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Jacobians among Ab elian threefolds: a formula of Klein and a question of Serre
Gilles Lachaud, Christophe Ritzenthaler and Alexey Zykin
Abstract Let k be a field and f be a Siegel modular form of weight h 0 and genus g > 1 over k . Using f , we define an invariant of the k -isomorphism class of a principally polarized abelian variety (A, a)/k of dimension g . Moreover when (A, a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions when g = 3 and k C we obtain (i) a new proof of a formula of Klein linking the modular form 18 to the square of the discriminant of plane quartics ; (ii) a proof that one can decide when (A, a) is a Jacobian over k by looking whether the value of 18 at (A, a) is a square in k . This answers a question of J.-P. Serre. Finally, we study the possible generalizations of this approach for g > 3. 1. Intro duction 1.1 Torelli theorem Let k be an algebraically closed field and g 1 be an integer. If X is a (nonsingular irreducible pro jective) curve of genus g over k , Torelli's theorem states that the map X (J acX, j ), associating to X its Jacobian together with the canonical polarization j , is injective. The determination of the image of this map is a long time studied question. When g = 3, the moduli space Ag of principally polarized abelian varieties of dimension g and the moduli space Mg of nonsingular algebraic curves of genus g are both of dimension g (g + 1)/2 = 3g - 3 = 6. According to Hoyt [12] and Oort and Ueno [25], the image of M3 is exactly the space of indecomposable principally polarized abelian threefolds. Moreover if k = C, Igusa [17] characterized the locus of decomposable abelian threefolds and that of hyperelliptic Jacobians, making use of two particular modular forms 140 and 18 on the Siegel upper half space of degree 3. Assume now that k is any field and g 1. J.-P. Serre noticed in [22] that a precise form of Torelli's theorem reveals a mysterious obstruction for a geometric Jacobian to be a Jacobian over k . More precisely, he proved the following: Theorem 1.1.1. Let (A, a) be a principally polarized abelian variety of dimension g 1 over k , and assume that (A, a) is isomorphic over k to the Jacobian of a curve X0 of genus g defined over k . The following alternative holds : (i) If X0 is hyperelliptic, there is a curve X/k isomorphic to X0 over k such that (A, a) is k isomorphic to (J acX, j ). (ii) If X0 is not hyperelliptic, there is a curve X/k isomorphic to X0 over k , and a quadratic character : Gal(k sep /k ) -- {±1} -
2000 Mathematics Subject Classification 14H42 ; 11G30 Keywords: genus 3 curves, Siegel modular form, obstruction, Jacobian


Gilles Lachaud, Christophe Ritzenthaler and Alexey Zykin such that the twisted abelian variety (A, a) is k -isomorphic to (J acX, j ). The character is trivial if and only if (A, a) is k -isomorphic to a Jacobian. Thus, only case (i) occurs if g = 1 or g = 2, with all curves being elliptic or hyperelliptic. 1.2 Curves of genus 3 Assume now again g = 3. Let there be given an indecomposable principally polarized abelian threefold (A, a) defined over k . In a letter to J. Top [28], J.-P. Serre asked a twofold question: -- How to decide, knowing only (A, a), that X is hyperel liptic ? -- If X is not hyperel liptic, how to compute the quadratic character ? Assume that k C. The first question can easily be answered using the forms 140 and 18 . As for the second question, roughly speaking, Serre suggested that is trivial if and only if 18 is a square in k â (see Th.4.1.2 for a more precise formulation). This assertion was motivated by a formula of Klein [20] relating the modular form 18 (in the notation of Igusa) to the square of the discriminant of plane quartics, which more or less gives the `only if ' part of the claim. In [21], two of the authors justified Serre's assertion for a three dimensional family of abelian varieties and in particular determined the absolute constant involved in Klein's formula. In this article we prove that Serre's assertion is valid for any abelian threefold. In order to do so, we start by taking a broader point of view, valid for any g > 1. (i) We look at the action of k -isomorphisms on Siegel modular forms defined over k and we define invariants of k -isomorphism classes of abelian varieties over k . (ii) We link Siegel modular forms, Teichmuller modular forms and invariants of plane curves. ¨ Once these two goals are achieved, Serre's assertion can be rephrased as the following strategy -- use (ii) to find a Siegel modular form whose `values' are a suitable power in k on the Jacobian locus; -- use (i) to distinguish between Jacobians and their twists. For g = 3, Klein's formula shows that the form 18 is a square on the Jacobian locus and that this is enough to characterize this locus. On the other hand, we show that this is no longer the case for the natural generalization h , h = 2g-2 (2g + 1), when g > 3. The relevance of Klein's formula in this problem was one of Serre's insights. We would like to point out that we do not actually need the full strength of Klein's formula to work out our strategy. Indeed, we do not go all the way from Siegel modular form to invariants. We use instead a formula due to Ichikawa relating 18 to the square of a Teichmuller modular form (see Rem.4.1.3). However ¨ we think that the connection between Siegel modular forms and invariants is interesting enough in its own, besides the fact that it gives a new proof of Klein's formula. The paper is organized as follows. In §2, we review the necessary elements from the theory of Siegel and Teichmuller modular forms. Only §2.4 is original: we introduce the action of isomorphisms and ¨ see how the action of twists is reflected on the values of modular forms. In §3, we link modular forms and certain invariants of ternary forms. Finally in §4 we deal with the case g = 3. We give first a proof of Klein's formula and then we justify the validity of Serre's assertion. Finally we explain the reasons behind the failure of the obvious generalization of the theory in higher dimensions and state some natural questions. 2


