Jacobians among Abelian threefolds: a formula of Klein and a question of Serre
aa r X i v : . [ m a t h . N T ] F e b JACOBIANS AMONG ABELIAN 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 ≥ g > k . Using f , we define an invariant of the k -isomorphism classof a principally polarized abelian variety ( A, a ) /k of dimension g . Moreoverwhen ( A, a ) is the Jacobian of a smooth plane curve, we show how to associateto f a classical plane invariant. As straightforward consequences of theseconstructions when g = 3 and k ⊂ C we obtain (i) a new proof of a formula ofKlein linking the modular form χ to the square of the discriminant of planequartics ; (ii) a proof that one can decide when ( A, a ) is a Jacobian over k bylooking whether the value of χ at ( A, a ) is a square in k . This answers aquestion of J.-P. Serre. Finally, we study the possible generalizations of thisapproach for g > Introduction
Torelli theorem.
Let k be an algebraically closed field. If X is a (nonsingularirreducible projective) curve of genus g over k , Torelli’s theorem states that themap X (Jac X, j ) , associating to X its Jacobian together with the canonicalpolarization j , is injective. The determination of the image of this map is a longtime studied question.When g = 3, the moduli space A g of principally polarized abelian varieties ofdimension g and the moduli space M g of nonsingular algebraic curves of genus g are both of dimension 3 g − g ( g + 1) / M is exactly the space of indecomposable principallypolarized abelian threefolds. Moreover if k = C , Igusa [17] characterized the locusof decomposable abelian threefolds and that of hyperelliptic Jacobians making useof two particular modular forms Σ and χ on the Siegel upper half space ofdegree 3.Assume now that k is any field and g ≥
1. J.-P. Serre noticed in [22] that a preciseform of Torelli’s theorem reveals a mysterious obstruction for a geometric Jacobianto 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 > over k , and assume that ( A, a ) is isomorphic over k to the Jacobian of acurve X of genus g defined over k . The following alternative holds : (i) If X is hyperelliptic, there is a curve X/k isomorphic to X over k suchthat ( A, a ) is k -isomorphic to (Jac X, j ) . (ii) If X is not hyperelliptic, there is a curve X/k isomorphic to X over k ,and a quadratic character ε : Gal( k sep /k ) −−−−→ {± } such that the twisted abelian variety ( A, a ) ε is k -isomorphic to (Jac X, j ) .The character ε is trivial if and only if ( A, a ) is k -isomorphic to a Jacobian. Date : November 20, 2018.
Thus, only case (i) occurs if g = 1 or g = 2, with all curves being elliptic orhyperelliptic.1.2. Curves of genus . Assume now k ⊂ C and g = 3. Let there be given anindecomposable principally polarized abelian threefold ( A, a ) defined over k . In aletter to J. Top [28], J.-P. Serre asked a twofold question:— How to decide, knowing only ( A, a ) , that X is hyperelliptic ? — If X is not hyperelliptic, how to find the quadratic character ε ? Moreover, he suggested a strategy in order to compute the twisting factor ε . Thisstrategy is based on a formula of Klein [20] relating the modular form χ (in thenotation of Igusa), to the square of the discriminant of plane quartics, see Th.4.1.2for a more precise formulation. In [21], two of the authors justified Serre’s strategyfor a three dimensional family of abelian varieties and in particular determined theabsolute constant involved in Klein’s formula.In this article we prove that Serre’s strategy can be applied to any abelian threefolds.More precisely, we take a broader point of view.(i) We look at the action of k -isomorphisms on Siegel modular forms definedover k and we define invariants of k -isomorphism classes of abelian varietiesover k .(ii) We link Siegel modular forms, Teichm¨uller modular forms and invariants.Then we derive a proof of Klein’s formula based on moduli spaces.Once these two goals achieved, Serre’s strategy can be rephrased as finding a Siegelmodular form whose locus has a good multiplicity on the Jacobian locus and thenusing point (i) to distinguish between Jacobians and their twists. For g = 3, theform χ fulfills the criterion as can be seen thanks to Klein’s formula. On the otherhand, we show that this is no longer the case for χ h when g >
3. We would liketo point out that we do not actually need Klein’s formula to prove Serre’s strategy.Indeed we do not need to go the full way from Siegel modular form to invariantsand could instead use a formula due to Ichikawa relating χ to the square of aTeichm¨uller modular form (see Rem. 4.2.2). However we think that the connectionbetween Siegel modular forms and invariants is interesting enough in its own, be-sides 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 fromthe theory of Siegel and Teichm¨uller modular forms. Only § §
3, we link modular forms and certain invariantsof ternary forms. Finally in § g = 3. We give first a proofof Klein’s formula and then we justify the validity of Serre’s strategy. Finally weexplain the reasons behind the failure of the obvious generalization of the theoryin higher dimensions and state some natural questions. Acknowledgements.
We would like to thank J.-P. Serre and S. Meagher forfruitful discussions and Y. F. Bilu and X. Xarles for their help in the final part ofSec.4.3. 2.
