the standard 3 3 −→ an coordinates on Z + τZ . Using this we identify a 2-torsion point p A [2](h3) 3 3 ∈ with a unique element m + τm′ 1/2(Z + τZ ) such that m and m′ have entries equal to either 0 or 1/2. ∈

The analytic theta nulls are the holomorphic functions on h3 given by the formulas 26 TWISTS OF GENUS THREE JACOBIANS

m ϑ (0; τ) := ϑ(m + τm′; τ). m′ The function p 4m m′ mod 2 deﬁnes a quadratic form with values in Z/(2) which is zero if and7→ only· if p is a point in the divisor of ϑ(z; τ) of even multiplicity. Thus an analytic theta null

m ϑ (0; τ) m′ is called even if 4m m′ is even. · We therefore deﬁne

an m χ18 (τ) := ϑ (0; τ). m′ 4m m′ 2Z ·Y∈ 5.2. Moret-Bailly’s isomorphism.

5.2.1. Consider the analytic isomorphism on h3

18 ω h3 ( H) = h3 A/h3 O − ∼ O given by sending O

(2π√ 1)54χan(τ)(dz dz dz )18 − 18 1 ∧ 2 ∧ 3 to the constant function 1. Theorem 0.5 of [30] shows that this isomorphism is compatible with the isomorphism

18 ωA/ 3,1,4 3,1,4 ( H) = 3,1,4 , A OA − ∼ OA over Z[ 1 , √ 1] of section 3.4.4. O 2 − Let A/S be an Abelian scheme of relative dimension 3 with a principal polarisation a. Let Ui be an open cover of S over which the functions ∆i of 4.2 exist, and over which the sheaf of K¨ahler diﬀerentials is free with basis ξ1, ξ2, ξ3. Then Proposition 7 guarantees that there is a constant c Z[1/2]∗, independent of S and i, so that ∈ c∆ (ξ ξ ξ )18 i 1 ∧ 2 ∧ 3 is mapped to the constant function 1 by the isomorphism of section 3.3.1.

Let S′ be an ´etale cover of S such that AS′ has a level 4 structure; by abuse of language we write ∆i for the composition of ∆i with the morphism from S′ to S.

Thus, tensoring S′ with C and embedding the analytification of S′ C into h3, we have N c∆ (ξ ξ ξ )18 = (2π√ 1)54χan(τ)(dz dz dz )18. i 1 ∧ 2 ∧ 3 − 18 1 ∧ 2 ∧ 3 an 5.2.2. Consider the relative differential forms dz1, dz2 and dz3 on A . They can be expressed in terms of algebraic invariant differential forms ξ1, ξ2, ξ3 by means of the period matrix Ω1

ξ1 dz1 ξ =Ω dz .  2  1  2  ξ3 dz3     6. THEOREM 1F 27

The matrix Ω1 is obtained in the following way: let γ1,γ2,γ3 denote the homology classes of A = C3/Z3+τZ3 corresponding to the lattice vectors (1, 0, 0), (0, 1, 0), (0, 0, 1). Then the entries of Ω1 are obtained by integrating the ξi along the γj .

5.2.3. The last formula of 5.2.1 combined with 5.2.2 yields

54 an (2π√ 1) χ18 c∆i = − 18 . det(Ω1) We note the the quantity on the right hand side is invariant under symplectic trans- formations, and it depends on A/S, as Ω1 depends on A/S. In particular if S is the spectrum of a field k, and Atw is the [ 1] twist of A over the field k(√D) then 18 18 27 − det(Ω1)A and det(Ω1)Atw differ by D .

Explicit numerical calculation8 with the analytic theta nulls determines that this is the case for c = 1. And this shows Theorem 1 part e). − 6. Theorem 1f The definition of Discr is made in two steps. It is first defined in the classical set- ting; then via pulling back, it is defined as a Katz-Teichmüller modular form.

6.1. Construction of the discriminant. 15 6.1.1. We work over the base Z[1/2]. Give P coordinates aijl with i,j,l 0, , 4 and i + j + l = 4. Consider the classical family of plane quartics: f : C∈ { P· ·15 · } 2 −→ where C is the subscheme of PP15 deﬁned by the equation i j l (F := aijlX Y Z )=0. Xi,j,l Let FX , FY and FZ be the derivatives of F with respect to X, Y and Z.

15 The discriminant of f : C P is defined to be the resultant of FX , FY and FZ . We will now explain what−→ this means. 6.1.2. Let U be an affine open subset of P15 whose coordinate ring R has the unique factorization property. Consider the restriction CU of C to U = Spec(R). 2 The scheme PU is then obtained as the Proj of the ring R[X,Y,Z]. For each positive integer m, let Im denote the R module of degree m homogeneous forms in X, Y and Z with coefficients in R. Then Im is free of rank (m+2)(m+1)/2 over R.

The polynomials F , F and F have a common zero at a closed point (a ) P15 X Y Z ijl ∈ if and only if the corresponding quartic F(aijl ) is singular. Equivalently the polynomials FX , FY and FZ have a common zero at a closed point (aijl) if and only if for all (γ ,γ ,γ ) I3 the polynomials 1 2 3 ∈ 4 γ1FX + γ2FY + γ3FZ have a common zero.

3 We have an R matrix M taking I4 to I7 which is given by multiplying the monomial elements of I3 by (F , F , F ). Evaluating M at a closed point (a ) P15, we 4 X Y Z ijl ∈ 8Christophe Ritzenthaler showed the author Maple code which does this: given an affine quartic, Maple computes the period matrix Ω1. Then computation with the analytic theta nulls and the algebraic invariant differentials shows in specfic examples that the right hand side of the above formula is a rational square when multiplied by −1. 28 TWISTS OF GENUS THREE JACOBIANS

ﬁnd that F(aijl) is smooth if and only if the maximal minors of M(aijl) are all non zero. The resultant (cf. [27] p3, cf. [23] pp388-404) of FX , FY and FZ is then the highest common factor of the maximal minors of M. It is thus zero at precisely the singular ﬁbres of C (cf. [27] p13).

Remark Let Mij denote the maximal minors of M and let Res denote the highest common factor of the M . Ideal theoretically we have (Res) (M ) and so ij ⊃ ij ij scheme theoretically we have, Z(Res) Z(M ). ⊂ ij ij P Let B be an invertible 3 3 matrix withS entries in R; then × 1 1 1 9 Res(F B− , F B− , F B− ) = det(B) Res(F , F , F ) X ◦ Y ◦ Z ◦ X Y Z (cf. [27] pp8-9).

15 Let V be the open subscheme of P where the resultant of FX , FY and FZ does not vanish.

Since smooth plane quartics are embedded into projective space via their canonical bundle, we have defined a weight 9 Teichmüller modular form DiscrC/V . . 6.1.3 We define DiscrC/ 3,4 by mapping C to Proj(ΩC/ 3,4 ) and using the above procedure. M M

Comparing DiscrC/ 3,4 with DiscrC/V we see that it is independent of the level 4 structure, and thusM descends to a weight 9 Katz-Teichm¨uller modular form.

Let C′/S be family of genus three curves and let x be a geometric point of S whose ﬁbre is hyperelliptic. Let S′ be an ´etale cover S so that the C′′ := C′ S′ has a ×S Jacobian with symplectic level 4 structure and let y be a point of S′ lying above x. By the universal property of , to show that Discr(C′′/S′) vanishes at y we M3,4 need only show that Discr(C/ 3,4) vanishes at the image z of y in 3,4. Let Z be a smooth curve intersectingM the hyperelliptic locus at z transversallyM and let I be the stalk of the sheaf of ideals of Z at z. Consider the discrete valuation ring

R := 3,1,4,z/I and let K be its fraction ﬁeld and let t be a uniformizer. Taking the canonicalOM embedding of the restriction of the universal curve to R we have a family of quartics over R with equation

F 2 + t2H =0 with F a conic and H a quartic, which can be normalized to obtain a smooth curve C over Spec(R) whose special ﬁbre is hyperelliptic. Thus DiscrC/Spec(R) vanishes on the hyperelliptic locus. And hence DiscrC′/S vanishes on the hyperelliptic locus of S.

Considered as a Katz-Teichmüller modular form, χ18 vanishes with multiplicity 2 on the hyperelliptic locus and has weight 18. Thus up to a unit of Z[1/2] it is 2 2 equal to Discr : for over 3,4 the modular form χ18 is equal to Discr modulo M 2 a unit in Z[1/2, √ 1] whereas over 3,3 it is equal to Discr modulo a unit in − M 2 Z[1/2, 1/3, ζ3]. Thus over 3,12 we see that Discr is equal to χ18 modulo a unit of Z[1/2]. M Twists of genus three curves over finite fields

How many points can a genus three curve over a ﬁnite ﬁeld have? What about the twists of such a curve? The latter question is intimately related with the geometric Frobenius endomorphism of a Jacobian and how it changes with twisting.

In this chapter we explain how to use the geometric Frobenius of a smooth projective plane quartic to compute the geometric Frobenius of its twists. We give an explicit formula which uses a matrix representation of Frobenius and a matrix representation of the 1-cocycle corresponding to the twist to obtain the “twisted Frobenius”.

We then explicitly compute the twists of genus three curves and their twisted Frobe- niuses in terms of 1-cocycles.

Two highlights are treatment of the Fermat and Klein quartics. In particular, we discuss the moduli theoretic meaning of twisting of Klein quartic which is isomorphic to X(7), and prove that the Fermat quartic is a twist of the modular curve X0(64).

We note Brünjes [2] has made a complete study of forms of Fermat-type varieties over finite fields, and Gouvêa and Yui [11] have studied the diagonal forms of hypersurfaces over finite fields. Duursma [8] has studied twists of the Klein quartic modulo 2 and Poonen, Schaefer and Stoll [41] have studied twists of the Klein quartic over number fields.

1. Forms and automorphisms 1.1. Definition of a Form. Let k be a field and let l be a field containing k and let V be a k variety; we will write V or V l for the l variety V l. l ⊗ ×k

s Definition 1. Let k be a ﬁeld and let V and V ′ be varieties over k; let k be a separable closure of k. We call V ′ a form of V over k if there is an isomorphism

φ : V s V ′s . k −→ k If the isomorphism φ is deﬁned over k itself, then V ′ is a trivial form of V . Two forms are considered equivalent if they are isomorphic over k. As we will see in Section 1.2, for a projective variety V to have non-trivial forms it is necessary that s V k have a non-trivial automorphism group. We will normally write ‘V ′ is a form⊗ of V ’ with k being understood implicitly.

We now look at an example to illustrate the deﬁnition of a form. Consider the 2 Fermat quartic C0 whose closed points in P (k) are given by the equation

X4 + Y 4 + Z4 =0.

