aa r X i v : . [ m a t h . AG ] M a y TORELLI REVISITED
STEPHEN MEAGHER
Abstract.
Assume given an Abelian variety which is a geometric Jacobian.We give an invariant of the moduli point of the Abelian variety which deter-mines the minimal extension of its ground field over which it is a Jacobian. Forgenus 3, we express this invariant in terms of weight 18 Siegel and Teichm¨ullermodular forms and periods of the Abelian variety. These results answer ques-tions of Serre posed in letters to Top from 2003 about Abelian 3-folds withindecomposable principal polarisations. The results we obtain hold in anycharacteristic other than 2 and thus generalise previous work of Ritzenthaler,Lachaud and Zykin on the questions in Serre’s letters. Introduction
Let (
A, a ) be an Abelian g -fold with an indecomposable principal polarisation a over a field k ; let k s be the separable closure of k . Assume that there is a smoothirreducible genus g curve C over k s whose Jacobian variety satisfies(Jac( C ) , λ Θ ) ∼ = ( A, a ) ⊗ k s , where λ Θ is the polarisation of Jac( C ) induced by the theta divisor. (For exampleif g ≤ C always exists [OU73]).Now assume k is not of characteristic 2. For any character ǫ : Gal( k s /k ) −→{± } we let ( A, a ) ǫ be the indecomposably principally polarised Abelian varietyobtained by twisting ( A, a ) over ( k s ) ker( ǫ ) by the [ −
1] automorphism. Thus ifker( ǫ ) = Gal( k s /k ) then ( A, a ) ǫ = ( A, a ). Serre has observed ([Lau03] Th´eor`emes4 & 5) the following refinement of the Torelli theorem: there is a unique character ǫ : Gal( k s /k ) −→ {± } and a unique C/k so that(Jac( C ) , λ Θ ) ∼ = k ( A, a ) ǫ . Moreover ǫ is trivial if C is hyperelliptic. Thus there is a D ( A,a ) ∈ k such that k ( p D ( A,a ) ) = ( k s ) ker( ǫ ) is the minimal field extension of k over which ( A, a ) is aJacobian. This motivates
Definition 1.
Let R be a Z [1 /
2] algebra. Let (
A, a ) / Spec( R ) be a family of prin-cipally polarised Abelian g -folds whose geometric fibres are Jacobians. A function∆ ∈ R is called a twisting function if for all fields k and all points x ∈ Spec( R )( k )we have:i) ∆( x ) = 0 if and only if ( A, a ) x is the Jacobian of a hyperelliptic curve over k ii) ( A, a ) x is a Jacobian if and only if ∆( x ) is a square in k .Our main result is i.e. a commutative group scheme which is smooth and proper over Spec( R ) and whose geo-metric fibres are connected Abelian g -folds. Theorem 2.
Let S be a scheme over which is invertible. Let ( A, a ) /S be afamily of principally polarised Abelian g -folds whose geometric fibres are Jacobians.If either g = 3 or the geometric fibres of ( A, a ) /S are all non-hyperelliptic Ja-cobians, then there exists an affine open cover { U i } of S and twisting functions ∆ i ∈ H ( U i , O U i ) for ( A, a ) U i . Remark:
For g > M be the moduli stack of smooth genus three curves and let A i , bethe moduli stack of Abelian threefolds equipped with an indecomposable principalpolarisation. The Torelli morphism t : M −→ A i , associates to a family of genus three curves C/S its family of principally polarisedJacobians (Jac( C ) /S, λ Θ ). The classical Torelli theorem implies that this map isa surjection on k s valued points. The refined Torelli theorem states that the fibreof t above an Abelian threefold ( A, a ) defined over k has splitting field given by( k s ) ker( ǫ ) for the character ǫ : Gal( k s /k ) −→ µ as above. In particular Theorem 2means that the Torelli morphism is “2 to 1 and ramified along the hyperellipticlocus over Z [1 / Theorem 3.
Let χ be the modular form obtained by the product of the eventheta nulls. Then χ extends to a modular form on A i , with coefficients in Z .Moreover χ vanishes on the hyperelliptic locus of A i , . Let Discr be the weight Katz-Teichm¨uller modular form defined in Section 5. Then
Discr vanishes on thehyperelliptic locus of M and t ∗ χ = γ Discr for some γ ∈ Z ∗ . Combining Theorems 2 and 3 we obtain
Theorem 4.
Let R ⊂ C be a ring containing / and let ( A, a ) be a principallypolarised Abelian threefold over R . Assume that Ω A/R is free and that there is atwisting function ∆ ∈ R for ( A, a ) . Let { η , . . . , η } be a basis for H ( A an C , Z ) andlet ξ , ξ , ξ be a basis for Ω A/R ( R ) . Let Ω be the matrix of integrals of the ξ i alongthe η j with j ∈ { , , } and let Ω be the matrix of integrals of the ξ i along the η j with j ∈ { , , } . Let τ = Ω · Ω − and let χ hol18 ( τ ) denote the product of the even theta nulls evaluated at τ . Let γ be as in Theorem 3. Then for each maximalideal m ⊂ R (2 π ) χ hol18 ( τ )(det(Ω )) + m = − γ ∆( A, a ) + m , up to a square unit in R/ m . Remarks: 1)
The hypotheses of Theorem 4 hold automatically for R ⊂ C alocal Z [1 /
2] algebra. The modular forms χ and Discr have long histories going back to Klein [Kle73]and Igusa [Igu67]. Lachaud, Ritzenthaler and Zykin [LR07, LRZ] have also given i.e. C/S is a smooth, proper morphism whose geometric fibres are irreducible genus 3 curves.