Siegel and Teichm¨uller modular forms
Geometric Siegel modular forms.
The references are [4], [5], [7], [10]. Let g > n > A g,n be the moduli stack of principallypolarized abelian schemes of relative dimension g with symplectic level n structure.Let π : V g,n −→ A g,n be the universal abelian scheme, fitted with the zero section ACOBIANS AMONG ABELIAN THREEFOLDS 3 ε : A g,n −→ V g,n , and π ∗ Ω V g,n / A g,n = ε ∗ Ω V g,n / A g,n −−−−→ A g,n the rank g bundle induced by the relative regular differential forms of degree oneon V g,n over A g,n . The relative canonical bundle over A g,n is the line bundle ω = ∧ g ε ∗ Ω V g,n / A g,n . For a projective nonsingular variety X defined over a field k , we denote byΩ k [ X ] = H ( X, Ω X ⊗ k )the finite dimensional k -vector space of regular differential forms on X defined over k . Hence, the fibre of the bundle Ω V g,n / A g,n over A ∈ A g,n ( k ) is equal to Ω k [ A ], andthe fibre of ω is the one-dimensional vector space ω [ A ] = ∧ g Ω k [ A ] . We put A g = A g, and V g = V g, . Let R be a commutative ring and h a positiveinteger. A geometric Siegel modular form of genus g and weight h over R is anelement of the R -module S g,h ( R ) = Γ( A g ⊗ R, ω ⊗ h ) . Note that for any n ≥
1, we have an isomorphism A g ≃ A g,n / Sp g ( Z /n Z ) . If n ≥
3, as shown in [24], from the rigidity lemma of Serre [27] we can deducethat the moduli space A g,n can be represented by a smooth scheme over Z [ ζ n , /n ].Hence, for any algebra R over Z [ ζ n , /n ], the module S g,h ( R ) is the submodule ofΓ( A g,n ⊗ Z [ ζ n , /n ] R, ω ⊗ h )consisting of the elements invariant under Sp g ( Z /n Z ).Assume now that R = k is a field. If f ∈ S g,h ( k ), A is a p.p.a.v. of dimension g defined over k and α is a basis of ω k [ A ], define(1) f ( A, α ) = f ( A ) /α ⊗ h . In this way such a modular form defines a rule which assigns the element f ( A, α ) ∈ k to every such pair ( A, α ) and such that:(i) f ( A, λα ) = λ − h f ( A, α ) for any λ ∈ k × .(ii) f ( A, α ) depends only on the k -isomorphism class of the pair ( A, α ).Conversely, such a rule defines a unique f ∈ S g,h ( k ). This definition is a straight-forward generalization of that of Deligne-Serre [6] and Katz [19] if g = 1.2.2. Complex uniformisation.
Assume R = C . Let H g = (cid:8) τ ∈ M g ( C ) | t τ = τ, Im τ > (cid:9) be the Siegel upper half space of genus g and Γ = Sp g ( Z ). As explained in [4, § A g ( C ) can be expressed as the quotient Γ \ H g where Γ actsby M.τ = ( aτ + b ) · ( cτ + d ) − if M = (cid:18) a bc d (cid:19) ∈ Γ . The group Z g acts on H g × C g by v. ( τ, z ) = ( τ, z + τ m + n ) if v = (cid:18) mn (cid:19) ∈ Z g . GILLES LACHAUD, CHRISTOPHE RITZENTHALER, AND ALEXEY ZYKIN If U g = Z g \ ( H g × C g ), the projection π : U g −−−−→ H g defines a universal principally polarized abelian variety with fibres A τ = π − ( τ ) = C g / ( Z g + τ Z g ) . Let j ( M, τ ) = cτ + d and define the action of Γ on H g × C g by M. ( τ, ( z , . . . , z g )) = ( M.τ, t j ( M, τ ) − · ( z , . . . , z g )) if M ∈ Γ . The map t j ( M, τ ) − : C g → C g induces an isomorphism: ϕ M : A τ −−−−→ A M.τ . Hence, V g ( C ) ≃ Γ \ U g and the following diagram is commutative:Γ \ U g ∼ −−−−→ V g ( C ) π y π y Γ \ H g ∼ −−−−→ A g ( C )As in [7, p. 141], let ζ = dq q ∧ · · · ∧ dq g q g = (2 iπ ) g dz ∧ · · · ∧ dz g ∈ Γ( H g , ω )with ( z i , . . . , z g ) ∈ C g and ( q i , . . . , q g ) = ( e iπz , . . . e iπz g ). This section of thecanonical bundle is a basis of ω [ A τ ] for all τ ∈ H g and the relative canonicalbundle of U g / H g is trivialized by ζ : ω U g / H g = ∧ g Ω U g / H g ≃ H g × C · ζ. The group Γ acts on ω U g / H g by M. ( τ, ζ ) = ( M.τ, det j ( M, τ ) · ζ ) if M ∈ Γ , in such a way that ϕ ∗ M ( ζ M.τ ) = det j ( M, τ ) − ζ τ . Thus, a geometric Siegel modular form f of weight h becomes an expression f ( A τ ) = e f ( τ ) · ζ ⊗ h , where e f belongs to the well-known vector space R g,h ( C ) of analytic Siegel modularforms of weight h on H g , consisting of complex holomorphic functions φ ( τ ) on H g satisfying φ ( M.τ ) = det j ( M.τ ) h φ ( τ )for any M ∈ Sp g ( Z ). Note that by Koecher principle [10, p. 11], the condition ofholomorphy at ∞ is automatically satisfied since g >
1. The converse is also true[7, p. 141]:
Proposition 2.2.1. If f ∈ S g,h ( C ) and τ ∈ H g , let e f ( τ ) = f ( A τ ) /ζ ⊗ h = (2 iπ ) − gh f ( A τ ) / ( dz ∧ · · · ∧ dz g ) ⊗ h . Then the map f e f is an isomorphism S g,h ( C ) ∼ −→ R g,h ( C ) . (cid:3) ACOBIANS AMONG ABELIAN THREEFOLDS 5
Teichm¨uller modular forms.