29 30 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

If k = F13, the ﬁeld with 13 elements, then C0 has 32 points. The plane quartic C0′ whose projective equation is

X4 +4Y 4 X2Y 2 +7Z4 =0 − has 8 points, and hence these curves can not be isomorphic over F13. However, over the ﬁeld F13(√2) we have the isomorphism

(X : Y : Z) (X + √2Y : X √2Y : Z). 7→ − Then C0′ is a form of C0; in fact, using the results of section 3.2, it can be shown that it is a non-diagonal form of C0. This means that it is not isomorphic to a curve with equation

aX4 + bY 4 + cZ4 =0 for a,b,c F . ∈ 13 1.2. Classiﬁcation of Forms. The standard reference for the material in this section is Serre [52] p131. However Serre does not explain why the quotient of a projective variety by a twisted Galois action exists and so we give a sketch of it existence (in fact the same hold for a quasi-projective variety).

Let k be a ﬁeld and let ks be a separable closure of k. Let V be a variety over k; let V ′ be a form of V and let φ : V s V ′s k −→ k be an isomorphism of ks varieties.

s Each element σ of the Galois group Γ = Gal(k /k) deﬁnes an automorphism of Vks by 1 σ : V s V s . × k −→ k Thus we have a natural action of Γ on Vks . Similarly, we have an action of Γ on Vk′s . The isomorphism φ then induces a “twisted” action of Γ on Vks by the rule tw 1 σ := φ− (1 σ) φ. ◦ × ◦ One can show that this twisted action does not depend on φ: that is, given any s other isomorphism ψ : Vks Vk′s over k , we obtain an action isomorphic to the action induced by φ. In other−→ words, equivalent forms of V yield isomorphic actions of Γ on Vks .

tw To recover a form V ′ from a twisted action σ we take the quotient of Vks by this action. That the quotient exists and has the desired properties is a consequence of V being projective1. We give a sketch of the construction: if V is projective then it has a very ample line bundle L; and there is a natural action of Γ on the corresponding very ample line bundle Lks of Vks . Consider Γ as a constant group s scheme over k and let p,m : :Γ V s V s be the projection and multiplication × k −→ k maps respectively. The data of the natural action of Γ on Lks is given by an tw isomorphism α between m∗Lks and p∗Lks . Now the twisted action σ defines a continuous 1-cocycle 1 tw a :Γ Aut s (V s ) : σ (1 σ− ) σ . σ −→ k k 7→ × ◦ 1For a smooth curve this is equivalent to a statement about the existence of quotients of the function field. Indeed, for some finite extension l ⊃ k we have l(C) = l(C′). Then Gal(ks/k) acts on l(C) through Gal(l/k). The quotient of l(C) under the twisted action of Gal(l/k) is then k(C′′) for some form C′′ of C; using the definition of the twisted action it is easy to check that ′′ ′ C =∼ C . 1. FORMS AND AUTOMORPHISMS 31

This twisted 1-cocycle induces a twisted action of Γ on Lks by pulling back the isomorphism α via the morphism

s :Γ V s Γ V s : (σ, v) (σ, a v). aσ × k −→ × k 7→ σ It is easy to see that the graded coordinate ring of Vks corresponding to the embedding deﬁned by Lks has a ring of invariants under this twisted action which is naturally a ﬁnitely generated graded k algebra. The Proj of this ring of invariants is the quotient of Vks by the twisted action.

We call two 1-cocycles aσ and bσ equivalent when there is an element g Autks (Vks ) so that ∈ σ gaσ = bσg. 1 The set of equivalence classes of 1-cocycles is denoted by H (Γ, Autks (Vks )) and is called the 1st Galois cohomology set of the Galois module Autks (Vks ).

It is easy to see that isomorphic Galois actions correspond to equivalent 1-cocycles. Once the existence of a quotient by a Galois action is established, it is also easy to see that equivalent forms of V correspond to isomorphic Galois actions.

We have therefore established the existence of a bijection

1 Forms of V / / H (Γ, Aut s (V s )) , { } ∼ k k (cf. Proposition 5 of Serre [52] p131). To paraphrase, the set of 1-cocycles

a :Γ Aut s (V s ) { σ −→ k k } modulo equivalence classifies the set of forms of V . In the next section we show how to explicitly compute representatives of these 1-cocycles when k is a finite field.

1 1.3. How to compute H (Γ, Autks (Vks )) for k finite. Let k be a finite field; to reduce notation we will now write

Aut(V ) for Autks (Vks ). Given an automorphism g Aut(V ) and an element of the Galois group σ Γ we will write σg for the image∈ of g under the left action of σ. We let Fr denote∈ the Frobenius automorphism in Γ.

We deﬁne an equivalence on Aut(V ) by Fr 1 g g′ g = hgh− for some h Aut(V ). ∼ ⇔ ∈ In particular, if the action of Γ on Aut(V ) is trivial then the equivalence is the same as conjugacy on the group Aut(V ). ∼

The computation of H1(Γ, Aut(V )) over k is reduced to something reasonable by

Proposition 2. The following map is an injection

H1(Γ, Aut(V )) Aut(V )/ : a a . → ∼ σ 7→ Fr Proof: A 1-cocycle aσ is deﬁned by the relations

−1 1+Fr+ +Frn aFrn = ··· aFr and 32 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

−1 1+Fr+ +Frm 1= ··· aFr for m = ord(aFr).

Thus in order to compute H1(Γ, Aut(V )) we ﬁrst compute Aut(V )/ . Then for each equivalence class [g] Aut(V )/ , we check to see if there is∼ an element ∈ ∼ g′ [g] which satisﬁes the cocycle relation. It turns out that there need not always be∈ such an element, as our calculation with the forms of the Klein quartic demon- strates.

As a matter of practice, it often turns out that Aut(V ) has a large subgroup on which Γ acts trivially. Such a situation can be exploited to simplify the computation of Aut(V )/ . Indeed, deﬁne an auxiliary equivalence by ∼ ∼0

1 g g′ g′ = hgh− ∼0 ⇔ Γ for some h Aut(V ) . Since g g′ implies g g′ we have a surjective map ∈ ∼0 ∼

Aut(V )/ 0 / Aut(V )/ : [g] 0 [g] . ∼ ∼ ∼ 7→ ∼ As we will see in the case of the Fermat quartic, there are practical reasons why Aut(V )/ 0 may have been already computed. The above observation can therefore be of real∼ value.

These remarks raise the following question: is the number of elements of twists of V over a ﬁnite ﬁeld bounded by the number of conjugacy classes of Aut(V )?

In fact the answer is yes, and after a lecture on this topic Hendrik Lenstra pointed out the following simple proof to the author.

Proposition 3.

H1(Gal(F¯ /F , Aut(V )) conjugacy classes of Aut(V ) . | q q |≤|{ }| Proof: The key ingredient is the formula of Burnside which states: let X be a finite set acted upon by a finite group G, let Xh denote the elements of X fixed by h G and let X/G denote the set of orbits, then ∈

1 #X/G := #Xh. #G h G X∈ In our situation we have two actions of Aut(V ) on itself: one by conjugation and one by twisted conjugation

h Fr 1 g := hgh− . Let X := Aut(V ) and write Xh for the elements of X ﬁxed by h under conjugation h and write XFr for the elements of X ﬁxed by twisted conjugation by h.

If there is a g Xh then there is a bijection from Xh to Xh given by ∈ Fr Fr

g′ g g′. 7→ · 1. FORMS AND AUTOMORPHISMS 33

Hence 1 H1(Gal(F¯ /F , Aut(V )) = #Xh | q q | #G Fr h G X∈ 1 #Xh ≤ #G h G X∈ = conjugacy classes of Aut(V ) . |{ }|

1.4. How to recognise the 1-cocycle of a form. Our calculation of the forms of a curve is essentialy a computation of representatives of 1-cocycles. It is therefore interesting to ask the question: suppose we have a plane curve C and we know its equation, then given an equation of a form of C can we work out a representative of the 1-cocycle? We include an answer here, not because we use it later on, but because it might be of interest to the reader.

The basic method is this: suppose given two homogenous degree d polynomials i j l i j l f = ΣaijlX Y Z and g = ΣbijlX Y Z such that g is a form of f. In practice, to know that g is a form of f is equivalent to having polynomials φi(X,Y,Z) and s ψi(X,Y,Z) with coeﬃcients in k such that the association

(X : Y : Z) (φ (X,Y,Z) : φ (X,Y,Z) : φ (X,Y,Z)) 7→ 1 2 3 takes the zero locus of f to that of g and

(ψ (φ (X,Y,Z), φ (X,Y,Z), φ (X,Y,Z)) : : ) = (X : Y : Z). 1 1 2 3 · · · · · · The polynomials φi and ψi induce an isomorphism between the function ﬁelds ks(X)[Y ]/(f(X,Y, 1)) and ks(X)[Y ]/(g(X,Y, 1)).

The Galois group Γ = Gal(ks/k) acts on both ﬁelds by acting on the coeﬃcients of any expression. Then, the 1-cocycle associated to g relative to f is given by the map

−1 σ (σ ψ (σφ (X,Y,Z),σ φ (X,Y,Z),σ φ (X,Y,Z)) : : ), 7→ 1 1 2 3 · · · · · · for σ Γ. ∈ David Kohel pointed out to the author that Hess [14] has a given an algorithm for determining an isomorphism between forms.

1.5. Splitting the Jacobian of a curve. Suppose that C is a smooth genus g curve with large automorphism group Aut(C). It often turns out that the Ja- cobian JC of C is isogenous to a product of lower dimensional Abelian varieties. Indeed, this occurs in the cases we consider in this chapter. We now give a heuristic explanation of this phenomenon although we do need this later.

The diﬀerentials Ω1(C) give a representation of Aut(C), and hence a g dimensional representation of the group algebra k[Aut(C)] of Aut(C). If there are orthogonal idempotents in k[Aut(C)], then the factors of Ω1(C) corresponding to these idempotents yield Abelian sub-varieties of Jac(C).

This is more fully discussed and explained in the article [18]. 34 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

2. The points on a form From now on we assume that our variety V is in fact a smooth curve C over a finite field k. Under this assumption we give explicit methods for computing the number of points on a form of C assuming the geometric Frobenius of the Jacobian of C is known. 2.1. The formula of Weil. Let p be a prime number and let q = pn. Let C be a smooth genus g curve over a finite field Fq of cardinality q. Let F be the geometric Frobenius endomorphism of C. On the function field K = Fq(C), F yields the map F ∗ given by

q F ∗ : K K : f f . −→ 7→ Furthermore, F induces a morphism J(F ) on the Jacobian JC of C which also raises functions of JC to their qth powers. That is, J(F ) is the geometric Frobenius endomorphism of JC . From now on we will write π for J(F ).

Since π is an endomorphism of a g dimensional Abelian variety it is a zero of a unique monic polynomial f Z[X] of degree 2g ∈ 2g 2g 1 f(X)= X tr(π)X − + + c. − · · · We call f the characteristic polynomial of Frobenius and tr(π) the trace of Frobe- nius.