ORELLI REVISITED 3 proofs of Theorems 3 and 4 in characteristic 0 without reference to twisting func-tions. They calculate explicitly that γ = −
1. The main novelty of our treatment isa proof of Theorem 2 in genus 3 which does not refer to modular forms. We give asecond proof of Theorem 2 for g = 3 as a corollary of Theorem 3. Theorems 3 and4 answer questions of Serre in letters to Top [Ser03]. Explicit formulae for specialtypes of twisting functions have been given in [AT02] and [HLP00]. A suggestionabout candidate modulur forms for analogues of Theorems 3 and 4 in genus 4 isgiven in the final section of [LRZ].2. Moduli spaces
Let M g be the category fibred in groupoids over schemes whose objects aresmooth proper morphisms of relative dimension 1 with smooth connected genus g curves for geometric fibres. Let A g, be the category fibred in groupoids overschemes whose objects are Abelian schemes of relative dimension g equipped witha principal polarisation. The Torelli morphism is the morphism given by t : M g −→ A g, : C/S (Jac( C ) /S, λ Θ ) . The categories M g and A g, are Deligne-Mumford stacks ([DM69, FC90]) and theTorelli morphism is a morphism of fibred categories.For each N ∈ Z we fix a primitive N th root of unity ζ N ∈ ¯ Q such that if M and N are coprime integers, ζ MN = ζ M ζ N . We consider only symplectic level N structures which send the determinant of a standard basis to ζ N . Let M g,N be thefunctor from Z [1 / N, ζ N ]-schemes to sets given by S
7→ { ( C, α ) | C ∈ M g ( S ) and α : Jac( C )[ N ] ∼ = ( Z /N ) gS × ( Z /N ) gS is symp. } / ∼ = S . If N ≥ M g,N is represented by a smooth irreducible scheme over Z [1 / N, ζ N ]([OS80] Theorem 1.8, [Pop77] p134 Theorem 10.10, p142 Remark 2, p104 Theorem8.11; for smoothness [HM98] p103 Lemma 3.35; for irreducibility [DM69]).Likewise for N ≥ Z [1 / N, ζ N ]-schemes to sets given by S
7→ { ( A, a, α ) | ( A, a ) ∈ A g, ( S ) and α : A [ N ] ∼ = ( Z /N ) gS × ( Z N ) gS is symp. } / ∼ = S , is represented by an irreducible smooth projective scheme A g, ,N over Z [1 / N, ζ N ]([GIT] p139 Theorem 7.9; for smoothness see [Oor70] p242 Theorem 2.33; for irre-ducibility see [FC90]).For lack of reference we prove Proposition 5.
Let ( A, a ) /S be a principally polarised Abelian scheme of relativedimension g . Let N ≥ be an integer which is invertible over S and let S ( A, a, N ) be the the functor from S -schemes to sets whose T -valued points are given by S ( A, a, N )( T ) = { α : A T [ N ] ∼ = ( Z /N ) gT | α is symplectic } . Let Sp g ( Z /N ) be constant group scheme associated to the group of symplectic auto-morphisms of ( Z /N ) g × ( Z /N ) g . Then S ( A, a, N ) is a Sp g ( Z /N ) torsor in the fppftopology. It is therefore represented by a scheme which is finite ´etale and Galoiswith structure group Sp g ( Z /N ) over S .Proof. The cover A [ N ] → S is fppf and S ( A, a, N )( A [ N ]) is non-empty. The actionof Sp g ( Z /N ) on T -valued points is clearly transitive and free. Moreover S ( A, a, N )is an fppf sheaf for A [ N ] and ( Z /N ) are fppf sheaves and so symplectic isomorphisms STEPHEN MEAGHER between them form an fppf sheaf. By ([GaDe70] p363) it is represented by a finite´etale scheme over S . (cid:3) Remark:
By construction there is a canonical morphism S ( A, a, N ) → A g, ,N .The induced morphism A g, ⊗ Z [1 /N ] → [ A g, ,N / Sp g ( Z /N )] : ( A, a ) /S ( S ( A, a, N ) /S, S ( A, a, N ) → A g, ,N )is an isomorphism of stacks as can be seen by using descent, although we do notmake explicit use of this fact.3. Proof of Theorem 2
In this section all schemes will be over Z [1 / Proposition 6.