Let g > n > M g,n denote the moduli stack of smooth and proper curves of genus g withsymplectic level n structure [5]. Let π : C g,n −→ M g,n be the universal curve, andlet λ be the invertible sheaf associated to the Hodge bundle , namely λ = ∧ g π ∗ Ω C g,n / M g,n . For an algebraically closed field k the fibre over C ∈ M g,n ( k ) is the one dimensionalvector space λ [ C ] = ∧ g Ω k [ C ].Let R be a commutative ring and h a positive integer. A Teichm¨uller modular form of genus g and weight h over R is an element of T g,h ( R ) = Γ( M g ⊗ R, λ ⊗ h ) . These forms have been thoroughly studied by Ichikawa [13], [14], [15], [16]. As inthe case of the moduli space of abelian varieties, for any n ≥ M g ≃ M g,n / Sp g ( Z /n Z ) , and M g,n can be represented by a smooth scheme over Z [ ζ n , /n ] if n ≥
3. Then,for any algebra R over Z [ ζ n , /n ], the module T g,h ( R ) is the submodule ofΓ( M g,n ⊗ Z [ ζ n , /n ] R, λ ⊗ h )invariant under Sp g ( Z /n Z ).Let C/k be a genus g curve. Let λ , . . . , λ g be a basis of Ω k [ C ] and λ = λ ∧ · · · ∧ λ g a basis of λ [ C ]. As for Siegel modular forms in (1), for a Teichm¨uller modular form f ∈ T g,h ( k ) we define f ( C, λ ) = f ( C ) /λ ⊗ h ∈ k. Ichikawa proves the following proposition:
Proposition 2.3.1.
The Torelli map θ : M g −→ A g , associating to a curve C itsJacobian Jac C with the canonical polarization j , satisfies θ ∗ ω = λ , and inducesfor any commutative ring R a linear map θ ∗ : S g,h ( R ) = Γ( A g ⊗ R, ω ⊗ h ) −−−−→ T g,h ( R ) = Γ( M g ⊗ R, λ ⊗ h ) , such that [ θ ∗ f ]( C ) = θ ∗ [ f (Jac C )] . Fixing a basis λ of λ [ C ] , this is f (Jac C, α ) = [ θ ∗ f ]( C, λ ) if θ ∗ α = λ. (cid:3) Action of isomorphisms.
Suppose φ : ( A ′ , a ′ ) −→ ( A, a ) is a k -isomorphismof principally polarized abelian varieties, then by definition f ( A, α ) = f ( A ′ , β )where β i = φ ∗ ( α i ) is a basis of Ω k [ A ′ ] and β = β ∧ · · · ∧ β g ∈ ω [ A ′ ]. If α ′ , . . . , α ′ g is another basis of Ω k [ A ′ ] and α ′ = α ′ ∧ · · · ∧ α ′ g , we denote by M φ ∈ GL g ( k ) thematrix of the basis ( β i ) in the basis ( α ′ i ). We can easily see that Proposition 2.4.1.
In the above notation, f ( A, α ) = det( M φ ) h · f ( A ′ , α ′ ) . (cid:3) First of all, from this formula applied to the action of − , we deduce that, if k is afield of characteristic different from 2, then S g,h ( k ) = { } if gh is odd. From nowon we assume that gh is even and char k = 2. GILLES LACHAUD, CHRISTOPHE RITZENTHALER, AND ALEXEY ZYKIN
Corollary 2.4.2.
Let ( A, a ) be a principally polarized abelian variety of dimension g defined over k and f ∈ S g,h ( k ) . Let α , . . . , α g be a basis of Ω k [ A ] , and put α = α ∧ · · · ∧ α g ∈ ω [ A ] . Then the quantity ¯ f ( A ) = f ( A, α ) mod × k × h ∈ k/k × h does not depend on the choice of the basis of Ω k [ A ] . In particular ¯ f ( A ) is aninvariant of the k -isomorphism class of A . (cid:3) Corollary 2.4.3.