The Weil conjectures imply the following ﬁxed point formula for the number of points of C over Fq:

#C(F )= q +1 tr(π ). q − C The theory of Honda and Tate tells us that isogenous Abelian varieties have equal characteristic polynomials of Frobenius and therefore equal traces of Frobenius. Following section 1.5, it is often the case that a curve with large automorphism group has a Jacobian which is isogenous to a product of lower dimensional Abelian varieties. Then the characteristic polynomial of Frobenius of the Jacobian is the product of the characteristic polynomials of Frobenius of the factors and hence the trace of Frobenius is just the sum of the traces of the Frobeniuses of the lower dimensional factors. We will make crucial use of this phenomenon in our computa- tions of the forms of the symmetric quartics in section 3.

2.2. Frobenius of a form. Consider a ﬁnite extension Fqm of Fq. Let C′ be a form of C so that we have an isomorphism φ : C F m C′ F m . Let π de- ⊗ q −→ ⊗ q note the geometric Frobenius endomorphism of JC and π′ the geometric Frobenius endomorphism of JC′ .

We recall from section 1.2 that a form C′ of C determines a twisted action of the Galois group Γ = Gal(F¯ /F ) on C F¯ and that the quotient of C F¯ by this q q ⊗ q ⊗ q action is in fact the form C′. If aσ is the 1-cocycle corresponding to C′ then σ acts 1 ‘twistedly’ on CF¯ as a σ− . This induces a twisted action of Γ on J F¯ via q σ × C ⊗ q

J(a ) F σ. σ × q Since π =1 F Fr we have × q

π′ = J(a ) π . Fr ◦ C Hence 3. THE ZOO OF SYMMETRIC QUARTICS 35

tr(π′) = tr(J(a ) π). Fr ◦ As we have remarked in section 1.5, JC is often isogenous to a product of lower dimensional Abelian varieties, and in all the cases we consider the lower dimensional Abelian varieties turn out to be elliptic curves. We therefore have J(aFr) as an element of a direct sum of matrix algebras over a quadratic imaginary ﬁeld or a quarternion algebra. The trace of J(aFr) can be given in terms of an embedding of the group algebra of Aut(C) into the endomorphism algebra of JC and the trace of the corresponding representation. This leads to closed formulas for tr(πC′ ) using the characters of Aut(C); for example, as we will see, this is the case with the Fermat quartic.

3. The zoo of symmetric quartics We now compute the forms and the corresponding Frobenius endomorphisms for two families of plane quartics, namely:

4 4 4 2 2 Ca := X + Y + Z + aX Y =0 and

4 4 4 2 2 2 2 2 2 Da := X + Y + Z + a(X Y + Y Z + Z X )=0. With the exception of part 6, we exclude characteristic 2; for otherwise the equations above deﬁne a non-reduced curve. In part 6 we study the forms of the Klein quartic with reference to a model which is deﬁned in characteristic 2.

For each family we first describe the automorphism group of a generic fibre. We then give a description of the Jacobian of the generic fibre. We compute the forms of the curve as 1-cocycles whose value of Frobenius is given by a certain element of the automorphism group of the curve. This calculation can be used to obtain the number of points on each form for a fixed fibre. Finally we combine the calculation of the 1-cocycles with information about how the automorphism group acts on the Jacobian’s isogeny factors to compute the trace of Frobenius of the forms in terms of the trace of Frobenius of the original quartic.

Both these families have the Fermat quartic as their fibre for a = 0; the family Da has the Klein quartic as its fibre over a = 3( 1+√ 7)/2. We pay special attention to the Fermat and Klein quartics, in particular− we− compute the number of points on them for small fields. Away from characteristic 3, the fibre of Ca at a =2ζ3 +1 (where ζ3 is a primitive third root of unity in the separable closure of the field of definition) is also unusual in that it has 48 automorphisms; but we have not made any attempt to compute twists for this curve.

We note, just for fun, that the Klein and Fermat quartics have extremal properties with respect to flex and hyperflex points. A flex of a smooth quartic is a zero of the quartic whose tangent line intersects the quartic with multiplicity 3; a hyperflex is a zero of the quartic whose tangent line intersects the quartic with multiplicity 4. The number of flexes plus twice the number of hyperflexes is independent of the quartic and is equal to 24. It turns out that the Klein quartic has no hyperflexes whereas the Fermat quartic has the maximal number of hyperflexes. More concep- tually, flexes and hyperflexes are Weierstrass points: these are points which appear in some representative of the canonical class with multiplicity larger than 2. 36 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

Also the Klein and Fermat quartics are both modular curves and the Klein is in fact a ﬁne moduli space. We describe the moduli theoretic meaning of the twists of the Klein in section 6.

The families Ca and Da make up a complete list of plane quartics with more than 15 automorphisms for characteristic 0 and characteristic p > 3+1 = 4. For characteristic 0 this can be seen by the theorem of Hurwitz [29] p82 and the list of Henn (see for example Vermeulen [57] p63-64); for characteristic p> 4 Theorem 1 of Roquette [43] shows that results for characteristic 0 remain valid in characteristic p.

4 4 4 2 2 3.1. The family Ca := X + Y + Z + aX Y = 0. Here a is any element of the ﬁeld k other than 2, in which case the curve is singular.

3.1.1. The automorphisms of Ca. For a = 0 the curve has 96 automorphisms, and in characteristic not 3 for a = (2ζ3 + 1) (with ζ3 a primitive third root of unity) the curve has 48 automorphisms;± we treat the ﬁbre a = 0 in 3.2.

Let DH4 denote the dihedral group of order 8 - i.e. the symmetry group of a square in the Euclidean plane. This group has presentation

DH = s,h s2 = (sh)4 = (shs)4 . 4 h | i For each a ks 0, 2ζ +1, 2 the automorphism group of C is a non-split ∈ −{ 3 } a extension of (Z/2) by DH4, i.e.

1 / DH4 / Aut(Ca) / (Z/2) / 0 .

Rather than give a presentation of Aut(Ca), we give a faithful representation φ of Aut(Ca). Let g be any element mapping to 1 in (Z/2). We deﬁne φ by

i 0 φ(g) = , 0 i 0 1 φ(h) = , 1 0 1 0 φ(s) = . −0 1 Adding φ to the trivial representation χ1 we obtain the three dimensional representation ρ = φ χ given by ⊕ 1

i 0 0 ρ(g) = 0 i 0 ,  0 0 1   0 1 0  ρ(h) = 1 0 0 ,  0 0 1   1 0 0 − ρ(s) = 0 10 .  0 01    Let X, Y and Z denote the linear functionals on (ks)3 dual to the vectors (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively. Then under the action of Aut(Ca), ρ leaves the symmetric quartic 3. THE ZOO OF SYMMETRIC QUARTICS 37

X4 + Y 4 + Z4 + aX2Y 2 invariant.

The centre of Aut(Ca) is generated by g. The inner automorphism group Inn(Aut(Ca)) 2 of Aut(Ca) is therefore isomorphic to the Klein Vierergruppe (Z/2) .

Let S3 denote the symmetric group on 3 letters. The outer automorphism group of Aut(C) is of order 12 and isomorphic to (Z/2) S3; the (Z/2) factor is generated by the automorphism ×

(g,h,s)Inn(Aut(C )) (g3,h,s)Inn(Aut(C )) a 7→ a and the S3 factor is generated by the following two automorphisms

(g,h,s)Inn(Aut(C )) (g,hsh,g2h)Inn(Aut(C )), a 7→ a (g,h,s)Inn(Aut(C )) (g,h,gsh)Inn(Aut(C )). a 7→ a The group Aut(C) has 7 subgroups of order 8 each isomorphic with either DH4 or the quaternion group Q or (Z/2) (Z/4). 8 ×

We now give the character table of Aut(Ca). The top row gives representatives of the conjugacy clases.

rep 1 g2 gsh s h g3 g sh gs gh χ1 11 111 1 1111 χ2 1 1 -1 1 1 -1 -1 1-1 -1 χ3 1 1 -1-1-1 -1 -1 1 1 1 χ4 1 1 1-1-1 1 1 1-1 -1 χ5 1 1 1-1 1 -1 -1 -1 1 -1 χ6 1 1 1 1-1 -1 -1 -1-1 1 χ7 1 1 -1-1 1 1 1-1-1 1 χ8 1 1 -1 1-1 1 1-1 1 -1 tr(χ2φ) 2-2 000 2i 2i 0 0 0 tr(φ) 2-2 0 0 0 2i −2i 0 0 0 −

Figure 1. Character table of Aut(Ca).

There are 10 conjugacy classes; all with 2 elements except those represented by an element of the centre. There are 7 non-trivial representations of dimension 1 given by taking the quotient of Aut(Ca) by the 7 normal subgroups of order 8.

Thus the 1 dimensional representations form a group isomorphic with (Z/2)3, which acts transitively on the set of isomorphism classes of the 2 dimensional irreducible representations [φ], [χ φ] . { 2 ⊗ }

3.1.2. The Jacobian of Ca. The Jacobian of Ca is isogenous to the following product of elliptic curves

E E E′ ; a × a × a here Ea is the curve with model

a2 y2 = x3 + ( 1)x 4 − 38 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS and automorphism group µ4 generated by i (y, x) := (iy, x); Ea′ is the curve with model · −

y2 = x3 ax2 + x. − The isogeny is given as follows: the quotient of Ca by the involution s : X X 1 7→ − or its conjugate hsh− : Y Y has model 7→ − y2 +1+ x4 + ax2y =0 and Jacobian isogenous to Ea. Whereas the quotient of Ca by the involution g2 : (X, Y ) ( X, Y ) has model 7→ − − x4 + y2 +1+ ax2 =0 and Jacobian isogenous to Ea′ . Let φj : C C/ j denote the quotient morphism 2 −→ h i for j g ,s,hsh ; and let γj denote the isogeny from Ea respectively Ea′ to the Jacobian∈ { of C/ j }. We then have a morphism h i

J(φ ) γ + J(φ ) γ + J(φ ) γ : E E E′ J . 1 ◦ 1 2 ◦ 2 3 ◦ 3 a × a × a −→ C To check this is an isogeny it suffices to show that the induced map on the differentials has no kernel. One way of seeing this is to show that the map on differentials is in fact an isomorphism of representations of Aut(C) up to projective equivalence. In fact lifting the invariant differential of Ea′ to C gives the one-dimensional representation χ5 of Aut(C). Lifting the invariant differential of Ea via φs and φhsh−1 gives the irreducible 2 dimensional representation φ of Aut(C). The first lift of the invariant differential of Ea corresponds to a 1 dimensional representation of the subgroup

g,s h i and the second lift to a 1 dimensional representation of the subgroup

g,hsh . h i 3.1.3. Geometric Frobenius of J . Since J is isogenous to E E E′ , the C C a × a × a endomorphism algebra End(JC ) Q of JC equals End(Ea Ea Ea′ ) Q, which is ⊗ × × ⊗

[Mat (End(E )) End(E′ )] Q, 2 a ⊕ a ⊗ if Ea′ is not isogenous to Ea; otherwise it is

Mat (End(E )) Q. 3 a ⊗ In particular the geometric Frobenius π of JC is Q conjugate to the matrix

πa 0 0 0 π 0  a  0 0 πa′   where πa and πa′ are the geometric Frobeniuses of Ea and Ea′ respectively.