Let R ⊂ R be a finite extension of local Noetherian regular rings.Then R is free as an R module.Proof. As R is regular, the global dimension of R is equal to the dimension of R ([Ser00] p77 Corollary 1). The projective dimension of R is therefore equal tothe dimension of R minus the depth of R over R ( ibid. p75 Proposition 21).Now R and R are Cohen-Macaulay as they are regular ( ibid. p77 Corollary 3)and therefore the depth of R over R is equal to the dimension of R ( ibid. p63Proposition 12). Thus the projective dimension of R over R is zero and R isfree. (cid:3) Let N ≥ t N : M g,N → A g, ,N : ( C, α ) (Jac( C ) , λ Θ , α )admits an involution τ : M g,N −→ M g,N : ( C, α ) ( C, − α ) . Let k be a field, the Torelli theorem [Wei57] states that C/k admits an automor-phism mapping to the [ −
1] automorphism of Jac( C ) if and only if C is hyperelliptic,i.e. ( C, α ) ∼ = ( C, − α )if and only if C is hyperelliptic. That is the fixed locus of τ is equal to the hy-perelliptic locus. Let V g,N be the geometric quotient of M g,N by the constantgroup scheme associated to h τ i . We note that V g,N exists and yields a finite sur-jective morphism q g,N : M g,N −→ V g,N and a morphism ι N : V g,N −→ A g, ,N suchthat t N = ι N ◦ q g,N ([AbV08] III. 12). For every field k not of characteristic 2 themorphism ι N ⊗ k is an immersion ([OS80] Theorem 3.1). Thus ι is unramified; let Z be the closure of its image in A g, ,N . Then let x ∈ V g,N and let y ∈ Z be itsimage and p ∈ Spec( Z [1 /N, ζ n ]) be the image of x under the structure morphism.The rings O V g,N ,x and O Z,y are local Z [1 /N, ζ n ] p algebras, and as ι ⊗ k ( p ) is animmersion and V g,N is integral, by the local criterion for flatness applied to theunique prime p ∈ p ∩ Z the ring O V g,N ,x is flat as an O Z,y module. Thus the mor-phism V g,N → Z is ´etale of degree 1 and therefore an open immersion, hence ι isan immersion. In particular given ( A, a ) / Spec( R ) as in Theorem 2 the canonicalmorphism from Spec( R ) to A g, ,N factors through V g,N . We remark that V g,N isNoetherian for an ascending chain of sheaves of ideals on V g,N corresponds to anascending chain of τ invariant sheaves of ideals on M g,N , and M g,N is Noetherian. ORELLI REVISITED 5
Proposition 7.
Let H g,N denote the image of the hyperelliptic locus in V g,N . Thesheaf ( q g,N ) ∗ O M g,N \ q − g,N ( H g,N ) is a locally free O V g,N \ H g,N -module of rank . Thesheaf ( q ,N ) ∗ O M ,N is a locally free O V ,N -module of rank Proof. As V g,N is the geometric quotient of M g,N and τ is fixed point free awayfrom q − g,N ( H g,N ), the scheme V g,N \ H g,N is Noetherian as remarked, and regular as M g,N is regular and Noetherian and so the first claim follows from Proposition 6.From the classical Torelli theorem we deduce that the map from V ,N to A i , ,N isan isomorphism. The moduli spaces M ,N and A i , ,N are regular and Noetherianand therefore V ,N is regular and Noetherian and the proposition follows fromProposition 6. (cid:3) Remarks: 1)
The second part of Proposition 7 is false for g > V g,N is not regular and Theorem 13b) p83 of [Ser00] implies that Proposition 7 is trueif and only if V g,N is regular. Indeed Oort and Steenbrink have shown that thetangent space at a hyperelliptic point of V g,N has dimension g ( g + 1) /
2. Howeverthe Krull dimension of V g,N is clearly 3 g − We are indebted to Marius van der Put for pointing out Proposition 6 to us andthe fact that it implied Proposition 7 which is the basis of the construction of thetwisting functions of Theorem 2.We observe that Sp g ( Z /N ) acts on M g,N , V g,N and A g, ,N and that t N is Sp g ( Z /N ) equivariant. We write V sm g,N for the regular locus of V g,N . Thus V sm g,N = V g,N \ H g,N if g > V sm3 ,N = V ,N . Proof of Theorem 2. Step I