Assume that g is odd. Let f ∈ S g,h ( k ) and φ : A ′ −→ A a nontrivial quadratic twist. If ¯ f ( A ) = 0 then ¯ f ( A ) and ¯ f ( A ′ ) do not belong to the sameclass in k × /k × .Proof. Assume that φ is given by the quadratic character ε of Gal( k/k ). Then d σ = ε ( σ ) g · d, where d = det( M φ ) ∈ k, σ ∈ Gal( k/k ) . Assume that g is odd. Then by our assumption h is even, and d = ε ( σ ) g dd σ ∈ k .But d / ∈ k since there exists σ such that ε ( σ ) = −
1. Using Prop.2.4.1 we find that f ( A, α ) = ( d ) h/ f ( A ′ , α ′ ) . Since d is not a square in k , if ¯ f ( A ) = 0 then ¯ f ( A ) and ¯ f ( A ′ ) belong to twodifferent classes in k h/ /k × h ≃ k/k × . (cid:3) Let now (
A, a ) be a principally polarized abelian variety of dimension g defined over C . Let ω , . . . , ω g be a basis of Ω C [ A ] and ω = ω ∧ · · · ∧ ω g ∈ ω [ A ]. Let γ , . . . γ g be a symplectic basis (for the polarization a ). The period matrixΩ = [Ω Ω ] = R γ ω · · · R γ g ω ... ... R γ ω g · · · R γ g ω g belongs to the set R g ⊂ M g, g ( C ) of Riemann matrices, and τ = Ω − Ω ∈ H g . Proposition 2.4.4.
In the above notation, f ( A, ω ) = (2 iπ ) gh e f ( τ )det Ω h . Proof.
The abelian variety A is isomorphic to A Ω = C g / Ω Z g and ω ∈ ω [ A ] maps to ξ = dz ∧· · ·∧ dz g ∈ ω [ A Ω ] under this isomorphism. The linear map z Ω − z = z ′ induces the isomorphism ϕ : A Ω −−−−→ A τ = C g / ( Z g + τ Z g ) . Let us denote ξ ′ = dz ′ ∧ · · · ∧ dz ′ g = (2 iπ ) − g ζ in ω [ A τ ]. Thus, using Prop.2.4.1,Equation (1) and Prop.2.2.1, we obtain f ( A, ω ) = f ( A Ω , ξ ) = det Ω − h f ( A τ , ξ ′ )= det Ω − h f ( A τ ) /ξ ′⊗ h = (2 iπ ) gh det Ω − h f ( τ ) /ζ ⊗ h = (2 iπ ) gh e f ( τ )det Ω h , from which the proposition follows. (cid:3) Invariants and modular forms
In this section k is an algebraically closed field of characteristic different from 2. ACOBIANS AMONG ABELIAN THREEFOLDS 7
Invariants.
We review some classical invariant theory. Let E be a vectorspace of dimension n over k . The left regular representation ρ of GL( E ) on thevector space X d = Sym d ( E ∗ ) of homogeneous polynomials of degree d on E is givenby ρ ( u ) : F ( x ) ( u · F )( x ) = F ( u − x )for u ∈ GL( E ), F ∈ X d and x ∈ E . If U is an open subset of X d stable under ρ , westill denote by ρ the left regular representation of GL( E ) on the k -algebra O ( U ) ofregular functions on U , in such a way that ρ ( u ) : Φ( F ) ( u · Φ)( F ) = Φ( u − · F ) , if u ∈ GL( E ), Φ ∈ O ( U ) and F ∈ U . If h ∈ Z , we denote by O h ( U ) the subspaceof homogeneous elements of degree h , satisfying Φ( λF ) = λ h Φ( F ) for λ ∈ k × and F ∈ U . The subspaces O h ( U ) are stable under ρ . An element Φ ∈ O h ( U ) is an invariant of degree h on U if u · Φ = Φ for every u ∈ SL( E ) , and we denote by Inv h ( U ) the subspace of O h ( U ) of invariants of degree h on U .If Inv h ( U ) = { } , then hd ≡ n ), since the group µ n of n -th roots of unity isin the kernel of ρ . Hence, if Φ ∈ O ( U ), and if w and n are two integers such that hd = nw , the following statements are equivalent:(i) Φ ∈ Inv h ( U );(ii) u · Φ = (det u ) − w Φ for every u ∈ GL( E ) . If these conditions are satisfied, we call w the weight of Φ.The multivariate resultant Res( f , . . . , f n ) of n forms f , . . . f n in n variables withcoefficients in k is an irreducible polynomial in the coefficients of f , . . . f n whichvanishes whenever f , . . . f n have a common non-zero root. One requires that theresultant is irreducible over Z , i. e. it has coefficients in Z with greatest commondivisor equal to 1, and moreoverRes( x d , . . . , x d n n ) = 1for any ( d , . . . , d n ) ∈ N n . The resultant exists and is unique. Now, let F ∈ X d ,and denote q , . . . , q n the partial derivatives of F . The discriminant of F isDisc F = c − n,d Res( q , . . . , q n ) , with c n,d = d (( d − n − ( − n ) /d , the coefficient c n,d being chosen according to [28]. Hence, the projective hypersur-face which is the zero locus of F ∈ X d is nonsingular if and only if Disc F = 0. Thediscriminant 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 thatwe shall need. Proposition 3.1.1. If F ∈ X d is a ternary form, the discriminant Disc F = d − ( d − d − − · Res( q , q , q ) where q , q , q are the partial derivatives of F , is given by an irreducible polynomialover Z in the coefficients of F , and vanishes if and only if the plane curve C F definedby F is singular. The discriminant is an invariant of X d of degree d − andweight d ( d − . (cid:3) We refer to [8, p. 118] and [21] for a beautiful explicit formula for the discriminant,found by Sylvester.