Since the trace of Frobenius depends only upon it’s Q conjugacy class we can work with the matrix above rather than the geometric Frobenius π. 3. THE ZOO OF SYMMETRIC QUARTICS 39

3.1.4. Action of Aut(C ) on the Jacobian. The action of Aut(C) on E E E′ a a × a × a is given by a morphism from Aut(C ) into the unit group of M (End(E )) End(E′ ). a 2 a ⊕ a This turns out to give a three dimensional representation of Aut(Ca) isomorphic with χ φ χ . To check this, we compute the action of Aut(C ) on the lifts of 2 ⊗ ⊕ 2 a the invariant diﬀerentials coming from the two copies of Ea and Ea′ .

3.1.5. Forms of C . There are two cases: k = F with q 1 mod 4 and q 3 a q ≡ ≡ mod 4. In the ﬁrst case Γ = Gal(F¯ q/Fq) acts trivially on Aut(C). In the second case the Frobenius automorphism Fr acts one the Aut(Ca) through the image ρ, where it acts by conjugation on i.

So for q 1 mod 4 we have by section 1.2 ≡ H1(Γ, Aut(C)) = conjugacy classes of Aut(C) , { } and there are exactly 10 conjugacy classes of Aut(C) with representatives given in the following ﬁgure:

rep 1 g2 gsh s h g3 g sh gs gh

Figure 2. Representatives of 1-cocycles of Aut(C) when q 1mod 4 ≡ For q 3mod 4 there are 8 elements of H1(Γ, Aut(C)), with representatives as given in≡ the following ﬁgure:

rep 1 gsh s h g sh gs gh

Figure 3. Representatives of 1-cocycles of Aut(C) when q 3mod 4 ≡

This computation was made by ﬁrst computing Aut(C)/ 0; most of Aut(C)/ 0 was deducible from knowledge of the conjugacy classes of∼ Aut(C). We therefore∼ have: Proposition 4. For all odd integral prime powers q Figures 2 and 3 give a complete 1 list of representatives of 1-cocycles of in H (Gal(F¯ q/Fq), Aut(C)).

3.1.6. Forms of Frobenius. Each of the representatives gives aFr for some 1 - cocycle a . Let A [M (End(E )) End(E′ )] Q denote the automorphism of σ Fr ∈ 2 a ⊕ a ⊗ JC induced by aFr.

We can compute AFr as an element of [M2(End(Ea)) End(Ea′ )] Q by using the representation φ χ of section 3.1.4. Indeed we have⊕ ⊗ ⊕ 5 A = (φ χ )(a ). Fr ⊕ 5 Fr Up to Q conjugation, the geometric Frobenius of Jac(C) has the form

πa 0 0 0 π 0  a  0 0 πa′ as an element of [M (End(E )) End(E′ )] Q. From 2.2, we therefore deduce 2 a ⊕ a ⊗ that the geometric Frobenius corresponding to the twist of C given by aσ has the form

φ(a )π 0 Fr a . 0 χ (a )π′ 5 Fr a 40 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

2 If q 1 mod 4 then πa = c + di for some c, d Z. The following are all the traces of geometric≡ Frobenius for forms of C: ∈

rep 1 g2 gsh s h tr(tr(ρ(rep))π ) + tr(χ (rep)π′ ) 4c + tr(π′ ) 4c + tr(π′ ) tr(π′ ) tr(π′ ) tr(π′ ) a 5 a a − a a − a a

rep g3 g sh gs gh. tr(tr(ρ(rep))π ) + tr(χ (rep)π′ ) 4d tr(π′ ) 4d tr(π′ ) tr(π′ ) tr(π′ ) tr(π′ ) a 5 a − − a − a − a a − a

Figure 4. Traces of Frobenius for twists of Ca corresponding to 1-cocyles as computed in 3.1.5 when q 1 mod 4. ≡

If q 3 mod 4 then E is super-singular and End(E ) is a quaternion ring and ≡ a a iπa = iπa which implies that tr(πa) = 0. Hence the only traces of geometric Frobenius− are

rep 1 gsh s h . tr(tr(ρ(rep))π ) + tr(χ (rep)π′ ) tr(π′ ) tr(π′ ) tr(π′ ) tr(π′ ) a 5 a a a − a a

rep g sh gs gh. tr(tr(ρ(rep))π ) + tr(χ (rep)π′ ) tr(π′ ) tr(π′ ) tr(π′ ) tr(π′ ) a 5 a − a − a a − a

Figure 5. Traces of Frobenius for twists of Ca corresponding to 1-cocyles as computed in 3.1.5 when q 3 mod 4. ≡

Proposition 5. For a = 0, 2ζ +1, 2, then Figures 4 and 5 give a complete list 6 3 of traces of Frobenius for twists of Ca corresponding to the representatives of the 1-cocycles as in Figures 2 and 3 of Proposition 4.

4 4 4 3.2. The Fermat Quartic C = C0 := X + Y + Z . 3.2.1. Aut(C). Let µ4 be the group of 4th roots of unity and let S3 be the symmetric group acting on 3 letters.

Abstractly, the automorphism group of C is

3 (µ4/µ4) ⋊ S3, 3 3 here we consider µ4 as diagonally embedded into µ4 and S3 as acting on µ4 by permutation of coordinates. We will identify an element of Aut(C) with any of the representatives of its coset.

The group Aut(C) is generated by the cosets of µ4 with representatives

2 It is not clear if we have chosen i or −i in the endomorphism ring of Ea. But here a small miracle occurs: whenever iπa makes a contribution to the trace of one form −iπa makes a contribution in the same way to the trace of some other form. Thus we are not forced to resolve this ambiguity. In fact this miracle is forced on us: if we could determine d exactly, then we would be able to distinguish between i and its conjugate! This explains the symmetry in the traces with respect to d. 3. THE ZOO OF SYMMETRIC QUARTICS 41

g := ((i, 1, 1), Id)), h := ((1, 1, 1), (12)), s := ((1, 1, 1), (123)).

The action of Aut(C) on C0 is given by the formula

g(x : y : z) := (ix : y : z), h(x : y : z) := (y : x : z), s(x : y : z) := (z : x : y)

The group Aut(C) has 96 elements and 10 conjugacy classes. Conjugation yields an involution on µ4 which induces an involution on Aut(C) and a Magma calculation shows that this is the only outer automorphism of Aut(C). A Magma calculation also shows that Aut(C) has 24 subgroups and 5 normal subgroups. In particular it contains two copies of Aut(Ca) (deﬁned in 3.1.1) which are taken to each other by conjugation by (1, (23)).

To compute the representations of Aut(C) we use the method of “little groups” of Wigner and Mackey [53] p 62. Each representation is obtained in the following way: 3 first compute representatives φi for the orbits of Hom(µ4/µ4, C) under the action of S . For each φ let G be the subgroup S fixing φ . Let ρ be the set of 3 i i 3 i { i,j } irreducible representations of Gi. Then each irreducible representation of Aut(C) is obtained uniquely as an induced representation of (µ3/µ )⋉G of the form φ ρ . 4 4 i i ⊗ i,j 3 It is easy to see that there are 5 orbits of S3 acting on Hom(µ4/µ4, C). Indeed, if we make a homomorphic identification

3 3 (Z/4) /(Z/4) = Hom(µ4/µ4, C) then we have the following representatives of the orbits:

(0, 0, 0), (1, 0, 0), (1, 3, 3), (0, 0, 1), (1, 0, 2), which are ﬁxed respectively by the following subgroups of S3

S3, (23) , (23) , (12) , 1. 3 { } { } { } The index of (µ4/µ4) ⋉ Gi in Aut(C) is easily seen to be #(S3/Gi), and so the dimension of the representation induced from ψ ρ is i ⊗ i,j #(S /G ) tr(ρ (1)). 3 i × i,j The character table of Aut(C) is given in Figure 6. 3.2.2. The Jacobian of JC . Since C0 is in the family Ca (see 3.1 for a deﬁnition) we know its Jacobian is

E E E′ . 0 × 0 × 0 Here E0′ is

y2 = x3 + x, which is of course the curve E0. 42 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

rep 1 ( (1,-1,-1),1) ( (1,-1,-1), (12) ) (1, (123)) ((i,1,1),1) χ1 11 11 1 χ2 1 1 -1 1 1 χ3 2 2 0 -1 2 χ4 3 3 -1 0 -1 χ5 3 3 1 0 -1 χ6 3 -1 1 0 -1-2i χ7 3 -1 1 0 -1+2i χ8 3 -1 -1 0 -1+2i χ9 3 -1 -1 0 -1-2i χ10 6 -2 0 0 2

rep ((i,-1,-1),1) ( (i,1,-1),1) ((1,1,i), (12)) ((i,1,1), (12)) ((i,-1,-1), (12)) χ1 11 11 1 χ2 1 1 -1 -1 -1 χ3 22 00 0 χ4 -1 -1 -1 1 1 χ5 -1 -1 1 -1 -1 χ6 -1+2i 1 -1 -i i χ7 -1-2i 1 -1 i -i χ8 -1-2i 1 1 -i i χ9 -1+2i 1 1 i -i χ10 2 -2 0 0 0

Figure 6. Character table of Aut(C)

Hence J is isogenous to E E E . C 0 × 0 × 0 3.2.3. Geometric Frobenius of J . Since J is isogenous to E E E the C C 0 × 0 × 0 endomorphism algebra of JC is

Q M (End(E )) ⊗ 3 0 and the Frobenius endomorphism of JC is Q conjugate to a matrix of the form

π0 0 0 0 π 0 ,  0  0 0 π0   where π0 is the geometric Frobenius endomorphism of E0.

3.2.4. Action of Aut(C) on the Jacobian. The automorphism group Aut(C) acts on E E E through the representation with character χ . This is calcu- 0 × 0 × 0 7 lated by comparing with the action of Aut(Ca) on its Jacobian.