Jacobians among Abelian threefolds Acknowledgements We would like to thank J.-P. Serre and S. Meagher for fruitful discussions and Y. F. Bilu and X. Xarles for their help in the final part of Sec.4.3. 2. Siegel and Teichmuller mo dular forms ¨ 2.1 Geometric Siegel mo dular forms The references are [4], [5], [7], [10]. Let g > 1 and n > 0 be two integers and Ag,n be the moduli stack of principally polarized abelian schemes of relative dimension g with symplectic level n structure. Let : Vg,n - Ag,n be the universal abelian scheme, fitted with the zero section : Ag,n - Vg,n , and 1 g,n /Ag,n = 1 g,n /Ag,n -- Ag,n - V V the rank g bundle induced by the relative regular differential forms of degree one on Vg The relative canonical bundle over Ag,n is the line bundle
g ,n

over Ag,n .

=

1 V

g ,n

/A

g ,n

.

For a pro jective nonsingular variety X defined over a field k , we denote by 1 [X ] = H 0 (X, 1 k ) X k the finite dimensional k -vector space of regular differential forms on X defined over k . Hence, the fibre of the bundle 1 g,n /Ag,n over A Ag,n (k ) is equal to 1 [A], and the fibre of is the onek V dimensional vector space
g

[A] =

1 [A]. k

We put Ag = Ag,1 and Vg = Vg,1 . Let R be a commutative ring and h a positive integer. A geometric Siegel modular form of genus g and weight h over R is an element of the R-module Sg,h (R) = (Ag R, Note that for any n 1, we have an isomorphism Ag A
g ,n h

).

/S p2g (Z/nZ).

If n 3, as shown in [24], from the rigidity lemma of Serre [27] we can deduce that the moduli space Ag,n can be represented by a smooth scheme over Z[n , 1/n]. Hence, for any algebra R over Z[n , 1/n], the module Sg,h (R) is the submodule of (Ag,n Z[
n

,1/n]

R,

h

)

consisting of the elements invariant under S p2g (Z/nZ). Assume now that R = k is a field. If f Sg,h (k ), A is a p.p.a.v. of dimension g defined over k and is a basis of k [A], define f (A, ) = f (A)/
h

.

(1)

In this way such a modular form defines a rule which assigns the element f (A, ) k to every such pair (A, ) and such that: (i) f (A, ) = -h f (A, ) for any k â . (ii) f (A, ) depends only on the k -isomorphism class of the pair (A, ). 3


Gilles Lachaud, Christophe Ritzenthaler and Alexey Zykin Conversely, such a rule defines a unique f Sg,h (k ). This definition is a straightforward generalization of that of Deligne-Serre [6] and Katz [19] if g = 1. 2.2 Complex uniformisation Assume R = C. Let Hg = Mg (C) | t = , Im > 0 be the Siegel upper half space of genus g and = S p2g (Z). As explained in [4, §2], The complex orbifold Ag (C) can be expressed as the quotient \Hg where acts by M . = (a + b) · (c + d)-1 The group Z2g acts on Hg â Cg by v .( , z ) = ( , z + m + n) if v = If Ug = Z2g \(Hg â Cg ), the pro jection : Ug -- H - A =
-1 g

if M =

ab cd

.

m n

Z2g .

defines a universal principally polarized abelian variety with fibres ( ) = Cg /(Zg + Zg ).
-1

Let j (M , ) = c + d and define the action of on Hg â Cg by M .( , (z1 , . . . , zg )) = (M . , t j (M , ) The map t j (M , ) Hence, Vg (C)
-1

· (z1 , . . . , zg ))

if M .

: Cg Cg induces an isomorphism:
M

: A -- AM . . -


\Ug and the following diagram is commutative: \Ug -- Vg (C) - \Hg -- Ag (C) -


As in [7, p. 141], let = dqg dq1 ... = (2i )g dz1 · · · dzg (Hg , ) q1 qg

with (zi , . . . , zg ) Cg and (qi , . . . , qg ) = (e2iz1 , . . . e2izg ). This section of the canonical bundle is a basis of [A ] for all Hg and the relative canonical bundle of Ug /Hg is trivialized by :
g

The group acts on in such a way that
Ug /H
g

Ug /Hg

=



1 Ug /H

g

Hg â C · . if M ,

by

M .( , ) = (M . , det j (M , ) · )

(M . ) = det j (M , )-1 . M Thus, a geometric Siegel modular form f of weight h becomes an expression f (A ) = f ( ) · 4
h

,


Jacobians among Abelian threefolds where f belongs to the well-known vector space Rg,h (C) of analytic Siegel modular forms of weight h on Hg , consisting of complex holomorphic functions ( ) on Hg satisfying (M . ) = det j (M . )h ( ) for any M S p2g (Z). Note that by Koecher principle [10, p. 11], the condition of holomorphy at is automatically satisfied since g > 1. The converse is also true [7, p. 141]: Proposition 2.2.1. If f Sg,h (C) and Hg , let f ( ) = f (A )/
h

= (2i )-gh f (A )/(dz1 · · · dzg )h .
g ,h

Then the map f f is an isomorphism Sg,h (C)-R 2.3 Teichmuller mo dular forms ¨

(C).