GILLES LACHAUD, CHRISTOPHE RITZENTHALER, AND ALEXEY ZYKIN
Example . We recall some results whose proofs are givenin [21]. Let Sym ( k ) be the vector space of symmetric matrices of size 3 withcoefficients in k , and G m ( x, y, z ) = t v.m.v, v = ( x, y, z ) , the quadratic form associated to m ∈ Sym ( k ). Then F m ( x, y, z ) = G m ( x , y , z )is a ternary quartic, and the map m F m is an isomorphism of Sym ( k ) to thesubspace of F ∈ X which are invariant under the three involutions σ ( x, y, z ) = ( − x, y, z ) , σ ( x, y, z ) = ( x, − y, z ) , σ ( x, y, z ) = ( x, y, − z ) . If m = a b b b a b b b a ∈ Sym ( k ) , then F m ( x, y, z ) = a x + a y + a z + 2( b y z + b x z + b x y ) . For 1 ≤ i ≤
3, let c i = a j a k − b i the cofactor of a i . ThenDisc F m = 2 a a a ( c c c ) det( m ) . Note that the discrepancy between the powers of 2 here and in [21, Prop.2.2.1]comes from the normalization by c n,d .3.2. Geometric invariants for nonsingular plane curves.
Let E be a vectorspace of dimension 3 over k and G = GL( E ). The universal curve over the affinespace X d = Sym d ( E ) is the variety Y d = (cid:8) ( F, x ) ∈ X d × P | F ( x ) = 0 (cid:9) . The nonsingular locus of X d is the principal open set X d = ( X d ) Disc = { F ∈ X d | Disc( F ) = 0 } . If Y d is the universal curve restricted to the nonsingular locus, the projection is asmooth surjective k -morphism Y d −−−−→ X d whose fibre over F is the non singular plane curve C F .We recall the classical way to write down an explicit k -basis of Ω [ C F ] = H ( C F , Ω )for F ∈ X d ( k ) (see [3, p. 630]). Let η = f ( x dx − x dx ) q , η = f ( x dx − x dx ) q , η = f ( x dx − x dx ) q , where q , q , q are the partial derivatives of F , and where f belongs to the space X d − of ternary forms of degree d −
3. The forms η i glue together and define aregular differential form η f ( F ) ∈ Ω [ C F ]. Since dim X d − = ( d − d − / g ,the linear map f η f ( F ) defines an isomorphism X d − ∼ −−−−→ Ω [ C F ] . This implies that the sections η f ∈ Γ( X d , Ω Y d / X d ) lead to a trivialization X d × X d − ∼ −−−−→ Ω Y d / X d . An element u ∈ G acts on Y d by u · ( F, x ) = ( u · F, ux ) , and the projection Y d −→ X d is G -equivariant. ACOBIANS AMONG ABELIAN THREEFOLDS 9
We denote η , . . . , η g the sequence of sections obtained by substituting for f in η f the g members of the canonical basis of X d − , enumerated according to thelexicographic order, the classical basis of Γ( X d , Ω Y d / X d ). The section η = η ∧ · · · ∧ η g is a basis of the one-dimensional space Γ( X d , α ), where α = ∧ g π ∗ Ω Y d / X d , is the Hodge bundle of the universal curve over X d . For every F ∈ X d , an element u ∈ G induces by restriction an isomorphism ϕ u : C F −−−−→ C u · F , which itself defines a linear automorphism ϕ ∗ u of α .For any h ∈ Z , we denote by Γ( X d , α ⊗ h ) G the subspace of sections s ∈ Γ( X d , α ⊗ h )such that ϕ ∗ u ( s ) = s for every u ∈ G. If α ∈ Γ( X d , α ) and F ∈ X d , we define, in the same way as in Equation (1), s ( F, α ) = s ( F ) /α ⊗ h . Hence, s ∈ Γ( X d , α ⊗ h ) G if and only if for all u ∈ G and F ∈ X d , one has( ϕ ∗ u s )( F, α ) = s ( F, α ) . Proposition 3.2.1.