3.2.5. Forms of C. For q 1mod 4 there are 10 forms with representatives as in Figure 7. ≡

For q 3 mod 4 : there are 6 forms with representatives as in Figure 8. ≡

Proposition 6. Figures 7 and 8 give a complete list of representatives of 1-cocycles 1 in H (Gal(F¯ q/Fq), Aut(C)) for all odd integral prime powers q. 3. THE ZOO OF SYMMETRIC QUARTICS 43

rep 1 ( (1,-1,-1),1) ( (1,-1,-1), (12) ) (1, (123)) ((i,1,1),1) # rep 1 3 12 32 3 h repi ((i,-1,-1),1) ( (i,1,-1),1) ((i,1,1), (12)) ((i,-1,-1), (12)) ((-i,i,1), (12)) # rep 3 6 12 12 12 h i

Figure 7. Forms of Fermat quartic over Fq as given by 1-cocycle representatives when q 1 mod 4. ≡ rep 1 ( (1,-1,-1), (12) ) ((i,1,1),1) (1,(13)) ((i,1,1), (12)) (1, (123)) # rep 4 12 12 12 24 32 h i

Figure 8. Forms of Fermat quartic over Fq as given by 1-cocycle representatives when q 3 mod 4. ≡ 3.2.6. Forms of Frobenius. To compute the trace of Frobenius we do the following: let φ be the representation with character χ7. Let aσ be the 1-cocycle corresponding to a twist Ctw of C. Then if π is the geometric Frobenius of Jac(C), the geometric Frobenius of Jac(Ctw) is

tw π = φ(aFr)π. Thus

tw tr(π ) = tr(χ7(aFr)π). We write π = a + bi End(Jac(C)), then the possible traces of Frobenius are: For q 1 mod 4: we∈ may write π = a + bi and then the possible traces of twists are as≡ in Figure 9.

rep 1 ( (1,-1,-1),1) ( (1,-1,-1), (12) ) (1, (123)) ((i,1,1),1) # rep 6a -2a 2a 0 -2a-4b h repi ((i,-1,-1),1) ( (i,1,-1),1) ((i,1,1), (12)) ((i,-1,-1), (12)) ((-i,i,1), (12)) # rep -2a+4b 2a -2a -2b 2b h i Figure 9. Traces Frobenius of forms of the Fermat quartic represented by 1-cocycles over F with q 1 mod 4. q ≡ For q 3mod 4. In this case E is supersingular and q is not a square so tr(π)=0. ≡ Thus all models of the Fermat over Fq have q + 1 points.

From these calculations we deduce that a form of the Fermat quartic attains the Hasse-Weil-Serre upper bound if and only if it is isomorphic to the untwisted Fer- mat quartic. Moreover, the Fermat quartic attains Hasse-Weil-Serre upperbound over F if q = pn with n 2 mod 4 and p 3 mod 4. q ≡ ≡ The Fermat quartic also attains the Hasse-Weil-Serre upper bound for some non- square values of q as we will see in section 4.

Proposition 7. The forms of the Fermat quartic over Fq have traces as given in Figure 9 when q 1mod 4 and trace 0 when q 3mod 4. ≡ ≡ 4 4 4 2 2 2 2 2 2 3.3. The family Da := X + Y + Z + a(X Y + Y Z + Z X )=0. This family also contains the Fermat quartic for a = 0. This fact allows us to exploit our previous calculation for the Fermat quartic considerably. The curve Da is singular at a = 2, 1. ± − 44 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

3.3.1. The automorphisms of Da. Abstractly Aut(Da) is the symmetric group on 4 letters S4.

The character table of S4 is

rep 1 (12)(34) (12) (123) (1234) χ1 1 111 1 χ2 1 1-1 1 -1 χ3 2 20 -1 0 χ4 3 -1-1 0 1 χ5 3 -1 1 0 -1

Figure 10. Character Table for S4.

Aut(Da) acts on Da through the representation ρ with character χ5. The representation ρ is given by

0 0 1 ρ((123)) = 1 0 0  0 1 0   0 1 0 ρ((1234)) = 1− 0 0 .  0 0 1    3.3.2. The Jacobian of Da. Consider the involutions φ1 : X X, φ2 : Y Y and φ : Z Z. The quotient D / φ by each of these involutions7→ − has a Jacobian7→ − 3 7→ − a h ii which up to isogeny is the elliptic curve Ea

y2 = (a2 4)x3 + (2a2 4a)x2 + (a2 4)x; − − − Let qi : Da Da/ φi denote the quotient by φi morphism, and let γi denote the isogeny between−→ Eh andi the Jacobian of D / φ . We then have a morphism a a h ii J(q ) γ + J(q ) γ + J(q ) γ : E E E J ; 1 ◦ 1 2 ◦ 2 3 ◦ 3 a × a × a −→ C since this is an isogeny for a = 0 it is an isogeny generically. Alternatively, this can be seen by pulling back the diﬀerentials from the various Ea factors to Da and showing that the pulledback diﬀerentials are linearly independent.

3.3.3. Geometric Frobenius. Again the geometric Frobenius is conjugate to the matrix

πa 0 0 0 π 0  a  0 0 πa in M (End(E )) Q.   3 a ⊗ 3.3.4. Action of the automorphims on Jacobian. An explicit calculation shows that the embedding

Aut(D ) M (End(E )) a ⊂ 3 a gives the representation χ4 of Aut(Da). 3. THE ZOO OF SYMMETRIC QUARTICS 45

3.3.5. Forms of Da. The forms of Da correspond to the 5 conjugacy classes of S4:

rep 1 (12) (12)(34) (123) (1234) # rep 16 64 3 h i Figure 11. Forms of D (for a = 0) as represented by conjugacy a 6 classes of S4.

Proposition 8. Figure 11 gives a complete list of 1-cocycle representatives for forms of D over F for a =0, 2, 1. a q 6 ± − tw 3.3.6. Forms of Frobenius. Let aσ be a 1 cocycle corresponding to a twist Da of Da. Let ρ′ be the representation of Aut(Da) with character χ4. Let π be the tw geometric Frobenius of Ea. Then the geometric Frobenius of Jac(Da ) is

ρ′(aFr)π. Thus the possible traces of Frobenius are

rep 1 (12) (12)(34) (123) (1234) # rep 3tr(π) -tr(π) -tr(π) 0 tr(π) h i

Figure 12. Traces of Frobenius of forms of Da over Fq for generic a.

Proposition 9. The forms of Da over Fq have traces as given in Figure 12 when a =0, 2, 1. 6 ± −

It is interesting to ask if Da yields curves with many points. From our table we see this happens only for the untwisted Da. In fact we can show that the family of elliptic curves

2 2 3 2 2 2 E′ := (a + 2)y = (a 4)x + (2a 4a)x + (a 4)x a − − − is isogenous to the Legendre family

E := y2 = x(x 1)(x λ). λ − − Moreover, an elliptic curve E is a member of the Legendre family if and only if E has rational 2 torsion and no 4 torsion element P has P in its Galois orbit. Such − elliptic curves Eλ can be constructed with a very large and negative c - within 7 of [2√q]. Thus when Eλ has the form Ea, we can construct a curve whose number of points is within 21 of the Hasse-Weil upperbound; in any case we always get a curve whose number of points is either within 21 of the upperbound or within 21 of the lowerbound. This fact is due to Roland Auer and Jaap Top [1].

3.4. The Klein quartic X := X3Y + Y 3Z + Z3X. 46 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

3.4.1. Notation. We have tried to keep our notation consistent with Elkies’ ex- cellent survey [9]. The only exception is that we write F for the ﬁeld Q(√ 7) instead of k. −

However our notation for groups, rings, and their elements is identical with Elkies’.

We will write for the ring of integers of F . OF We will write ζ Q¯ for an arbitrary but fixed primitive 7th root of unity and put ∈ 1+ √ 7 α := ζ + ζ2 + ζ4 = − − . 2 We will always work with curves defined over a finite field of characteristic = 7. 6 In particular, we will abuse notation and write ζ and α for the image of ζ and α in any Z[ζ] algebra. For example a finite field Fp contains α if and only if p 1, 2, 4 mod 7. ≡

3.4.2. The group G168. The group G168 is the unique simple group with 168 elements. It has three generators g,h and s of orders 7, 3 and 2 respectively. Rather than give the relations amongst g,h and s we give their images under a faithful representation.

The representation:

ρ : G SL (Q¯ ), 168 −→ 3 is given by

ζ4 0 0 ρ(g) := 0 ζ2 0 ,  0 0 ζ    0 1 0 ρ(h) := 0 0 1 ,  1 0 0  and  

ζ ζ6 ζ2 ζ5 ζ4 ζ3 1 − − − ρ(s) := − ζ2 ζ5 ζ4 ζ3 ζ ζ6 . √ 7  ζ4 − ζ3 ζ −ζ6 ζ2− ζ5  − − − − The representation ρ acts on Q¯[X,Y,Z] on the left and the Klein quartic

X3Y + Y 3Z + Z3X is invariant under this action.

Consider the isomorphism

φ : G PSL (F ), 168 −→ 2 7 given by

1 1 φ(g)= 1 , ± 0 1 3. THE ZOO OF SYMMETRIC QUARTICS 47

2 0 φ(h)= 1 , ± 0 4 and

0 1 φ(s)= 1 − . ± 1 0 Realised as PSL2(F7) the group G168 has automorphism group PGL2(F7) and we have the following short exact sequence

det 2 1 / PSL2(F7) / PGL2(F7) / F7∗/(F7∗ ) / 1.

Using this realisation as PSL2(F7) we can show that every element of G168 can be written uniquely as either hmgn or hmgnsgl with m =0, 1, 2 and n,l =0, 1, 2, 3, 4, 5, 6. Alternatively we can use Magma to show that every element of G168 can be written in this way.

The character table over Q¯ for G168 is as follows:

c 1A 2A 3A 4A 7A 7B # c 1 21 56 42 24 24 χ1 111111 χ3 3 -1 0 1 α α¯ χˆ3 3 -1 0 1α ¯ α χ6 6 2 0 0 -1 -1 χ7 7 -1 1 -1 0 0 χ8 8 0-1 0 1 1

Figure 13. Character table for the simple group with 168 elements.

Here c is the name of a conjugacy class. The digit in the name represents the order of any element of that class. The classes 7A and 7B are permuted by any outer automorphism. We use χi to represent a character of degree i: the character of ρ is thus χ3.

3.4.3. Jacobian of X. The action of G168 on X is given in 3.4.2. Consider the 1 CM elliptic curve EF over Z[ 7 ] with Weierstrass form

y2 + yx = x3 +5x2 +7x.

The Endomorphism ring of EF is F . Let k be the ﬁeld of deﬁnition of X. Using the tensor notation3 of Serre ([51],O [6], also see [28] for a variation involving Galois twists) we may write

Jac(X) k(ζ) = (E k(ζ)) L ×k ∼ F ×k ⊗ where L is the lattice OF 3Let G be a commutative group scheme with endomorphism ring R and let M be an R module of finite presentation. Then G defines a functor from schemes to R modules and M defines a constant functor from schemes to R modules. It makes sense to tensor these functors and ask if the result is representable. In our case G = EF ×k k(ζ) and M = L. It is easy to see that G ⊗ M is representable in this case by tensoring a finite presentation of M with G and using the fact that the cokernel of a morphism of Abelian varieties is representable. 48 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

1 (x,y,z) 3 y 2x, z 4x (√ 7) and x +2y +4z (7) . √ 7{ ∈ OF | − − ∈ − ∈ } − Here (√ 7) and (7) are considered as ideals of and as embedded into Q¯ . In − OF OF these coordinates G168 acts on L via ρ and this action induces the action on Jac(X).