Let g > 1 and n > 0 be positive integers and let Mg,n denote the moduli stack of smooth and proper curves of genus g with symplectic level n structure [5]. Let : Cg,n - Mg,n be the universal curve, and let be the invertible sheaf associated to the Hodge bund le, namely
g

=



1 Cg,n /M

g ,n

.

For an algebraically closed field k the fibre over C Mg,n (k ) is the one dimensional vector space [C ] = g 1 [C ]. k Let R be a commutative ring and h a positive integer. A Teichmul ler modular form of genus g and ¨ weight h over R is an element of T
g ,h

(R) = (Mg R,

h

).

These forms have been thoroughly studied by Ichikawa [13], [14], [15], [16]. As in the case of the moduli space of abelian varieties, for any n 1 we have Mg M
g ,n

/S p2g (Z/nZ), 3. Then, for any algebra R

and Mg,n can be represented by a smooth scheme over Z[n , 1/n] if n over Z[n , 1/n], the module Tg,h (R) is the submodule of (Mg,n invariant under S p2g (Z/nZ).
Z[n ,1/n]

R,

h

)

Let C /k be a genus g curve. Let 1 , . . . , g be a basis of 1 [C ] and = 1 . . . g a basis of [C ]. k As for Siegel modular forms in (1), for a Teichmuller modular form f Tg,h (k ) we define ¨ f (C, ) = f (C )/ Ichikawa asserts the following proposition: Proposition 2.3.1. The Torelli map : Mg - Ag , associating to a curve C its Jacobian J acC with the canonical polarization j , satisfies = , and induces for any commutative ring R a linear map : Sg,h (R) = (Ag R,
h h

k.

) -- Tg,h (R) = (Mg R, h ), - if = .

such that [ f ](C ) = [f (J acC )]. Fixing a basis of [C ], this is f (J acC, ) = [ f ](C, )

5


Gilles Lachaud, Christophe Ritzenthaler and Alexey Zykin 2.4 Action of isomorphisms Suppose : (A , a ) - (A, a) is a k -isomorphism of principally polarized abelian varieties. Let 1 , . . . , g 1 k [A] and = 1 . . . g [A]. Then by definition f (A, ) = f (A , ) where i = (i ) is a basis of 1 k [A ] and = 1 . . . g [A ]. If 1 , . . . , g is another basis of 1 k [A ] and = 1 . . . g , we denote by M GLg (k ) the matrix of the basis (i ) in the basis (i ). We can easily see that Proposition 2.4.1. In the above notation, f (A, ) = det(M )h · f (A , ). First of all, from this formula applied to the action of -1, we deduce that, if k is a field of characteristic different from 2, then Sg,h (k ) = {0} if g h is odd. From now on we assume that g h is even and chark = 2. Corollary 2.4.2. Let (A, a) be a principally polarized abelian variety of dimension g defined over k and f Sg,h (k ). Let 1 , . . . , g be a basis of 1 [A], and let = 1 . . . g [A]. Then the k quantity ¯ f (A) = f (A, ) modâ k âh k /k âh ¯ does not depend on the choice of the basis of 1 [A]. In particular f (A) is an invariant of the k -isomorphism class of A.
k

Corollary 2.4.3. Assume that g is odd. Let f Sg,h (k ) and : A - A a non trivial quadratic ¯ ¯ ¯ ¯ ¯ twist. There exists c k \ k 2 such that f (A) = ch/2 f (A ). Thus, if f (A) = 0 then f (A) and f (A ) â /k âh . do not belong to the same class in k Proof. Assume that is given by the quadratic character of Gal(k /k ). Then d = ( )g · d, where d = det(M ) k , Gal(k /k ).

Assume that g is odd. Then by our assumption h is even, and d2 = ( )g dd k . But d k since / there exists such that ( ) = -1. Using Prop.2.4.1 we find that f (A, ) = (d2 )h/2 f (A , ). ¯ ¯ ¯ Since d2 is not a square in k , if f (A) = 0 then f (A) and f (A ) belong to two different classes. Let now (A, a) be a principally polarized abelian variety of be a basis of 1 [A] and = 1 . . . g [A]. Let C polarization a). The period matrix 1 1 · · . . = [1 2 ] = . 1 g · · belongs to the set Rg Mg
,2g

dimension g defined over C. Let 1 , . . . , g 1 , . . . 2g be a symplectic basis (for the · ·
2g

1 g



. . .