We first construct the twisting functions:Put S = Spec( R ). Fix an integer N ≥ R is invertible, forexample N = 4, and set T N = S ( A, a, N ) as in Proposition 5. We note that T N is affine as T N /S is finite and we write R N for the affine coordinate ring of T N .The Abelian scheme ( A, a ) T N has a universal level N symplectic structure α bydefinition of T N and thus there is a morphism ψ : T N −→ V sm g,N corresponding to ( A T N , a T N , α ).Put M N = T N × V sm g,N M g,N . By construction there is a family of genus g curves C/M N with a symplectic level N structure α on the Jacobian Jac( C ) /M N and anisomorphism (( A, a ) T N ) M N ∼ = (Jac( C ) , λ Θ ) which is Sp g ( Z /N ) equivariant.Then the morphism M N → T N is affine as t N is affine and the base change ofan affine morphism is affine. Thus M N is affine and we write R ′ N for its coordinatering. Consider an open affine cover { U i } of V sm g,N over which ( q g,N ) ∗ O M g,N ( U i ) isfree as an O V sm g,N ( U i ) module. Consider also an open affine cover { W ij } of ψ − ( U i ).Then the preimages X ij of the W ij in M N are affine and cover M N and moreover O M N ( X ij ) = O T N ( W ij ) ⊗ O V sm g,N ( U i ) ( q g,N ) ∗ O M g,N ( U i ) . Therefore R ′ N is locally free of rank 2 as an R N module. Furthermore the involution τ acts on R ′ N so that R N = ( R ′ N ) τ and the inclusion R N ⊂ R ′ N is Sp g ( Z /N ) STEPHEN MEAGHER equivariant. By definition of T N we have R = R Sp g ( Z /N ) N . Set R ′ = ( R ′ N ) Sp g ( Z /N ) . Then the map R ′ ⊗ R R N → R ′ N : a ⊗ b ab, is an isomorphism, as the category of R N modules with an Sp g ( Z /N ) action isequivalent to the category of R modules and the functor which takes invariants isinverse to the functor ⊗ R R N ([AbV08] III. 12). In particular M N / Spec( R ′ ) is ´etaleas the base change of an ´etale morphism is ´etale.Now R ′ is a locally free R module of rank 2 ( ibid. ). Without loss, we may assumethat R ′ is free as an R module. Then there exists a function F ∈ R ′ such that R ′ = R · h , F i and F = aF + b for some a, b ∈ R . Put √ ∆ = F − a/ . Then ∆ = F − aF + a / ∈ R . Moreover as R ′ N = R ′ ⊗ R R N we have R ′ N = R N [ √ ∆]and τ ( √ ∆) = −√ ∆ . Step II
We now show that ∆ is a twisting function for (
A, a ) T N .Let pr ψ be the morphism from M N to M g,N induced by ψ . If g = 3 and x ∈ M N ( k ) is hyperelliptic; then τ ( x ) = x and therefore √ ∆( x ) = 0. Likewise forany x ∈ M N ( k ) if √ ∆( x ) = 0 then τ ( x ) = x and therefore x ∈ M N ( k ) is hyperel-liptic.Now given x ∈ T N ( k ), if ∆( x ) is a square then M N /T N is split above x and thereexists a y ∈ M N ( k ) lying above x . Then t N (pr ψ ◦ y ) = ψ ◦ x and from the moduliinterpretation (Jac( C ) , λ Θ ) y = (( A, a ) T N ) x .Let x ∈ T N ( k ) and assume there is a curve C /k such that (( A, a ) T N ) x =(Jac( C ) , λ Θ ). This means that Jac( C ) admits a level N structure α and thus t − N ( ψ ◦ x )( k ) = { ( C, α ) , ( C, − α ) } and therefore t − N ( ψ ◦ x ) and hence M N × T N x is split over k . This implies that∆( x ) is a square. Step III
We now show that ∆ is a twisting function for (
A, a ) /S .Given σ , σ ∈ Sp g ( Z /N ) there is a canonical isomorphism θ σ ,σ : σ ∗ C ∼ = σ ∗ C given by pulling back the corresponding isomorphism on the universal curve over M g,N . Thus the pullbacks to M N × Spec( R ′ ) M N of σ ∗ C and σ ∗ C are isomorphicvia θ σ ,σ ,M N and moreover θ σ ,σ ,M N × M N = θ σ ,σ ,M N × M N ◦ θ σ ,σ ,M N × M N . ORELLI REVISITED 7
In other words the σ ∗ C/M N and the isomorphisms θ σ ,σ form descent datum withrespect to the ´etale cover M N / Spec( R ′ ). As M g is a stack, descent datum areeffective and there exists a unique curve (up to isomorphism) ˜ C/ Spec( R ′ ) whosepull back to M N is isomorphic with C/M N . Similarly as A g, is a stack the isomor-phism (( A, a ) T N ) M N ∼ = (Jac( C ) , λ Θ ) descends to an isomorphism ( A, a ) Spec( R ′ ) ∼ =(Jac( ˜ C ) , λ ˜Θ ).Now given x ∈ S ( k ) if ∆( x ) is a square then there is a point y ∈ Spec( R ′ )( k )lying above x and (Jac( ˜ C ) , λ ˜Θ ) y = ( A, a ) x .Assume there is a curve C /k such that (Jac( C ) , λ Θ ) ∼ = ( A, a ) x . As T N /S is´etale, the scheme ( T N ) x is isomorphic to the spectrum of a direct sum of fields; as T N /S is Galois these fields are all isomorphic to a given field k N . That is for somepositive integer s we have ( T N ) x = Spec( ⊕ si =1 k N ) . By Step II ∆ is a twisting function for T N and therefore ∆( x ) is a square in k N .Therefore the scheme ( M N ) Spec( R ′ ) x has exactly two connected components, eachof which is isomorphic with ( T N ) x . In other words( M N ) Spec( R ′ ) x = Spec( ⊕ j =1 ⊕ si =1 k N ) . The scheme Spec( R ′ ) x is isomorphic to the spectrum of k [ X ] / ( X − ∆( x )). Wetherefore have an inclusion of rings ⊕ j =1 ⊕ si =1 k N ⊕ si =1 k N o o k [ X ] / ( X − ∆( x )) O O k o o O O Now both vertical arrows are Galois extensions of ´etale k -algebras with Galoisgroup