To obtain a model for the Jacobian of X over k we make the following observations. Over k(ζ) we may make the coordinate change

1 1+ ζα ζ2 + ζ6 1+ ζα ζ2 + ζ6 1 .  ζ2 + ζ6 1 1+ ζα 

Then the Klein quartic takes the form D3α. We have seen that the Jacobian of D3α is isogenous to a third power of EF . Moreover X is a cubic twist of D3α over the ﬁeld extension k(α, ζ)/k(α), and we now describe the 1-cocycle corresponding to this twist. The Galois group of k(α, ζ)/k(α) is isomorphic with (Z/3) by identifying the ﬁeld automorphism ζ ζ2m with m mod 3. Then the 1-cocycle is given by 7→ 1 h. 7→ Then the geometric Frobenius of Jac(X) is represented by the matrix

π 0 0 ho(q) 0 π 0 ,  0 0 π  where o(q) is deﬁned by the rule  

0 if q 1, 6 mod 7 ≡ o(q)= 1 if q 2, 3 mod 7  ≡ 2 otherwise. ¯ 3.4.4. Galois action on G168over ﬁnite ﬁelds. The action of Gal(Fq/Fq) on the primitive 7th root of unity ζ and thus on G168 depends only on q mod 7. As noted before we exclude q 0 mod 7. ≡

To specify the action of the Galois group on G168 we introduce some notation: Since q is an integer not divisible by 7 there is a unique element x 1, 2, 3, 4, 5, 6 so that ∈{ }

q x mod 7. ≡ We will write q mod 7 for this unique integer. h i We then deﬁne

q mod 7 if q mod 7 < 4, q := h i h i | | 7 q otherwise. ( − Thus for example 2 = 5 = 2 and 1 = 6 = 1. | | | | | | | |

The action of Gal(F¯ q/Fq) is then given on the generators of G168 by the formulas

q q Fr(g)= g , Fr(h)= h and Fr(s)= h| |s. 3. THE ZOO OF SYMMETRIC QUARTICS 49

1 3.4.5. Representatives of 1-cocycles. In this section we describe H (Gal(F¯ q/Fq), G168). See section 1.3 for an explanation of the method. The description is as follows:

For each value of q modulo 7 we give representatives of the equivalence classes in G168/ (see 1.3) and underneath it, we give the number of elements in the given equivalence∼ class. Then for each equivalence class [x], we count the number of 1-cocycles f such that f(Fr) [x]. This often turns out to be zero! When it is non-zero the representative given∈ actually yields a 1-cocycle.

In general there are always between 4 and 6 equivalence classes.

rep 1 s hh2g3sg6 g g6 # rep 1 21 56 42 24 24 # 1-cocyclesh i 1212 000

Figure 14. Forms of the Klein quartic over Fq with q 1mod 7 as given by representatives of the 1-cocycles. The second≡ row gives the number of elements in a twisted conjugacy class with representative in the ﬁrst row of the same column. The third row gives the number of elements in the twisted conjugacy class which actually yield a 1-cocycle.

Remark: In this case the Galois action is trivial, and thus we are merely computing the conjugacy classes of G168.

rep 1 hg6sg2 h2g5 hg2sg h2g4 h2 # rep 56 21 24 42 24 1 # 1-cocyclesh i 2 66 061

Figure 15. Forms of the Klein quartic over Fq with q 2mod 7 as given by representatives of the 1-cocycles. ≡

rep 1 g2sg6 h2g3sg6 hg2sg3 # rep 56 28 42 42 # 1-cocyclesh i 27 0 0

Figure 16. Forms of the Klein quartic over Fq with q 3mod 7 as given by representatives of the 1-cocycles. ≡

rep 1 s h h2g3sg6 hsg4 hg5s # rep 56211 24 42 24 # 1-cocyclesh i 201 0 0 0

Figure 17. Forms of the Klein quartic over Fq with q 4mod 7 as given by representatives of the 1-cocycles. ≡

Proposition 10. Figures 14 to 19 give a complete list of represenatives of 1- cocycles corresponding to twists of the Klein quartic. 50 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

rep 1 h h2g3sg6 hgsg6 # rep 5628 42 42 # 1-cocyclesh i 21 0 0

Figure 18. Forms of the Klein quartic over Fq with q 5mod 7 as given by representatives of the 1-cocycles. ≡

rep 1 h h2g4sg5 h2g6s # rep 2856 42 42 # 1-cocyclesh i 48 0 0

Figure 19. Forms of the Klein quartic over Fq with q 6mod 7 as given by representatives of the 1-cocycles. ≡

3.4.6. Forms of Frobenius. We note that EF is supersingular over Fp if p 3, 5, 6 mod 7 and therefore has endomorphism algebra ≡

End(E F¯ ) Q = Q 1,π,β,βπ F ⊗ p ⊗ h i where π2 = p, β2 = 7 and πβ = βπ. − − Now End(E F¯ ) and we may choose β so that α = (β 1)/2. OF ⊂ F ⊗ p − In particular we have4

Z 1, π, α, απ End(E F¯ ). h i⊂ F ⊗ p Thus given v = a + bπ + cα + dαπ with a,b,c,d Z we have ∈ tr(v)= v +ˆv =2a c. − When EF is not supersingular the endomorphism ring (over F¯ p) is just F and the geometric Frobenius endomorphism π has the form O

π = a + bα. These facts allow us to compute the trace of a twist in the following way: let x G be a representative of a 1-cocycle. Then if q 2, 3 mod7 ∈ 168 ≡ xhπ End(Jac(X F¯ )) ∈ ⊗ p is the geometric Frobenius but if q 4, 5 mod 7 then ≡ xh2π End(Jac(X F¯ )) ∈ ⊗ p is the geometric Frobenius endomorphism of Jac(X) - the power of h is needed as 3 3 Jac(X) is not literally equal to EF but rather the twist of EF by a 1-cocycle (see 3.4.3). Otherwise q 1, 6 mod 7 and ≡ xπ is the geometric Frobenius endomorphism.

We ﬁrst list all possible tr(gho(q)) (By abuse of language we write tr for the trace of both a matrix and the trace of an element of End(EF ).) for representatives g of

4The division ring Zh1,π,α,απi has index 7 in its integral closure. This can be seen by computing the discriminant det(tr(eiej )) of a basis ei; we do not know when the extra elements of the integral closure represent endomorphisms of EF . 3. THE ZOO OF SYMMETRIC QUARTICS 51 the various cocycyles, according to the value of q modulo 7. Underneath tr(gho(q)) o(q) n we give tr(gh π). We note that if EF is supersingular over Fq and q = p with n odd then tr(π)=0.

We write π = a + b√ 7 for the Frobenius endomorphism of EF and give the trace of Frobenius in terms− of a and b. We note that it is only possible to determine b up to sign. This fact is reﬂected in the symmetry with respect to the sign of b that we see in the following tables.

rep 1 s h tr (rep ) 3 1 0 tr(tr (rep π )) 6a −2a 0 −

Figure 20. Traces of forms of the Klein quartic over Fq with q 1 mod 7 as given by representatives of the 1-cocycles. The second≡ row gives the trace of automorphism which twists Frobe- nius. The third row gives the trace of the twisted Frobenius endomorphism.

rep 1 hg6sg2 h2g5 h2g4 h2 tr (rep h) 0 1α ¯ α 3 tr(tr (rep h π)) 0 −2a (a 7b) (a +7b) 6a − − − −

Figure 21. Traces of forms of the Klein quartic over Fq with q 2 mod 7 as given by representatives of the 1-cocycles. ≡

rep 1 g2sg6 tr (rep h ) 0 1 tr (rep h π) 0 −2a −

Figure 22. Traces of forms of the Klein quartic over Fq with q 3 mod 7 as given by representatives of the 1-cocycles. ≡

rep 1 h tr (rep h2) 0 3 tr(tr (rep h2 π)) 0 6a

Figure 23. Traces of forms of the Klein quartic over Fq with q 4 mod 7 as given by representatives of the 1-cocycles. ≡

Proposition 11. Figures 20 to 25 give a complete list of traces of Frobenius of twists of the Klein quartic over a ﬁnite ﬁeld. We deduce from these calculations the following amusing fact: a prime power q is 6 mod 7 if and only if over Fq2 the Klein quartic is maximal - i.e. the number of points on it is equal to the Hasse-Weil upper bound. 52 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

rep 1 h tr (rep h2) 0 3 tr (rep h2π) 0 6a

Figure 24. Traces of forms of the Klein quartic over Fq with q 5 mod 7 as given by representatives of the 1-cocycles. ≡ rep 1 h tr (rep ) 3 0 tr (rep π) 6a 0

Figure 25. Traces of forms of the Klein quartic over Fq with q 6 mod 7 as given by representatives of the 1-cocycles. ≡

4. Points on Fermat Using the calculation of 3.2 we obtain the following table for points on the Fermat quartic: For all prime powers q< 100 which are 1 mod 4, we have: q a + bi min q + 1 − 6a q + 1 − 2a q + 1 q +1+2a q +1+2b q + 1 − 2b max 5 1 ± 2i 0 0 4 6 8 10 218 9 −3 0 28 16 10 4 10 10 28 13 −3 ± 2i 0 32 20 14 8 18 10 32 17 1 ± 4i 0 12 16 18 20 26 10 40 25 −3 0 44 32 26 20 34 18 56 29 5 ± 2i 0 0 20 30 40 34 26 60 37 1 ± 6i 2 32 36 38 40 26 50 72 41 5 ± 4i 6 12 32 42 52 34 50 78 49 −7 0 92 36 50 64 50 50 92 53 −7 ± 2i 12 96 68 54 40 58 50 96 61 5 ± 6i 17 32 52 62 72 50 74 107 73 −3 ± 8i 23 92 80 74 68 90 58 122 81 9 28 136 64 82 100 82 82 136 89 5 ± 8i 36 60 80 90 100 106 74 144 97 9 ± 4i 41 44 80 98 116 90 106 155 Here the column min denotes the Hasse-Weil lower bound for the number of points on a genus 3 curve over Fq. Similarly max denotes the known upper bound for the number of points on genus 3 curve over Fq, as it appears in [54] p276.

In general, one can exploit the structure of the curve elliptic curve E := E0 to obtain a formula for the number of points on C0. We now outline this argument.

If E is supersingular then tr(π) = ( 1)n2√q if q = p2n; otherwise tr(π)=0. If − E is ordinary then π = a+bi where a2+b2 = q. We now explain how to determine a.