2g

(C) of Riemann matrices, and = -1 1 Hg . 2

Proposition 2.4.4. In the above notation, f (A, ) = (2i )g 6
h

f ( ) . det h 2


Jacobians among Abelian threefolds Proof. The abelian variety A is isomorphic to A = Cg /Z2g and [A] maps to = dz1 · · · dzg [A ] under this isomorphism. The linear map z -1 z = z induces the isomorphism 2 : A -- A = Cg /(Zg + Zg ). - Let us denote = dz1 · · · dzg = (2i )-g in [A ]. Thus, using Prop.2.4.1, Equation (1) and Prop.2.2.1, we obtain f (A, ) = f (A , ) = det -h f (A , ) 2 = det -h f (A )/ 2 from which the proposition follows. 3. Invariants and mo dular forms In this section k is an algebraically closed field of characteristic different from 2. 3.1 Invariants We review some classical invariant theory. Let E be a vector space of dimension n over k . The left regular representation r of GL(E ) on the vector space Xd = Symd (E ) of homogeneous polynomials of degree d on E is given by r(u) : F (x) (u · F )(x) = F (ux) for u GL(E ), F Xd and x E . If U is an open subset of Xd stable under r, we still denote by r the left regular representation of GL(E ) on the k -algebra O(U ) of regular functions on U , in such a way that r(u) : (F ) (u · )(F ) = (u · F ), if u GL(E ), O(U ) and F U . If h Z, we denote by Oh (U ) the subspace of homogeneous elements of degree h, satisfying (F ) = h (F ) for k â and F U . The subspaces Oh (U ) are stable under r. An element Oh (U ) is an invariant of degree h on U if u · = for every u S L(E ), and we denote by Invh (U ) the subspace of Oh (U ) of invariants of degree h on U . If Invh (U ) = {0}, then hd 0 (mod n), since the group µn of n-th roots of unity is in the kernel of r. Hence, if O(U ), and if w and n are two integers such that hd = nw, the following statements are equivalent: (i) Invh (U ); (ii) u · = (det u)w for every u GL(E ). If these conditions are satisfied, we call w the weight of . The multivariate resultant Res(f1 , . . . , fn ) of n forms f1 , . . . fn in n variables with coefficients in k is an irreducible polynomial in the coefficients of f1 , . . . fn which vanishes whenever f1 , . . . fn have a common non-zero root. One requires that the resultant is irreducible over Z, i. e. it has coefficients in Z with greatest common divisor equal to 1, and moreover Res(xd1 , . . . , xdn ) = 1 n 1 for any (d1 , . . . , dn ) Nn . The resultant exists and is unique. Now, let F Xd , and denote q1 , . . . , qn the partial derivatives of F . The discriminant of F is DiscF = c-1 Res(q1 , . . . , qn ), n,d 7 with cn,d = d
((d-1)n -(-1)n )/d h

= (2i )gh det -h f ( )/ 2

h

= (2i )g

h

f ( ) , det h 2

,


Gilles Lachaud, Christophe Ritzenthaler and Alexey Zykin the coefficient cn,d being chosen according to [28]. Hence, the pro jective hypersurface which is the zero locus of F Xd is nonsingular if and only if DiscF = 0. The discriminant is an irreducible polynomial in the coefficients of F , see for instance [8, Chap. 9, Ex. 1.6(a)]. From now on we restrict ourselves to the case n = 3, i. e. we consider invariants of ternary forms of degree d, and summarize the results that we shall need. Proposition 3.1.1. If F Xd is a ternary form, the discriminant DiscF = d
-(d-1)(d-2)-1

· Res(q1 , q2 , q3 )

where q1 , q2 , q3 are the partial derivatives of F , is given by an irreducible polynomial over Z in the coefficients of F , and vanishes if and only if the plane curve CF defined by F is singular. The discriminant is an invariant of Xd of degree 3(d - 1)2 and weight d(d - 1)2 . We refer to [8, p. 118] and [21] for an explicit formula for the discriminant, found by Sylvester. Example 3.1.2 Ciani quartics. We recall some results whose proofs are given in [21]. Let a1 b3 b2 m = b3 a2 b1 Sym3 (k ), b2 b1 a3 and for 1 i 3, let ci = aj ak - b2 be the cofactor of ai . If i det(m) = 0, a1 a2 a3 = 0 and c1 c2 c3 = 0 then Fm (x, y , z ) = a1 x4 + a2 y 4 + a3 z 4 + 2(b1 y 2 z 2 + b2 x2 z 2 + b3 x2 y 2 ) defines a non singular plane quartic. Moreover DiscFm = 240 a1 a2 a3 (c1 c2 c3 )2 det(m)4 . Note that the discrepancy between the powers of 2 here and in [21, Prop.2.2.1] comes from the normalization by cn,d . 3.2 Geometric invariants for nonsingular plane curves Let E be a vector space of dimension 3 over k and G = GL(E ). The universal curve over the affine space Xd = Symd (E ) is the variety Yd = (F, x) Xd â P2 | F (x) = 0 . The nonsingular locus of Xd is the principal open set X0 = (Xd )D d
isc

= {F Xd | Disc(F ) = 0} .

0 If Yd is the universal curve over the nonsingular locus X0 , the pro jection is a smooth surjective d k -morphism 0 - : Yd -- X0 d

whose fibre over F is the non singular plane curve CF . We recall the classical way to write down an explicit k -basis of 1 [CF ] = H 0 (CF , 1 ) for F X0 (k ) d (see [3, p. 630]). Let 1 = f (x2 dx3 - x3 dx2 ) , q1 2 = f (x3 dx1 - x1 dx3 ) , q2 3 = f (x1 dx2 - x2 dx1 ) , q3

where q1 , q2 , q3 are the partial derivatives of F , and where f belongs to the space Xd-3 of ternary forms of degree d - 3. The forms i glue together and define a regular differential form f (F ) 8


Jacobians among Abelian threefolds 1 [CF ]. Since dim Xd
-3

= (d - 1)(d - 2)/2 = g , the linear map f f (F ) defines an isomorphism Xd
-3

-- 1 [CF ]. -
1 0 Yd /X0 d



This implies that the sections f (X0 , d X0 â Xd d
-3

) lead to a trivialization
0 d

-- 1 - Y

/X

0 d

.