Sp g ( Z /N ). Therefore given y N ∈ ( M N ) Spec( R ′ ) x and x N ∈ ( T N ) x and y ∈ Spec( R ′ ) x such that y N lies above y , the decomposition group of y N /y is equalto the decomposition group of x N . The residue field of y N and x N is k N and theresidue field of y is k ( √ ∆( x )). Thus the Galois groups of k N /k and k N /k ( √ ∆( x ))are equal, which implies that ∆( x ) ∈ k .Now assume g = 3 and assume k = k is a field, if y N ∈ M N ( k s ) lies above y ∈ Spec( R ′ )( k ) then C y N ⊗ k s ∼ = ˜ C y ⊗ k s . Moreover ∆( x ) is zero if and only if √ ∆( y N ) is zero. Thus the locus of vanishing of ∆ is the hyperelliptic locus on S . (cid:3) Line bundles on stacks and Modular forms
Definition 8 (Mumford [Pic63] p64) . A line bundle L on a stack X is given bythe following data: • for each section x : S −→ X an O S module L ( x ) on S • for each morphism f : S ′ −→ S an isomorphism ψ f,x : f ∗ L ( x ) −→ L ( x ◦ f ).subject to the compatibility condition ψ f ◦ g,x ◦ ξ = ψ g,x ◦ f ◦ g ∗ ψ f,x STEPHEN MEAGHER where ξ is the natural isomorphism between g ∗ f ∗ L ( x ) and ( f ◦ g ) ∗ L ( x ).A global section of a line bundle L on a stack X is given by the following data: foreach section x : S −→ X there is a section s x ∈ H ( S, L ( x )), subject to the condition ψ f,x ( f ∗ s x ) = s x ◦ f .For example O X is given by O X ( x : S −→ X ) = O S . We write H ( X , L ) for theH ( X , O X ) module of global sections of L .Given global sections s and s of a line bundle L on a stack X if for each section x : S −→ X the Cartier divisor of zeroes of s is equal to the Cartier divisor of zeroesof s then there is a function λ x ∈ O ∗ S ( S ) such that λ x s = s . In particular the λ x are unique as L ( x ) is a line bundle and therefore the λ x definea global section λ of the structure sheaf O X of X which is a unit.Given a scheme X and a morphism f : X −→ X which is initial in the categoryof maps from X to schemes we say that X is a coarse moduli space for X if theisomorphism classes of X ( k ) are in bijection via f with X ( k ) for every algebraicallyclosed field k . Then the natural injectionH ( X, O X ) ⊂ H ( X , O X )is an equality: an element of the right hand side corresponds to a morphism f : X −→ A which factors uniquely through X . The stacks A g, / Z and M g / Z both have coarse moduli spaces ([GIT] Theorem 7.10, Corollary 7.14), which areZariski locally quasi-projective and therefore separated and of finite type over Z .Let f : X −→ S be either an Abelian scheme or a family of genus g curves. Forlack of reference we prove Proposition 9.
Let f : X −→ S be as above, then the sheaf f ∗ Ω X/S is locally free.Proof.
In case
X/S is an Abelian scheme the natural morphism f ∗ Ω X/S → O ∗ Ω X/S is an isomorphism and the Proposition follows. In case
X/S is a family of curves(i.e. smooth proper with geometrically connected fibres of dimension 1), then O X is flat over O S . Thus higher direct images of O X under f commute with arbitrarybase change ([EGAIII.2] 7.7.5.3 for Y = S and P • equal to the complex supportedin degree 0 determined by O X ). Therefore R f ∗ O C is locally free ( ibid apply Propo-sitions 7.7.10a) & 7.8.4 to Proposition 7.8.5). By relative Serre duality f ∗ Ω X/S islocally free ([Liu02]). (cid:3)
The Hodge bundle of f is the sheaf ω X/S := det( f ∗ Ω X/S ) . Let ρ be a positive integer; a section s of ω ρX/S is called a modular form of weight ρ . We have line bundles ω ρA/ A g, and ω ρC/ M g on A g, and M g respectively definedby the corresponding sheaves of modular forms because of basic properties of dif-ferentials under base change. Sections of these line bundles are called Katz-Siegeland Katz-Teichm¨uller modular forms respectively (cf. [Kat73]). ORELLI REVISITED 9
Proposition 10. If f : C −→ S is a family of genus g curves and π : Jac( C ) −→ S is its Jacobian then ω C/S is isomorphic with ω Jac( C ) /S .Proof. As Jac( C ) /S is a group scheme we have an isomorphism π ∗ Ω Jac( C ) /S = O ∗ Ω Jac( C ) /S . Interpreting sections of R f ∗ O C as Cech 1-cocycles shows that R f ∗ O C ( U ) = Hom( O ∗ Ω Jac( C ) /S ( U ) , O S ( U )) . Then Serre duality for smooth and proper morphisms shows that ([Liu02]) π ∗ Ω Jac( C ) /S ∼ = f ∗ Ω C/S . (cid:3) Thus the pullback of a Katz-Siegel modular form via the Torelli morphism is aKatz-Teichm¨uller modular form.5.
The modular form
Discr
Definition 11.