2 3 The curve E′ with model y = x x is a twist of E. Moreover the kernel of the − isogenies [2] and [2 + 2i] have all their points deﬁned ove Fq: for assuming that q 1 mod 4 we have ≡ s s 1 s E′[2+2i](k )= E′[2](k ) [2]− (E′[1 + i](k )) ∪ 1 1 and a calculation shows [2]− E′[1 + i] = [2]− ( (0, 0), ) is rational. Therefore { ∞} s s E′[2+2i](k ) ker(π′ 1)(k ) ⊂ − 4. POINTS ON FERMAT 53

s which implies that in the endomorphism ring End(E′ k ) we have ⊗

π′ 1 0 mod2+2i − ≡ 2 2 moreover we know that π′ = c + di with c + d = q. Therefore, d is even, and this determines c and d up to sign. Then c is determined by the congruence π′ 1 0 mod 2+2i. − ≡

Finally E is isomorphic to E′ when q 1 mod 8 and π = π′; otherwise q 5 ≡ ≡ mod 8 and E is a quadratic twist of E′, in which case π = π′. − Using this we can determine for which non-square values of q the Fermat quartic attains the Hasse-Weil-Serre bound. If q 1 mod 8 and q = c2 +d2 with c+di 1 ≡ ≡ mod 2+2i and 2c = [2√q]. Or if q 5 mod 8 and q = c2 + d2 with c + di 1 − ≡ ≡− mod 2+2i and 2c = [2√q]. 54 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

5. Points on Klein We list the number of points on twists of X := x3y + y3z + z3x for all prime powers q less than 100 and coprime to 7. The first column represents the cardinality q of the finite field considered. The row corresponding to a given q lists of the number of points on the twists of the Klein quartic X in the same order as they appear in 3.4.6. Thus in particular, the second column gives the number of points on X itself. For example for q = 19 the table tell us that there are 2 forms of the Klein quartic and both have exactly 20 rational points and the first form has trivial 1-cocycle while the second form has a 1-cocycle whose value at Frobenius is h as in part (5) of section 3.4.6. When q = 37 the table tell us there are 5 forms of the Klein quartic with 38, 32, 21, 49 and 56 rational points respectively, and the respective values of the corresponding 1-cocycles of Frobenius are 1,hg6sg2,h2g5,h2g4 and h2. q 2 460 0 3 4 4 4 5 14 5 6 6 8 24 4 0 9 10 4 7 728 11 12 0 13 14 14 16 18 14 21 14 17 18 18 19 20 20 23 24 32 21 35 0 25 26 56 27 28 28 29 24 32 0 31 32 32 32 33 0 37 38 32 21 49 56 41 42 42 43 80 32 0 47 48 48 53 54 84 59 60 60 61 62 62 64 38 74 0 67 68 56 71 24 88 0 73 74 74 79 80 88 63 105 56 81 82 28 83 84 84 89 90 90 97 98 98

6. Moduli theoretic interpretation of twists of C and X The Klein quartic

X := X3Y + Y 3Z + Z3X 6. MODULI THEORETIC INTERPRETATION OF TWISTS OF C AND X 55 is isomorphic with the modular curve X(7) (see [9] p84-85). The Fermat quartic is a form of the modular curve X0(64). The curve X(7) represents a functor for elliptic curves with a symplectic level 7 structure; the modular curve X0(64) does not represent a functor, although it is a coarse moduli space for elliptic curves with a subgroup scheme of order 64.

It is natural to ask what functor the twists of X represent, and we explore this question in this section.

6.1. Deﬁnition of certain moduli problems. A family of elliptic curves is a smooth and proper morphism of schemes f : E S with a section O : S E whose geometric ﬁbres are elliptic curves with origin−→ given by O. −→

We write (Z/N)S for the constant group scheme over S with values in (Z/N).

Let ζN be a primitive Nth root of unity and assume that ζN Γ(S, S∗ ). A symplectic level N structure for a family of elliptic curves is an isomorphism∈ O

φ : (Z/N)2 E[N] S −→ 2 of group schemes which takes the standard symplectic form on (Z/N)S to the Weil N pairing on E.

We consider the functor from schemes to sets given by

(N)(S) := (f : E S, φ) φ is a level N structure on E/S . X { −→ | } 1 For N 3 this functor is represented by a smooth Z[ζN , N ] scheme X(N) of ﬁnite type. ≥

We note that the functorial interpretation of X(N) gives an obvious action of the constant group scheme PSL2(Z/N)Z[1/N,ζN ] on X(N).

It turns out that X is isomorphic to X(7). It is natural to ask: what functors do the twists of X(7) represent?

In 6.2 we answer this question for any twist of X(N).

We also note that the Fermat quartic is a twist of the modular curve X0(64). Indeed, X0(64) has three eigenforms, two of which are old forms and the other is a new form. The old forms have q expansion (here q is the nome q = Exp(2πiτ)) (see p136 [3])

f := q 2q5 3q9 +6q13 +2q17 16q25 + 1 − − − · · · and

f := q2 2q10 3q18 +6q26 + . 2 − − · · · The new form has expansion

f := q +2q5 3q9 6q13 +2q17 + . 3 − − · · · The expression f 4 f 4 + 16f 4 transforms is a cusp form of weight 8, and so 1 − 3 2 by Riemann-Roch, it must be zero. Thus X0(64) has smooth projective model X4 Y 4 + 16Z4 which is easily seen to be a twist of the Fermat quartic. − 56 TWISTS OF GENUS THREE CURVES OVER FINITE FIELDS

6.2. Moduli interpretation of twists of X(N) for N 3. In order to simplify considerations we consider X(N) as a variety over a ﬁeld≥5 k containing a primitive Nth root of unity ζN .

s The group PSL2(Z/N) acts on X(N)(k ) in the following way: let (E, φ) be an s elliptic curve over k with a level N structure φ, then g PSL2(Z/N) takes (E, φ) to (E,g φ). ∈ ◦ Similarly, Gal(ks/k) acts on the points of X(N)(ks) by the rule

(E, φ) (Eσ, φσ). 7→ Here Eσ denotes the ﬁbre product of E and Spec(σ):Spec(ks) Spec(ks); a sim- ilar remark holds for φσ. −→

s Now let f : Gal(k /k) PSL2(Z/N) be a 1-cocycle, then we may deﬁne a twisted Gal(ks/k) action on X−→(N) ks pointwise by ⊗ (E, φ) (Eσ,f(σ) φσ). 7→ ◦ Let X (N) be the k variety obtained by taking the quotient of X(N) ks by f ⊗ this twised Galois action. Then a k valued point in Xf (N)(k) consists of a pair (E, φ) where E is an elliptic curve over k and φ is a symplectic isomorphism of E ks[N](ks) with (Z/N)2. Moreover, the Galois action on (E ks)[N](k) can be speciﬁed⊗ on (Z/N)2 by the formula ⊗

σ(a,b)= f(σ)(a,b). s s Thus the k points of Xf (N) are just elliptic curves E with a basis of (E k )[N](k ) and an action of Gal(ks/k) on (E ks)[N](ks) given by f. ⊗ ⊗ We note that twisted modular curves were used be Wiles in his proof of the Taniyama-Shimura conjecture (see [44] p469).

5Otherwise, we are forced to consider what it means to twist over an arbitrary ´etale cover of the base scheme; this can be done, but it involves studying the ´etale fundamental group of the base scheme. Bibliography

[1] Auer, Roland; Top, Jaap, Some genus 3 curves with many points. Algorithmic number theory (Sydney, 2002), 163–171, Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002 [2] Brünjes, Lars, Forms of Fermat equations and their zeta functions, World Scientific, 2004 [3] Birch, Bryan; Kuyk Willem, Modular functions of one variable IV, Lecture Notes in Mathe- matics 476, Springer-Verlag, Berlin New York,1975 [4] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete 21. Springer-Verlag, Berlin, 1990. [5] Chai, Ching-Li, Compactification of Siegel moduli schemes. London Mathematical Society Lecture Note Series, 107. Cambridge University Press, Cambridge, 1985. [6] Conrad, Brian, Gross-Zagier revisited, in Heegner points and Rankin L-series, Math. Sci. Res. Inst. Pub. 49, Cambridge Univ. Press, Cambridge, 2004, 67163 [7] Deligne, Pierre; Mumford, David, The irreducibility of the space of curves of given genus. Inst. Hautes Etudes´ Sci. Publ. Math. No. 36 1969 75–109 [8] Duursma, Iwan, Twisted Klein curves modulo 2, arXiv:math/0302154v1 [9] Elkies, Noam D., The Klein quartic in number theory. The eightfold way, 51–101, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge, 1999 [10] Faltings, Gerd; Chai, Ching-Li, Degeneration of abelian varieties. With an appendix by David Mumford. Ergebnisse der Mathematik und ihrer Grenzgebiete 22. Springer-Verlag, Berlin, 1990 [11] Gouvêa, Fernando; Yui, Noriko, Arithmetic of diagonal hypersurfaces over finite fields (Lon- don Mathematical Society Lecture Note Series), Cambridge, 1995 [12] Hartshorne, Robin, Residues and Duality, Lecture notes in mathematics 20. Springer-Verlag 1966 [13] Harris, Joe; Morrison, Ian, Moduli of curves. Graduate Texts in Mathematics, 187. Springer- Verlag, New York, 1998 [14] Hess, F. An algorithm for computing isomorphisms of algebraic function fields. Algorithmic number theory, 263–271, Lecture Notes in Comput. Sci., 3076, Springer, Berlin, 2004. [15] Howe, Everett W.; Leprévost, Franck; Poonen, Bjorn, Large torsion subgroups of split Jaco- bians of curves of genus two or three. Forum Math. 12 (2000), no. 3, 315–364. [16] Igusa, Jun-ichi, Modular forms and projective invariants. Amer. J. Math. 89 1967, 817–855 [17] Katz, N.M., p-adic properties of modular schemes and modular forms. Modular functions of one variable III, Lecture Notes in Mathematics, 350 Springer-Verlag, Berlin-New York 1973 [18] Kani, E.; Rosen, M., Idempotent relations and factors of Jacobians. Math. Ann. 284 (1989), no. 2, 307–327 [19] Kempf, George, On the geometry of a theorem of Riemann. Ann. of Math. (2) 98 1973, 178–185 [20] Kleiman, Steven L., The Picard scheme. Fundamental algebraic geometry, 235–321, Math. Surveys Monogr., 123, Amer. Math. Soc., Providence, RI, 2005. [21] Klein, Felix, Gesammelte mathematische Abhandlungen. (German) Dritter Band: Elliptis- che Funktionen, insbesondere Modulfunktionen, hyperelliptische und Abelsche Funktionen, Riemannsche Funktionentheorie und automorphe Funktionen, Anhang verschiedene Verze- ichnisse. Herausgegeben von R. Fricke, H. Vermeil und E. Bessel-Hagen (von F. Klein mit ergänzenden Zusätzen versehen). Reprint der Erstauflagen [Verlag von Julius Springer, Berlin, 1923]. Springer-Verlag, Berlin-New York, 1973 [22] Lachaud, G.; Ritzenthaler, C. On a conjecture of Serre on abelian threefolds. Algebraic Geometry and its applications (Tahiti, 2007), 88–115. World Scientific, Singapore, 2008. [23] Lang, Serge, Algebra. Revised third edition, Graduate Texts in Mathematics, 211, Springer- Verlag, New York, 2002 [24] Lauter, Kristin The maximum or minimum number of rational points on genus three curves over finite fields. With an appendix by Jean-Pierre Serre. Compositio Math. 134 (2002), no. 1, 87–111.