We denote 1 , . . . , g the sequence of sections obtained by substituting for f in f the g members of the canonical basis of Xd-3 , enumerated according to the lexicographic order. Then = 1 . . . g is a section of =
g

X0 d

1 0 Yd /X0 d

,

the Hodge bundle of the universal curve over . Since the map u : x ux induces an isomorphism u : Cu·F -- C - u
1 1 F

it has a natural action : [CF ] [Cu·F ] on the differentials and hence, on the sections of h , for h Z. More specifically, if s (X0 , h ), one can write s = · h with O(X0 ) ; for d d F X0 , one has d u s(F ) = (F ) · (u (F ))h . Lemma 3.2.1. The section (X0 , ) satisfies for u G and F X0 , then d d u (F ) = det(u)w0 · (u · F ), with w0 =
â

d 3

=

dg N. 3

Proof. Since dim [F ] = dim [u · F ] = 1, there is c(u, F ) k and c is a "crossed character", satisfying

such that

u (F ) = c(u, F ) · (u · F ). c(uu , F ) = c(u, F ) c(u , u · F ). Now the regular function F c(u, F ) does not vanishes on X0 . By Lemma 3.2.2 below and the d irreducibility of the discriminant (Prop. 3.1.1), we have c(u, F ) = (u)(DiscF )n(
u)

with (u) k â and n(u) Z. The group G being connected, the function n(u) = n is constant. Since c(I3 , F ) = 1, we have (DiscF )n = (I3 )-1 , and this implies n = 0. Hence, c(u, F ) is independent of F and is a character of G. Since the group of commutators of G is S L3 (k ), we have (u) = det(u)
w0

for some w0 Z. It is therefore enough to compute (u) when u = I3 , with k â . In this case u · F = d F . Moreover, for all f Xd-3 , since the section f is homogeneous of degree -1 f (d F ) = -d · f (F ), and (d F ) = - Hence, as u is the identity on the curve CF = C u (F ) = (F ) = This implies det(u)w0 = 9
3w0 dg u·F dg

· (F ).

,

· (u · F ) = det(u)w0 · (u · F ). = dg ,


Gilles Lachaud, Christophe Ritzenthaler and Alexey Zykin and the result is proven. We made use of the following elementary lemma: Lemma 3.2.2. Let f k [T1 , . . . , Tn ] be irreducible and let g k (T1 , . . . , Tn ) be a rational function which has neither zeroes nor poles outside the set of zeroes of f . Then there is an m Z and c k â such that g = cf m . Proof. This is an immediate consequence of Hilbert's Nullstellensatz, together with the fact that the ring k [T1 , . . . , Tn ] is factorial. For any h Z, we denote by (X0 , d Proposition 3.2.3. Let h
h )G

the subspace of sections s (X0 , d

h

) such that

u s(F ) = s(u · F ) for every u G, F X0 . d 0 be an integer. The linear map () = · is an isomorphism : Invgh (X0 ) -- (X0 , - d d Proof. Let Invgh (X0 ), s = () = · d Lem.3.2.1,
h h ) G h

.

, and w = dg h/3, the weight of . Then using
h

u s(F ) = (F ) · (u (F )) = (F ) · det(u)w
0

h

· (u · F )h

= det(u)w (F ) · (u · F )h = (u · F ) · (u · F )h = s(u · F ). Hence, () (X0 , h )G . Conversely, the inverse of is the map s s/ d proposition.
h

, and this proves the

3.3 Mo dular forms as invariants 0 Let d > 2 be an integer and g = d . Since the fibres of Yd - X0 are nonsingular non hyperelliptic d 2 plane curves of genus g , by the universal property of Mg we get a morphism p : X0 -- M0 , - g g where M0 is the moduli stack of nonhyperelliptic curves of genus g and p = by construction. g This induces a morphism p : (M0 , g Moreover, for u G, since u : Cu diagram
·F h

) -- (X0 , - d

h

).

C

F

is an isomorphism, we get the following commutative
u

- [CF ] -- [Cu·F ] p p [F ] -- [u · F ]. - For any f (M0 , g Then u [(p f )(F )] = u [p (f (CF ))] = p [u f (CF )] = p [f (Cu·F )] = (p f )(u · F ), 10
h u

), the modular invariance of f means that u f (CF ) = f (Cu·F ).


Jacobians among Abelian threefolds and this means that p f (X0 , d
h )G

. Combining this result with Prop.3.2.3, we obtain: 0, the linear map = -1 p is a homomorphism:
h

Proposition 3.3.1. For any integer h such that

(M0 , g

) -- Invgh (X0 ) - d

(f )(F ) = f (CF , ) with = (p
)-1

, for any F

X0 d

and any section f (M0 , g

h

).