The section η ∈ Γ( X d , α ) satisfies the following properties. (i) If u ∈ G , then ϕ ∗ u η = det( u ) w η, with w = (cid:18) d (cid:19) = dg ∈ N . (ii) Let h ≥ be an integer. The linear map Φ τ (Φ) = Φ · η ⊗ h is an isomorphism τ : Inv gh ( X d ) ∼ −−−−→ Γ( X d , α ⊗ h ) G . Proof.
Let u ∈ G . Since dim α u · F = 1, there is c ( u, F ) ∈ k × such that( ϕ ∗ u η )( F, η ) = c ( u, F ) · η ( F, η ) = c ( u, F ) . and c is a “crossed character”, satisfying c ( uu ′ , F ) = c ( u, F ) c ( u ′ , u · F ) . Now the regular function F c ( u, F ) does not vanishes on X d . By Lemma 3.2.2below and the irreducibility of the discriminant (Prop. 3.1.1), we have c ( u, F ) = χ ( u )(Disc F ) n ( u ) with χ ( u ) ∈ k × and n ( u ) ∈ Z . The group G being connected, the function n ( u ) = n is constant. Since c ( I , F ) = 1, we have (Disc F ) n = χ ( I ) − , and this implies n = 0. Hence, c ( u, F ) is independent of F and χ is a character of G . Since thegroup of commutators of G is SL ( k ), we have χ ( u ) = det( u ) w for some w ∈ Z . It therefore suffices to calculate χ ( u ) when u = λ I , with λ ∈ k × .In this case u · F = λ − d F . Moreover, the section η f is homogeneous of degree − λ ∈ k × and F ∈ X d , then η f ( λ − d F ) /η f ( F ) = λ d , hence, ( ϕ ∗ u η )( F, η ) = λ dg = det( u ) w . This implies λ w = det( u ) w = λ dg , and we have proven (i).Let Φ ∈ Inv gh ( X d ) and s = τ (Φ) = Φ · η ⊗ h , let also w = dgh/ . Then( φ ∗ u s )( F, η ) = Φ( u · F ) · ( ϕ ∗ u η )( F, η ) h = det( u ) − d ( gh ) / Φ( F ) · det( u ) w h = Φ( F ) = s ( F, η ) , hence, τ (Φ) ∈ Γ( X d , λ ⊗ h ) G . Conversely, the inverse of τ is the map s s/η ⊗ h ,and this proves (ii). (cid:3) We made use of the following elementary lemma:
Lemma 3.2.2.
Let f ∈ k [ T , . . . , T n ] be irreducible and let g ∈ k ( T , . . . , T n ) be arational function which has neither zeroes nor poles outside the set of zeroes of f. Then there is an m ∈ Z and c ∈ k × such that g = cf m . Proof.
This is an immediate consequence of Hilbert’s Nullstellensatz, together withthe fact that the ring k [ T , . . . , T n ] is factorial. (cid:3) Modular forms as invariants.
Let d > g = (cid:0) d (cid:1) . Sincethe fibres of Y d −→ X d are nonsingular non hyperelliptic plane curves of genus g ,by the universal property of M g we get a morphism p : X g −−−−→ M g , where M g is the moduli stack of nonhyperelliptic curves of genus g and p ∗ λ = α by construction. This induces a morphism p ∗ : Γ( M g , λ ⊗ h ) −−−−→ Γ( X d , α ⊗ h ) . Let s ∈ Γ( M g , λ ⊗ h ). For every F ∈ X d , an element u ∈ G induces an isomorphism ϕ u : C F −−−−→ C u · F . By the universal property of M g , the diagram λ | p ( X d ) Id −−−−→ λ | p ( X d ) p ∗ y p ∗ y α ϕ ∗ u −−−−→ α is commutative. Hence ϕ ∗ u ◦ p ∗ ( s ) = p ∗ ( s ) , and this means that p ∗ s ∈ Γ( X d , α ⊗ h ) G . Combining this result with Prop.3.2.1(ii),we obtain: Proposition 3.3.1.
For any integer h ≥ , the linear map σ = τ − ◦ p ∗ is ahomomorphism: Γ( M g , λ ⊗ h ) −−−−→ Inv gh ( X d ) such that σ ( f )( F ) = f ( C F , ( p ∗ ) − η ) for any F ∈ X d and any section f ∈ Γ( M g , λ ⊗ h ) . (cid:3) ACOBIANS AMONG ABELIAN THREEFOLDS 11
We now make a link between invariants and Siegel modular forms. Let F ∈ X d and let η , . . . , η g be the basis of regular differentials on C F defined in Sec.3.2.Let γ , . . . γ g be a symplectic basis of H ( C, Z ) (for the intersection pairing). Thematrix Ω = [Ω Ω ] = R γ η · · · R γ g η ... ... R γ η g · · · R γ g η g belongs to the set R g ⊂ M g, g ( C ) of Riemann matrices, and τ = Ω − Ω ∈ H g . Corollary 3.3.2.