The classical family of plane quartics is the family of quartics f : Q −→ P where Q is the subscheme of P P defined by the equation( F := X i + j + l =4 a ijl X i Y j Z l ) = 0 . Definition 12 (Classical discriminant) . Let F X , F Y and F Z be the derivatives of F with respect to X, Y and Z . The classical discriminant of f : Q −→ P is definedto be the resultant of F X , F Y and F Z ; it is a section of ω Q/ P .We recall the main properties of the resultant (see [Lan02] pp388-404 for details).Let { U j } be the standard affine cover of P . The locus of ( a ijl ) ∈ U i such that F X , F Y and F Z have a common zero is defined by the single homogenous equationRes U i ( F X , F Y , F Z ) which generates a prime ideal. Moreover there is a canonicalchoice of Res U i ( F X , F Y , F Z ). Finally if B is a 3 by 3 matrix with entries in U i , thenRes( F X ◦ B − , F Y ◦ B − , F Z ◦ B − ) = det( B ) Res( F X , F Y , F Z ) . Therefore the Res U i ( F X , F Y , F Z ) glue to form a unique global section Res( F X , F Y , F Z )of ω Q/ P .Let f : C −→ S be any family of genus 3 curves. There is a affine open cover { V i } of S such that f ∗ Ω C Vi /V i and f ∗ Sym (Ω C Vi /V i ) are free (by analogous reasoning toProposition 9). Put V i = Spec( R i ) and let A i be the graded R i algebra given bythe direct sum of the tensor powers of the R i modules f ∗ Ω C Vi /V i ( V i ). We then havea morphism j i : C V i −→ Proj( A i )and Proj( A i ) ∼ = P V i . Let I i be the homogenous ideal corresponding to the image of C V i . The degree 4elements I (4) i of I i are equal to the kernel of the surjective R i linear morphismSym ( f ∗ Ω C Vi /V i )( V i ) → f ∗ Sym (Ω C Vi /V i )( V i ) . Now cohomology and base change commute for Sym (Ω C Vi /V i ) thus for each s ∈ V i ( k ) dim k H ( C s , Sym (Ω C s /k )) = dim k f ∗ Sym (Ω C Vi /V i ) s . Thus by Riemann-Roch for curves over fieldsrank R i ( I (4) i ) = 15 − dim k H ( C s , Sym (Ω C s /k )) = 1 . Thus there is a unique degree 4 element F i ∈ I i up to R ∗ i -scaling. There is amorphism ι i : V i → P such that F i the pullback of F with ι i (up to scalingby an element of R ∗ i ). The ι ∗ i Res( F X , F Y , F Z ) are independent of ι i and thereforeglue to form a global section Discr C/S of ω C/S . Moreover by construction givena morphism h : S ′ −→ S we have a canonical isomorphism ψ h : h ∗ ω C/S −→ ω C S ′ /S ′ and Discr C S ′ /S ′ = ψ h ( h ∗ Discr
C/S ). Definition 13.
The discriminant modular form Discr is the Katz-Teichm¨uller mod-ular given by the Discr
C/S as above.
Proposition 14.
Discr vanishes on the hyperelliptic locus.Proof. If C/k is hyperelliptic, then the hyperelliptic structure morphism f : C −→ P defines an element of the function field k ( C ) whose polar divisor is equal to halfthe canonical class. Therefore we have a basis for Ω C/k given byΩ
C/k = k h , f, f i . Thus the corresponding quartic for
C/k is the square of a conic and is thereforesingular. (cid:3) The modular form χ Let h be the Siegel upper half space of degree 3: that is the complex domainof 3 × π an : A an −→ h be the universal complex analytic Abelian threefold over h . Ana-lytic uniformisation provides a family f : C × h −→ h of complex vector spaces,and a family h : Λ −→ h of lattices of rank 6 whose fibre above a point τ is h − ( τ ) := Z + τ Z ⊂ f − ( τ ) = C × { τ } so that A an ∼ = ( C × h ) / Λas complex analytic manifolds above h .Let z = ( z ( τ ) , z ( τ ) , z ( τ )) denote the standard coordinates of C × { τ } ⊂ C × h . The Riemann theta function is the holomorphic function on C × h defined by the formula ϑ ( z ; τ ) := X n ∈ Z e π √− n t τn +2 π √− n · z . We have a canonical level 2 structure on π an : A an −→ h given by the standardcoordinates on Z + τ Z . Using this we identify a 2-torsion point p ∈ A an [2]( h )with a unique element m + τ m ′ ∈ / Z + τ Z ) such that m and m ′ have entries ORELLI REVISITED 11 equal to either 0 or 1 /
2. The analytic theta nulls are the holomorphic functions on h given by the formulas ϑ (cid:20) mm ′ (cid:21) (0; τ ) := ϑ ( m + τ m ′ ; τ ) . The function p m · m ′ mod 2 defines a quadratic form with values in Z / Z .Thus an analytic theta null ϑ (cid:20) mm ′ (cid:21) (0; τ )is called even if 4 m · m ′ is even. Definition 15.