0 We finally make a link between invariants and analytic Siegel modular forms. Let F Xd (C) and let 1 , . . . , g be the basis of regular differentials on CF defined in Sec.3.2. Let 1 , . . . 2g be a symplectic basis of H1 (C, Z) (for the intersection pairing). The matrix 1 1 · · · 2g 1 . . . . = [1 2 ] = . . 1 g · · · 2g g

belongs to the set Rg Mg

,2g

(C) of Riemann matrices, and = -1 1 Hg . 2

Corollary 3.3.2. Let f Sg,h (C) be a geometric Siegel modular form, f Rg,h (C) the corresponding analytic modular form, and = ( f ) the corresponding invariant. In the above notation, (F ) = (2i )g Proof. Let = (p )
-1 h

f ( ) . det h 2

( ) and = ( )-1 (). From Prop.2.3.1 and 3.3.1, we deduce (F ) = ( f )(CF , ) = f (J acCF , ).

On the other hand, by the canonical identifications 1 [CF ] = 1 [J acCF ], and Prop.2.4.4 we get f (J acCF , ) = (2i )g from which the result follows. 4. The case of genus 3 4.1 Klein's formula We recall the definition of theta functions with (entire) characteristics [] =
1 2 h

H1 (CF , Z) = H1 (J acCF , Z) f ( ) , det h 2

Zg Zg ,

following [2]. The (classical) theta function is given, for Hg and z Cg , by 1 2 (z , ) =
nZg

q

(n+1 /2) (n+1 /2)+2(n+1 /2)(z +2 /2)

.

The Thetanul lwerte are the values at z = 0 of these functions, and we write []( ) =
1 2

( ) = 11



1 2

(0, ).


Gilles Lachaud, Christophe Ritzenthaler and Alexey Zykin Recall that a characteristic is even if 1 · 2 0 (mod 2) and odd otherwise. Let Sg (resp. Ug ) be the set of even characteristics with coefficients in {0, 1}. For g 2, we put h = #Sg /2 = 2g-2 (2g + 1) and h ( ) = (2i )g
h Sg

[]( ).

In his beautiful paper [17], Igusa proves the following result [loc. cit., Lem. 10 and 11]. Denote by 140 the modular form defined by the thirty-fifth elementary symmetric function of the eighth power of the even Thetanullwerte. Recall that a principally polarized abelian variety (A, a) is decomposable if it is a product of principally polarized abelian varieties of lower dimension, and indecomposable otherwise. Theorem 4.1.1. If g 3, then h ( ) R
140 g ,h

(C). Moreover, If g = 3 and H3 , then: ( ) = 0.

(i) A is decomposable if 18 ( ) =

( ) = 0.
140

(ii) A is a hyperelliptic Jacobian if 18 ( ) = 0 and (iii) A is a non hyperelliptic Jacobian if 18 ( ) = 0.

Using Prop. 2.2.1, we define the geometric Siegel modular form of weight h h (A ) = (2i )gh h ( )(dz1 · · · dzg )h . Ichikawa [15], [16] proved that h Sg,h (Q). For g = 3, one finds 18 (A ) = -(2 )54 18 ( )(dz1 dz2 dz3 )18 . Now we are ready to give a proof of the following result [20, Eq. 118, p. 462]: Theorem 4.1.2 Klein's formula. Let F X0 (C) and CF be the corresponding smooth plane quartic. 4 Let 1 , 2 , 3 be the classical basis of 1 [CF ] from Sec.3.2 and 1 , . . . 6 be a symplectic basis of H1 (CF , Z) for the intersection pairing. Let 1 1 · · · 6 1 . . . . = [1 2 ] = . . 1 3 · · · 6 3 be a period matrix of J ac(C ) and =
-1 2

1 H3 . Then 1 18 ( ) (2 )54 . 228 det(2 )18 18 ( ) . det 18 2

Disc(F )2 =

Proof. Cor.3.3.2 shows that I = (18 )satisfies for any F X0 , 4 I (F ) = -(2 )54

Moreover Th. 4.1.1 (iii) shows that I (F ) = 0 for all F X0 . Thus I is a non-zero invariant of weight 4 54. Applying Lem. 3.2.2 for the discriminant, we find by comparison of the weights that I = cDisc2 with c C a constant. But if Fm is the Ciani quartic associated to a matrix m Sym3 (k ) as in Example 3.1.2, it is proven in [21, Cor. 4.2] that Klein's formula is true for Fm and c = -228 . Remark 4.1.3. The morphism defines an injective morphism of graded k -algebras S3 (k ) = h
0 S3,h

(k ) -- T3 (k ) = h -

0

T

3,h

(k ).