Let f ∈ S g,h ( C ) be a geometric Siegel modular form, e f ∈ R g,h ( C ) the corresponding analytic modular form, and Φ = σ ( θ ∗ f ) the corresponding invari-ant. In the above notation, Φ( F ) = (2 iπ ) gh e f ( τ )det Ω h . Proof.
Let λ = ( p ∗ ) − ( η ) and ω = ( θ ∗ ) − ( λ ). From Prop.2.3.1 and 3.3.1, we deduceΦ( F ) = ( θ ∗ f )( C F , λ ) = f (Jac C, ω ) . On the other hand, Prop.2.4.4 implies f (Jac C, ω ) = (2 iπ ) gh e f ( τ )det Ω h , from which the result follows. (cid:3) The case of genus
Klein’s formula.
We recall the definition of theta functions with (entire)characteristics [ ε ] = (cid:20) ε ε (cid:21) ∈ Z g ⊕ Z g , following [2]. The (classical) theta function is given, for τ ∈ H g and z ∈ C g , by θ (cid:20) ε ε (cid:21) ( z, τ ) = X n ∈ Z g q ( n + ε / τ ( n + ε / n + ε / z + ε / . The
Thetanullwerte are the values at z = 0 of these functions, and we write θ [ ε ]( τ ) = θ (cid:20) ε ε (cid:21) ( τ ) = θ (cid:20) ε ε (cid:21) (0 , τ ) . Recall that a characteristic is even if ε .ε ≡ odd otherwise. Let S g (resp. U g ) be the set of even characteristics with coefficients in { , } . For g ≥ h = | S g | / g − (2 g + 1) and e χ h ( τ ) = (2 iπ ) gh Y ε ∈ S g θ [ ε ]( τ ) . In his beautiful paper [17], Igusa proves the following result [ loc. cit. , Lem. 10and 11]. Denote by e Σ the modular form defined by the thirty-fifth elementarysymmetric function of the eighth power of the even Thetanullwerte. Recall thata principally polarized abelian variety ( A, a ) is decomposable if it is a productof principally polarized abelian varieties of lower dimension, and indecomposableotherwise.
Theorem 4.1.1. If g ≥ , then e χ h ( τ ) ∈ R g,h ( C ) . Moreover, If g = 3 and τ ∈ H ,then: (i) A τ is decomposable if e χ ( τ ) = e Σ ( τ ) = 0 . (ii) A τ is a hyperelliptic Jacobian if e χ ( τ ) = 0 and e Σ ( τ ) = 0 . (iii) A τ is a non hyperelliptic Jacobian if e χ ( τ ) = 0 . (cid:3) Using Prop. 2.2.1, we define the geometric modular form of weight hχ h ( A τ ) = (2 iπ ) gh e χ h ( τ )( dz ∧ · · · ∧ dz g ) ⊗ h . Then Ichikawa [15], [16] proved that χ h ∈ S g,h ( Q ). For g = 3, one finds χ ( A τ ) = − (2 π ) e χ ( τ )( dz ∧ dz ∧ dz ) ⊗ . 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 be a plane quartic defined over C suchthat C F is nonsingular. Let η , η , η be the classical basis of Ω [ C F ] and γ , . . . γ be a symplectic basis of H ( C F , Z ) for the intersection pairing. Let Ω = [Ω Ω ] = R γ η · · · R γ η ... ... R γ η · · · R γ η be a period matrix of Jac( C ) and τ = Ω − Ω ∈ H . Then Disc( F ) = 12 (2 π ) e χ ( τ )det(Ω ) . Proof.
The Cor.3.3.2 shows that I = σ ◦ θ ∗ ( χ ) is an invariant of weight 54, andfor any F ∈ X , I ( F ) = − (2 π ) e χ ( τ )det Ω . Moreover Th. 4.1.1(iii) shows that I ( F ) = 0 for all F ∈ X . Applying Lem. 3.2.2for the discriminant, we find by comparison of the weights that I = c Disc with c ∈ C a constant. But if F m is the Ciani quartic associated to a matrix m ∈ Sym ( k )as in Example 3.1.2, and if Disc F m = 0, then it is proven in [21, Cor. 4.2] thatKlein’s formula is true for F m and c = − . (cid:3) Remark . The morphism θ ∗ defines an injective morphism of graded k -algebras S ( k ) = ⊕ h ≥ S ,h ( k ) −−−−→ T ( k ) = ⊕ h ≥ T ,h ( k ) . In [14], Ichikawa proves that if k is a field of characteristic 0, then T ( k ) is generatedby the image of S ( k ) and a primitive Teichm¨uller form µ , ∈ T , ( Z ) of weight 9,which is not of Siegel modular type. He also proves in [16] that(2) θ ∗ ( χ ) = − · ( µ , ) . Th. 4.1.2 implies that µ , is actually equal to the discriminant up to a sign. Thismight probably be deduced from the definition of µ , , although we did not sort itout (see also [18, Sec. 2.4]). Remark . Besides [23] and [11] where an analogue of Klein’s formula is derivedin the hyperelliptic case, there exists a beautiful algebraic Klein’s formula, linkingthe discriminant with irrational invariants [9, Th.11.1].