57 58 BIBLIOGRAPHY

[25] Liu, Qing, Algebraic geometry and arithmetic curves, Oxford Univeristy Press, 2002 [26] Lønsted, Knud; Kleiman, Steven L., Basics on families of hyperelliptic curves. Compositio Math. 38 (1979), no. 1, 83–111. [27] Macaulay, F. S., The algebraic theory of modular systems. Revised reprint of the 1916 original. With an introduction by Paul Roberts. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1994 [28] Mazur, B.; Rubin, K.; Silverberg, A., Twisting commutative algebraic groups. J. Algebra 314 (2007), no. 1 [29] Miranda, Rick, Algebraic curves and Riemann surfaces, American Mathematical Society, 1995 [30] Moret-Bailly, Laurent, Sur l’équation fonctionnelle de la fonction thêta de Riemann. Compo- sitio Math. 75 1990, 203–21 [31] Mumford, David, Geometric invariant theory. Ergebnisse der Mathematik und ihrer Gren- zgebiete, Neue Folge, Band 34 Springer-Verlag, Berlin-New York 1965 [32] Mumford, David, On the equations defining abelian varieties. I. Invent. Math. 1 1966 287–354 [33] Mumford, David, On the equations defining abelian varieties. II. Invent. Math. 3 1967 75–135 [34] Mumford, David, On the equations defining abelian varieties. III. Invent. Math. 3 1967 215– 244. [35] Mumford, David Picard groups of moduli problems. 1965 Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) pp. 33–81 Harper & Row, New York [36] Mumford, David, Theta characteristics of an algebraic curve. Ann. Sci. Ecole´ Norm. Sup. (4) 4 1971, 181–192 [37] Oort, Frans; Steenbrink, Joseph, The local Torelli problem for algebraic curves. Journes de Géometrie Algébrique d’Angers, Juillet 1979, Sijthoff & Noordhoff, Alphen aan den Rijn— Germantown, Md., 1980. [38] Oort, Frans, Commutative group schemes. Lecture Notes in Mathematics, 15 Springer-Verlag, Berlin-New York 1966 [39] Oort, Frans; Ueno, Kenji, Principally polarized abelian varieties of dimension two or three are Jacobian varieties. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 377–381 [40] Oort, Frans, Finite group schemes, local moduli for abelian varieties, and lifting problems, in Algebraic geometry, Oslo, 1970. Proceedings of the Fifth Nordic Summer School in Math- ematics held in Oslo, August 5–25, 1970. Edited by F. Oort. Wolters-Noordhoff Publishing, Groningen, 1972 [41] Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael Twists of X(7) and primitive solutions to x2 + y3 = z7. Duke Math. J. 137 (2007), no. 1, 103–158. [42] Popp, Herbert, Moduli theory and classification theory of algebraic varieties. Lecture Notes in Mathematics, Vol. 620. Springer-Verlag, Berlin-New York, 1977 [43] Roquette, Peter, Abschätzung der automorphismenanzahl von Funktionenkörpern bei Primzahlcharakteristik. Math. Z. 117 1970 [44] Rubin, Karl, Modularity of mod 5 representations. Modular forms and Fermat’s last theorem (Boston, MA, 1995), 463–474, Springer, New York, 1997 [45] Sekiguchi, Tsutomu, Erratum: “On the fields of rationality for curves and for their Jacobian varieties” [Nagoya Math. J. 88 (1982), 197–212 [46] Sekiguchi, Tsutomu, On the fields of rationality for curves and for their Jacobian varieties. Nagoya Math. J. 88 (1982), 197–212. [47] Serre, Jean-Pierre, Letter to Jaap Top (Private communication), 2003 [48] Serre, Jean-Pierre, Nombres de points des courbes algébriques sur Fq . Seminar on number theory, 1982–1983 , Exp. No. 22, Univ. Bordeaux I, Talence, 1983, [49] Serre, Jean-Pierre, Sur le nombre des points rationnels d’une courbe algébrique sur un corps fini. C. R. Acad. Sci. Paris Sr. I Math. 296 1983, 397–402 [50] Serre, Jean-Pierre, Lectures on curves over finite fields, notes taken by Fernando Gouvêa, Harvard 1985, [51] Serre, Jean-Pierre, Complex multiplication. 1967 Algebraic Number Theory (Proc. Instruc- tional Conf., Brighton, 1965) pp. 292–296 Thompson, Washington, D.C. [52] Serre, Jean-Pierre, Cohomolgie Galoisienne, Cinquième édition, Lecture Notes in Mathemat- ics, Springer-Verlag 1994, [53] Serre, Jean-Pierre, Linear representations of finite groups. Translated from the second French edition by Leonard L. Scott. Graduate Texts in Mathematics, Vol. 42. Springer-Verlag, New York-Heidelberg, 1977 [54] Top, Jaap, Curves of genus 3 over small finite fields, Indag. Mathem., N.S., 14 (2), 275-283, 2003, [55] Tsuyumine, Shigeaki, Factorial property of a ring of automorphic forms. Trans. Amer. Math. Soc. 296 (1986), no. 1, 111–123. BIBLIOGRAPHY 59

[56] Tsuyumine, Shigeaki, Thetanullwerte on a moduli space of curves and hyperelliptic loci. Math. Z. 207 (1991), no. 4, 539–568. [57] Vermeulen, Alexius Maria, Weierstrass points of weight two on curves of genus three, Ph.D Thesis University of Amsterdam, 1983 [58] Waterhouse, William C. Abelian varieties over ﬁnite ﬁelds. Ann. Sci. cole Norm. Sup. (4) 2 1969 521–560

Samenvatting

Stel C is een kromme van geslacht g over een eindig lichaam Fq. Uit werk van Hasse en van Weil (met een kleine verbetering door Serre) weten we, dat het aantal Fq rationale punten op C een niet-negatief geheel getal is in het interval [q +1 g[2√q], q +1+ g[2√q]] . − Een voor de hand liggende vraag is, welke getallen in dit gegeven interval daadw- erkelijk voorkomen als het aantal punten op een kromme van geslacht g over Fq.

In dit proefschrift behandelen we twee vragen die voortkomen uit een poging, boven- staand probleem voor geslacht 3 krommen te beantwoorden. Hier volgt een uitleg van deze vragen:

Uit werk van Lauter-Serre en van Auer-Top is bekend, dat een constante c bestaat, onafhankelijk van q, zodat voor elke q er een geslacht 3 kromme C over Fq is waarvoor ofwel #C(F ) > q +1+3[2√q] c, q − ofwel #C(F ) < q +1 3[2√q]+ c. q − De ambigu¨ıteit hier heeft de volgende oorzaak: om bovenstaande bewering aan te tonen, construeren we eerst een constante c zodat voor elke q een indecomposabel hoofdgepolariseerd abels drievoud (A, a) over Fq bestaat met de eigenschap, dat het Frobenius morﬁsme πA voldoet aan de ongelijkheid tr(π ) > 3[2√q] c. − A − De stelling van Torelli levert ons een geslacht 3 kromme C over Fq waarvan de Jacobiaan Jac(C) ofwel isomorf is met (A, a), ofwel met een [ 1] twist van (A, a). − Dit heeft tot gevolg dat het Frobenius morﬁsme πJ van Jac(C) voldoet aan π = π of π = π . J A J − A Bovenstaande bewering van Auer-Top en Lauter-Serre volgt nu uit de spoorformule van Weil #C(F )= q +1 tr(π ). q − J Serre opperde in een brief aan Top een manier om de gegeven ambigu¨ıteit te behandelen- m.a.w. om te bepalen of Jac(C) isomorf is met (A, a) of met een [ 1] − twist van (A, a). Hij stelde voor, dat met χ18 het product van de even algebra¨ısche theta nulls, opgevat als “algebra¨ısche modulaire vormen van halftallig gewicht”, Jac(C) isomorf is met (A, a) over Fq( χ18(A, a)). Feitelijk opperde Serre, dat dit het geval zou moeten zijn over ieder lichaam van karakteristiek ongelijk aan p 2. Deze suggestie wordt in hoofdstuk 1 van dit proefschrift bewezen. In wat meer hoogdravend taalgebruik kan het hoofdresultaat van dit hoofdstuk als volgt gefor- muleerd worden: de moduliruimte van krommen van geslacht 3 wordt verkregen uit de moduliruimte van indecomposabele hoofdgepolariseerde abelse drievouden door “de wortel uit χ18 toe te voegen”.

61 62 SAMENVATTING

De tweede vraag die in dit proefschrift wordt behandeld is de volgende. Zij C een geslacht 3 kromme over Fq met veel automorfismen. Dan is bekend dat er geslacht 3 krommen C′ over Fq bestaan met de eigenschap dat C′ en C niet isomorf zijn over Fq maar wel over een uitbreiding van Fq. Een natuurlijke vraag is: kunnen we de mogelijke C′ classificeren, en daarbij de getallen #C′(Fq) in termen van deze classificatie? Hoofdstuk 2 geeft een antwoord op deze vraag. In het bijzonder worden in detail de speciale gevallen van de Klein en de Fermat krommen van graad 4 behandeld. Acknowledgements

This thesis would not have been possible without the kind and thoughtful supervi- sion of Jaap Top and Marius van der Put. I was very fortunate to have such good supervisors and it is an honour to thank them both.

The comments of the reading commitee were very useful in polishing the original version of this manuscript, and it is a pleasure and a privilege to be able to thank Gerard van der Geer, Ching-Li Chai and David Kohel for their work.

In writing Chapter 1 of this thesis, I beneﬁted immensely from conversations or correspondence with, or work by: Christophe Ritzenthaler, Gilles Lachaud, Jean- Pierre Serre, Robert Carls, Everett Howe, Frans Oort, Bas Edixhoven, Laurent Moret-Bailly and Bert van Geemen.

The questions treated in Chapter 2 were the brainchild of Jaap Top. Although the work was mine, the direction and inspiration was Jaap’s. And I thank him again separately for providing such a good problem from which to learn about algebraic curves, Jacobian varieties and Galois cohomology.

I thank Laurentiu Paunescu, King-Fai Lai, Amnon Neeman, Christopher Durrant, Daniel Daners, Terry Gagen and Robert Welsh for moral and intellectual encour- agement.

During the ﬁrst 3 years of my time as a Ph.D. student at Groningen I also proﬁted a lot of from conversations with Lenny Taelman.

I thank all the Ph.D. students at Groningen for the warm and friendly environment which they provided.

Finally, I thank my parents and Olga for all their support.