In [14], Ichikawa proves that if k is a field of characteristic 0, then T3 (k ) is generated by the image of S3 (k ) and a primitive Teichmuller form µ3,9 T3,9 (Z) of weight 9, which is not of Siegel modular ¨ 12


Jacobians among Abelian threefolds type. He also proves in [16] that (18 ) = -228 · (µ3,9 )2 . (2) Th. 4.1.2 implies that µ3,9 is actually equal to the discriminant up to a sign. This might probably be deduced from the definition of µ3,9 , although we did not sort it out (see also [18, Sec. 2.4]). Remark 4.1.4. Besides [23] and [11] where an analogue of Klein's formula is derived in the hyperelliptic case, there exists a beautiful algebraic Klein's formula, linking the discriminant with irrational invariants [9, Th.11.1]. 4.2 Jacobians among ab elian threefolds Let k C be a field and let g = 3. We prove the following theorem which allows us to determine whether a given abelian threefold defined over k is k -isomorphic to a Jacobian of a curve defined over the same field. This settles the question of Serre recalled in the introduction. Theorem 4.2.1. Let (A, a) be a principally polarized abelian threefold defined over k C. Let 1 , 2 , 3 be a basis of 1 [A] and 1 , . . . 6 a symplectic basis of H 1 (A, Z), in such a way that k 1 1 · · · 6 1 . . . . = [1 2 ] = . . 3 · · · 6 3 1 is a period matrix of (A, a). Put = -1 1 H3 . 2 (i) If 140 ( ) = 0 then (A, ) is decomposable. In particular it is not a Jacobian. (ii) If 140 ( ) = 0 and 18 ( ) = 0 then there exists a hyperelliptic curve X/k such that (J acX, j ) (A, a). (iii) If 18 ( ) = 0 then (A, a) is isomorphic to a Jacobian if and only if -18 (A, 1 2 3 ) = (2 )54 is a square in k . Proof. The first and second points follow from Th.4.1.1 and Th.1.1.1. Suppose now that (A, a) is isomorphic over k to the Jacobian of a non hyperelliptic genus 3 curve C /k and let = 1 2 3 . Using Prop.2.3.1, we get -18 (A, ) = (-18 )(C, ) with = . The left hand side is (Prop.2.4.4) -18 (A, ) = -(2i )54 18 18 ( ) = (2 )54 . 18 det(2 ) det(2 )18
2 3,9

18 ( ) det(2 )18

According to Rem.4.1.3, the right hand side of the equality is (-18 )(C, ) = 228 · µ (C, ) = (214 · µ3,9 (C, ))
2

so the desired expression is a square in k . On the contrary, Cor.2.4.3 shows that if (A, a) is a quadratic twist of a Jacobian (A , a ) then there exists a non square c k such that -18 (A) = c9 · (-18 (A )). ¯ ¯ As we have just proved that -18 (A ) is a non-zero square in k /k ¯ is not. 13
â18

, -18 (A) (and then -18 (A, )) ¯


Gilles Lachaud, Christophe Ritzenthaler and Alexey Zykin Corollary 4.2.2. In the notation of Th.4.2.1, the quadratic character of Gal(ksep /k ) introduced in Theorem 1.1.1 is given by ( ) = d/d , where d= with an arbitrary choice of the square root. 4.3 Beyond genus 3 It is natural to try to extend our results to the case g > 3. The first question to ask is -- Does there exist an analogue of Klein's formula for g > 3 ? Here we can give a partial answer. Using Sec.2.3, we can consider the Teichmuller modular form ¨ (h ) with h = 2g-2 (2g + 1). In [16, Prop.4.5] (see also [29]), it is proven that for g > 3 the element (h )/2 has as a square root a primitive element µg we find the following amazing formula
,h/2 2g
-1

(2 )54

18 ( ) , det(2 )18

(2g -1)

T

g ,h/2

(Z). If g = 4, in the footnote, p. 462 in [20] (3)

68 ( ) = c · (X )2 · T (X )8 . det(2 )68

Here = -1 1 , with = [1 2 ] a period matrix of a genus 4 non hyperelliptic curve X given 2 in P3 as an intersection of a quadric Q and a cubic surface E . The elements (X ) and T (X ) are defined in the classical invariant theory as, respectively, the discriminant of Q and the tact invariant of Q and E (see [26, p.122]). No such formula seems to be known in the non hyperelliptic case for g > 4. Let us now look at what happens when we try to apply Serre's approach for g > 3. To begin with, when g is even, we cannot use Cor.2.4.2 to distinguish between quadratic twists. In particular, using the previous result, we see that h (A, k ) is a square when A is a principally polarized abelian variety defined over k which is geometrically a Jacobian. A natural question is: -- What is the relation between this condition and the locus of geometric Jacobians over k ? Let us assume now that g is odd. Corollary 2.4.3 shows that there exists c k \ k 2 such that h (A ) = ch/2 · h (A) for a Jacobian A and a quadratic twist A . What enabled us to distinguish between A and A when g = 3 is that h/2 = 9 is odd. However as soon as g > 3, 2 | 2g-3 , the power g - 3 being the maximal power of 2 dividing h/2, so it is not enough for 18 (A) to be a square in k to make a distinction ¯ g -2 between A and A . It must rather be an element of k 2 . It can be easily seen from the proof of [29, g -2 Th.1] that (h ) does not admit a fourth root. According to [1] or [30] this implies h (A) k 2 / for infinitely many Jacobians A defined over number fields k . So we can no longer use the modular form h to easily characterize Jacobians over k . The question is: -- Is it possible to find a modular form to replace h in our strategy when g > 3 ? References
1 Bilu, Y. F. Personal communication. 2 Birkenhake C.; Lange, H. Complex abelian varieties. Second edition. Grundlehren der Mathematischen Wissenschaften, 302 Springer-Verlag, Berlin, 2004.