ACOBIANS AMONG ABELIAN THREEFOLDS 13
Jacobians among abelian threefolds.
Let k ⊂ C be a field and let g = 3.We prove the following theorem which allows to determine whether a given abelianthreefold defined over k is k -isomorphic to a Jacobian of a curve defined over thesame 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 ω , ω , ω be a basis of Ω k [ A ] and γ , . . . γ a symplectic basis of H ( A, Z ) , in such a way that Ω = [Ω Ω ] = R γ ω · · · R γ ω ... ... R γ ω · · · R γ ω is a period matrix of ( A, a ) . Put τ = Ω − Ω ∈ H . (i) If e Σ ( τ ) = 0 then ( A, λ ) is decomposable. In particular it is not a Jaco-bian. (ii) If e Σ ( τ ) = 0 and e χ ( τ ) = 0 then there exists a hyperelliptic curve X/k such that (Jac
X, j ) ≃ ( A, a ) . (iii) If e χ ( τ ) = 0 then ( A, a ) is isomorphic to a Jacobian if and only if − χ ( A, ω ∧ ω ∧ ω ) = (2 π ) e χ ( τ )det(Ω ) is a square in k .Proof. The first and second points follow from Th.4.1.1 and Th.1.1.1. Suppose nowthat (
A, a ) is isomorphic over k to the Jacobian of a non hyperelliptic genus 3 curve C/k . Let F ∈ X be a plane model of C . Using Th.4.1.2 we get that − χ ( A, ω ∧ ω ∧ ω ) = (2 π ) e χ ( τ )det(Ω ) = 2 Disc( F ) so it is a square in k . On the contrary, Cor.2.4.2 shows that if ( A ′ , a ′ ) is a quadratictwist of a Jacobian ( A, a ) then the expression − f ( A ′ , ω ′ ∧ ω ′ ∧ ω ′ ) = (2 π ) e χ ( τ ′ )det(Ω ′ ) is not a square. (cid:3) Remark . Note that one does not really need Klein’s formula. Alternatively,we could use (2) which also proves that θ ∗ ( − χ ) is a square. Corollary 4.2.3.
In the notation of Th.4.2.1, the quadratic character ε of Gal( k sep /k ) introduced in Theorem 1.1.1 is given by ε ( σ ) = d/d σ , where d = s (2 π ) e χ ( τ )det(Ω ) , with an arbitrary choice of the square root. Beyond genus . It is natural to try to extend our results to the case g > g > θ ∗ ( χ h ) with h = 2 g − (2 g + 1). In [16, Prop.4.5] (see also [29]), it isproven that for g > θ ∗ ( χ h ) / g − (2 g − has as a square root a primitive element µ g,h/ ∈ T g,h/ ( Z ). If g = 4, in thefootnote, p. 462 in [20] we find the following amazing formula(3) e χ ( τ )det(Ω ) = c · ∆( X ) · T ( X ) . Here τ = Ω − Ω , with Ω = [Ω Ω ] beeing a period matrix of a genus 4 nonhyperelliptic curve X given in P as an intersection of a quadric Q and a cubicsurface E . The elements ∆( X ) and T ( X ) are defined in the classical invarianttheory 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 casefor g > g > . To begin with, when g is even, we cannot use Cor.2.4.2 to distinguish betweenquadratic twists. In particular, using the previous result, we see that χ h ( A, ω k ) isa square when A is a principally polarized abelian variety defined over k which isgeometrically a Jacobian. A natural question is:— What is the relation between this condition and the locus of geometricJacobians over k ?Let us assume now that g is odd. As we pointed out in Rem.4.2.2, the existence ofthe square root is almost sufficient to answer Serre’s questions when g = 3. This isnot the case when g >
3. The proof of the corollary 2.4.3 shows that χ h ( A ′ ) = ( d ) h/ χ h ( A )for a Jacobian A and a quadratic twist A ′ . What enables us to distinguish between A and A ′ when g = 3 is the following: if A is the Jacobian of a curve then χ h ( A ) isa square whereas d is not and h/ g >
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Gilles LachaudInstitut de Math´ematiques de LuminyUniversit´e Aix-Marseille - CNRSLuminy Case 907, 13288 Marseille Cedex 9 - FRANCE
E-mail address : [email protected] Christophe RitzenthalerInstitut de Math´ematiques de Luminy
E-mail address : [email protected] Alexey ZykinInstitut de Math´ematiques de LuminyMathematical Institute of the Russian Academy of SciencesLaboratoire Poncelet (UMI 2615)
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