Put χ hol18 ( τ ) := Y m · m ′ ∈ Z ϑ (cid:20) mm ′ (cid:21) (0; τ ) , and χ = χ hol18 (2 π ) ( dz ∧ dz ∧ dz ) . Igusa has shown ([Igu67] Lemmas 10 and 11) that χ hol18 transforms as a weight18 Siegel modular form under the group PSp ( Z ), and vanishes on the hyperellip-tic locus. By the Koecher principle, this means that χ is a modular form on asmooth toroidal compactification of A , ⊗ C and therefore an algebraic modularform by GAGA (cf. [FC90] p141). Moreover χ is a modular form on A , / Z bythe q -expansion principle ([FC90] p140).For the reader unhappy with applying GAGA to a stack, despite the authority ofFaltings-Chai, we offer an alternative argument. First of all χ defines a modularform on the analytic moduli space A an3 , , , which by the Koecher principle, extendsto a smooth toroidal compactification of A an3 , , and is therefore analytic by GAGA.Now if ( A, a ) /S is any principally polarised Abelian scheme over a C scheme S , thecover S ( A, a, → S (Proposition 5) is Galois with Galois group Sp ( Z / χ to T is Sp ( Z /
4) invariant and sodescends to S . Thus χ defines an algebraic modular form on A , ⊗ C .7. Proof of Theorem 3
Step I:
The global regular functions of M are equal to the global regular func-tions of its coarse moduli space M (see Section 4). Moreover M ⊗ Q is isomor-phic with the coarse moduli space of indecomposable principally polarised Abelian3-folds A i , ⊗ Q ([OS80] Theorem 3.2). The coarse moduli space A i , ⊗ Q is quasi-projective with boundary of codimension 2 under the Satake embedding. Thereforeusing the theorem of Bertini ([Jou83] p89, Th´eor`eme 6.10), any two closed pointson A i , ⊗ Q may be connected by a complete curve C lying within A i , ⊗ Q , andtherefore A i , ⊗ Q has only the field of constants Q as global regular functions. As Z ⊂ Q is flat, and M is separated and of finite type over Z , we have ([EGAIII.1]1.4.15) that H ( M , O M ) = Z . Step II:
The Katz-Teichm¨uller modular forms t ∗ χ and Discr both have weight18. Assume there is a section x : S −→ M such that for all γ ∈ Γ( S, O S ) we have( t ∗ χ ) x = γ Discr x . Then there exists an open affine Spec( R ) ⊂ S such that for all γ ∈ R we have ( t ∗ χ ) x,R = γ Discr x,R . Moreover, since for any two distinctprimes p, l ∈ Z the open affines Spec( R [1 /l ]) and Spec( R [1 /p ]) cover Spec( R ), wemay without loss assume that we have a rational prime p ≥ /p ∈ R .Now let C/ Spec( R ) be the family of genus three curves corresponding to R . Let T ′ = S (Jac( C ) , λ Θ , p ) as in Proposition 5. Then if the pullbacks of ( t ∗ χ ) x,R of Discr x,R to T ′ satisfy ( t ∗ χ ) T ′ = γ Discr T ′ for some γ ∈ Γ( T ′ , O T ′ ) then γ is invariant under the Sp ( Z /p ) and therefore lies in R . Therefore for all γ ∈ Γ( T ′ , O T ′ ) we have ( t ∗ χ ) T ′ = γ Discr T ′ . Now the Jacobian of C T ′ admits a level p structure, and therefore there is a morphism ψ : T ′ −→ M ,p and( t ∗ χ ) T ′ = ψ ∗ ( t ∗ χ ) M ,p and Discr T ′ = ψ ∗ Discr M ,p . Let C p be the universal curve over M ,p . As Z [1 /p, ζ p ] ⊂ C is flat and M ,p isseparated and of finite type over Z [1 /p, ζ p ] we have ( ibid. ) thatH ( M ,p , ω C p / M ,p ) ⊗ C = H ( M ,p ⊗ C , ω C p, C / M ,p ⊗ C ) . Now ( t ∗ χ ) C and Discr C have the same divisor of zeroes on M ⊗ C , namely thehyperelliptic locus. Thus there is a γ ∈ C such that( t ∗ χ ) M ,p ⊗ M ,p ⊗ γ. Therefore γ ∈ Z [1 /p, ζ p ] ∗ , and( t ∗ χ ) M ,p = γ Discr M ,p , which is a contradiction. Thus for all x : S −→ M there exists a γ x ∈ Γ( S, O S )such that ( t ∗ χ ) x = γ x Discr x . Thus the γ x define an element γ ∈ H ( M , O M ) ∗ such that( t ∗ χ ) = γ Discr M ,p . Therefore γ ∈ H ( M , O M ) ∗ = Z ∗ . Proof of Theorem 4
Let R be a ring containing 1 / R ⊂ C . Let ( A, a ) / Spec( R )be an Abelian scheme of relative dimension 3 with an indecomposable principal po-larisation. Let ∆ be a twisting function for ( A, a ). Assume that Ω