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Jacobians among Abelian threefolds
3 Brieskorn, E.; Kn¨ orrer, H. Plane Algebraic Curves. Birkh¨ auser Verlag, 1986. 4 Chai, Ching-Li. Siegel moduli schemes and their compactifications over C. Arithmetic geometry (Storrs, Conn., 1984), 231­251, Springer, New York, 1986. 5 Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus. Inst. Hautes tudes Sci. Publ. Math. 36 (1969), 75­109. ´ 6 Deligne, P.; Serre, J.-P. Formes modulaires de poids 1. Ann. Sci. Ecole Norm. Sup. (1974), 507­530; = Serre, J.-P. OEuvres, vol. III, No 101, 193­216. 7 Faltings, G.; Chai, C.-L. Degeneration of abelian varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 22. Springer, Berlin, 1990. 8 Gel'fand, I.M.; Kapranov, M.M.; Zelevinsky, A.V. Discriminants, resultants, and multidimensional determinants. Birkh¨ auser, Boston, (1994). 9 Gizatullin, M. On covariants of plane quartic associated to its even theta characteristic. Algebraic geometry, 37-74, Contemp. Math., 422, Amer. Math. Soc., Providence, RI, 2007. 10 Van Der Geer, G. Siegel modular forms. Preprint. arXiv: math/0605346v2 [math.AG] (2007). 11 Gu` ardia, G. Jacobian nullwerte and algebraic equations. J. Algebra 253 (2002), 112-132. 12 Hoyt, W.L. On products and algebraic families of Jacobian varieties. Ann. of Math. 77, (1963), 415-423. 13 Ichikawa, T. On Teichmuller modular forms. Math. Ann. 299 (1994), no. 4, 731­740. ¨ 14 Ichikawa, T. Teichmuller modular forms of degree 3. Amer. J. Math. 117 (1995), no. 4, 1057­1061. ¨ 15 Ichikawa, T. Theta constants and Teichmuller modular forms. J. Number Theory 61 (1996), no. 2, 409­ ¨ 419. 16 Ichikawa, T. Generalized Tate curve and integral Teichmuller modular forms. Amer. J. Math. 122 (2000), ¨ no. 6, 1139­1174. 17 Igusa, J.-I. Modular forms and pro jective invariants. Amer. J. Math, 89, (1967), 817-855. 18 de Jong, R. Local invariants attached to Weierstrass points. Preprint. arXiv:0710.5464v1 [math.AG] (2007). 19 Katz, N. M. p-adic properties of modular schemes and modular forms. Modular functions of one variable, III (Antwerp, 1972), 69­190. Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, 1973. 20 Klein, F. Zur Theorie der Abelschen Funktionen. Math. Annalen, 36 (1889-90); = Gesammelte mathematische Abhandlungen XCVI I, 388-474. 21 Lachaud, G.; Ritzenthaler, C. On a conjecture of Serre on abelian threefolds. Algebraic Geometry and its applications (Tahiti, 2007), 88­115. World Scientific, Singapore, 2008. arXiv: 0710.3303v1 [math.NT] 22 Lauter, K. Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, with an appendix by J. P. Serre. Journal of Algebraic Geometry 10 (2001), 19­36. 23 Lockhart, P. On the discriminant of a hyperelliptic curve. Trans. Amer. Math. Soc. 342, (1994), 729­752. 24 Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34. Springer-Verlag, Berlin, 1994. 25 Oort, F.; Ueno, K. Principally polarized abelian varieties of dimension two or three are Jacobian varieties. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20, (1973), 377-381. 26 Salmon, G. Trait´ de g´ ´ e eometrie analytique ` trois dimensions. Troisi` a eme partie. Ouvrage traduit de l'anglais sur la quatri` eme ´ edition, Paris, 1892. 27 Serre, J.-P. Rigidit´ du foncteur de Jacobi d'´ helon n 3. Appendix to: Grothendieck, A. Techniques e ec de construction en g´ ´ eometrie analytique, X. Construction de l'espace de Teichmuller. S´ ¨ eminaire Henri Cartan 13, No 2, exp. 17. Secr´ etariat Math´ ematique, Paris, 1960-61. 28 Serre, J.-P. Two letters to Jaap Top. Algebraic Geometry and its applications (Tahiti, 2007) 84­87. World Scientific, Singapore, 2008. 29 Tsuyumine, S. Thetanullwerte on a moduli space of curves and hyperelliptic loci. Math. Z. 207 (1991), 539-568.

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Jacobians among Abelian threefolds
30 Xarles, X. Personal communication.

Gilles Lachaud lachaud@iml.univ-mrs.fr Gilles Lachaud Institut de Math´ ematiques de Luminy Universit´ Aix-Marseille - CNRS e Luminy Case 907, 13288 Marseille Cedex 9 - FRANCE

Christophe Ritzenthaler ritzent@iml.univ-mrs.fr Christophe Ritzenthaler Institut de Math´ ematiques de Luminy

Alexey Zykin zykin@iml.univ-mrs.fr Alexey Zykin Institut de Math´ ematiques de Luminy Mathematical Institute of the Russian Academy of Sciences Laboratoire Poncelet (UMI 2615)

16