A/R free of rank 3and let { ξ , ξ , ξ } be a basis. Let s : Spec( R ) −→ A i , be the section correspondingto ( A, a ). Then ∆( ξ ∧ ξ ∧ ξ ) and ( χ ) s are both global sections of ω A/ A i , ( s ) with the same divisor of zeroes. Thereforethere is a κ ∈ R ∗ such that κ ∆( ξ ∧ ξ ∧ ξ ) = (2 π √− χ hol18 ( dz ∧ dz ∧ dz ) . ORELLI REVISITED 13
Let B ( A, a ) denote the set of isomorphisms between H (( A C ) an , Z ) and Z . Let M ( C ) be the set of 3 × C . We define functionsΩ : B ( A, a ) −→ M ( C )and Ω : B ( A, a ) −→ M ( C )as follows. For an element η ∈ B ( A, a ) we let η i = η − ( e i ) where e i is the i th stan-dard basis element of Z . Then the entries of Ω ( η ) are obtained by integratingthe ξ i along the η j with j ∈ { , , } . Likewise the entries of Ω ( η ) are obtained byintegrating the ξ i along the η j with j ∈ { , , } .Set τ ( η ) = Ω ( η ) · Ω ( η ) − . Then τ is a bijection between B ( A, a ) and the elements of h corresponding to( A C , a C ) an .Integration defines the non-degenerate Poincar´e pairingH dR (( A C ) an , R ) ⊗ (H (( A C ) an , Z ) ⊗ R ) → R and we let dz ηi denote the differential dual to η i ⊗ ξ ξ ξ = Ω ( η ) dz η dz η dz η . Let x be the image of 0 under the morphism Spec( C ) → Spec( R ); we note that x is the generic point of Spec( R ), as Spec( R ) is integral and therefore irreducible.The value of ∆ at x is equal to ∆ as R ⊂ R x . Therefore ∆ is given by evaluating(2 π √− χ hol18 ( dz ∧ dz ∧ dz ) ( ξ ∧ ξ ∧ ξ ) − at a choice of η ∈ B ( A, a ). Thus κ ∆ = (2 π √− χ hol18 ( dz ∧ dz ∧ dz ) ( ξ ∧ ξ ∧ ξ ) − ( η )= (2 π √− χ hol18 ( τ ( η ))( dz η ∧ dz η ∧ dz η ) ( ξ ∧ ξ ∧ ξ ) − = (2 π √− χ hol18 ( τ ( η ))det(Ω ( η )) We note the quantity on the last line is invariant under symplectic transfor-mations, and it depends on
A/R , as Ω ( η ) depends on A/R . If A tw is the [ − A over R [ √ D ] with D ∈ R \ R then the global differentials of A tw inΩ A ⊗ R [ √ D ] /R [ √ D ] ( A ⊗ R [ √ D ]) are invariant under the R -linear automorphism givenby ξ i
7→ − ξ i and √ D
7→ −√ D. Therefore the √ Dξ i ∈ Ω A ⊗ R [ √ D ] /R [ √ D ] ( A ⊗ R [ √ D ]) form an R basis for the differ-entials on A tw . Thus det(Ω ( η )) A = D − det(Ω ( η )) A tw . It remains to show that κ/γ is a square in k for each x ∈ Spec( R )( k ). If ∆( x ) isa square, then there exists a curve C/k in the fibre of t − (( A, a ) x ) and thus κ/γ ∆( x )( ξ ∧ ξ ∧ ξ ) x = Discr C/k . Thus if ∆( x ) is a square then κ/γ ∈ k . If ∆( x ) is not a square in k the [ −
1] twistof (
A, a ) x over k ( √ ∆( x )) is a Jacobian and thus as elements of k ∆ − (2 π √− χ hol18 ( τ ( η ))det(Ω ( η )) = γ Discr C/k . And therefore κ/γ ∆( x )( ξ ∧ ξ ∧ ξ ) x = ∆ − ( x )Discr C/k and therefore κ/γ is a square in k .9. Alternative Proof of Theorem 2 for genus R be a Z [1 /
2] algebra. Let (
A, a ) / Spec( R ) be an indecomposable principallypolarised Abelian scheme of relative dimension 3. Assume that Ω A/ Spec( R ) is freeand choose an isomorphism φ : ω A/R ( R ) −→ R. Fix a basis ξ i of Ω A/R ( A ). We claim the function∆ = γ − φ ( χ ) φ ( ξ ∧ ξ ∧ ξ ) − is a twisting function.In the first place χ vanishes on the hyperelliptic locus by Theorem 3.Given x ∈ Spec( R )( k ), if ( A, a ) x is a Jacobian, then∆( x ) = φ (Discr x ) φ ( ξ ∧ ξ ∧ ξ ) − , is a square. On the other hand, if ( A, a ) x is not a Jacobian, there is a D ∈ k \ k such that the [ −
1] twist (
A, a ) ǫ of ( A, a ) x over k [ √ D ] is a Jacobian. Then {√ Dξ i | i ∈ { , , }} is an k -basis of differentials for ( A, a ) ǫ in Ω ( A,a ) ⊗ k [ √ D ] ( A ⊗ k [ √ D ]).Therefore ∆( x ) = D ∆(( A, a ) ǫ )is not a square. Acknowledgements
We thank Christophe Ritzenthaler and Gilles Lachaud for sharing their workwith us at an early stage. We thank Robert Carls, Everett Howe, Ching-Li Chai,David Kohel, Gerard van der Geer, Bert van Geemen, Jaap Top, Marius van derPut and Jean-Pierre Serre for comments upon earlier versions of this work.
ORELLI REVISITED 15
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