aa r X i v : . [ m a t h . N T ] A p r Siegel modular forms mod p R.WeissauerLet R be a commutative ring with unit. Assume / ∈ R . For a prime p let F p denote the prime field of characteristic p and F p an algebraic closure. We alwaysassume p = 2 . The moduli stack A g,R . Let g ≥ be an integer. Consider the moduli stack A g,R of isomorphism classes of the following data ( S, A, λ ) a ring homomorphism R → S of commutative rings,an abelian scheme A → Spec ( S ) of relative dimension g together with itszero section e : Spec ( S ) → A ,a principal polarization λ : A → A ∨ over Spec ( S ) .To ( S, A, λ ) we associate the sheaf of invariant relative differential forms Ω A = e ∗ (Ω A/S ) , which is a projective rank 1 module ω = Λ g (Ω A ) on S functorial withrespect to base change. Modular forms . By definition a modular form of integral weight k and genus g over R is an assignment ( S, A, λ ) f ( S, A, λ ) ∈ Γ( Spec ( S ) , ω ⊗ k ) with the following functorial properties(i) Basechange compatibility: For ring homomorphisms of commutative rings S → S ′ one has f ( S ′ , A × Spec ( S ) Spec ( S ′ ) , λ × Spec ( S ) id ) = f ( S, A, λ ) ⊗ O S . (ii) For S -isomorphic data ( A, λ ) and ( A ′ , λ ′ ) one has f ( S, A, λ ) = f ( S, A ′ , λ ′ ) . Let M kg,R be the space of modular forms of weight k . The direct sum over allintegers k defines the graded R -algebra M • g,R = L k M kg,R of modular forms. Variants . One has variants for the definition of modular forms: a) additional level-N-structure over extension rings R of Z [ N , ζ N ] . This defines the stack A g,R,N . Forthis we assume p ∤ N later in case R is the spectrum of a field κ of characteristic p . Or b) tensor valued modular forms, i.e. sections f ( S, A, λ ) ∈ Γ( Spec ( S ) , Ω ⊗ kA ) with corresponding functorial properties. The classical case . M kg, C is isomorphic to the space of all holomorphic functions f : H g → C on the Siegel upper half space H g of genus g with the property f ( γ h Ω i ) = det ( C Ω + D ) k f (Ω) for all γ ∈ ( A BC D ) ∈ Γ g = Sp g ( Z ) . The isomorphism is given by f f (Ω) = f ( C , C / Γ Ω , λ can ) for periods Ω in the Siegel upper half space H g and the standard polarization λ can defined for the standard period lattice Γ Ω = Z g + Ω Z g . The Hasse invariant for R = F p . Let Lie ( A/S ) ∼ = Ω ∗ A denote the Lie algebra oftranslation invariant relative vectorfields X , and let denote D = D X the derivationassociated to X . Consider the Hasse-Witt map Φ :
Lie ( A/S ) → Lie ( A/S )Φ( D ) = D p . Notice Φ is p -linear, i.e. Φ( λv ) = λ p Φ( v ) for λ ∈ S . For an arbitrary p -linearendomorphism Φ : V → V of a locally free S -module V with pS = 0 put V ( p ) = V ⊗ S S such that vλ ⊗ S λ ′ = v ⊗ S λ p λ ′ . Then Φ extends to a S -linearmorphism Φ : V ( p ) → V . Let ST p ( V ) denote the p -symmetric elements in thetensor product. It contains the symmetrization Sym p ( ⊗ p V ) as a S -submodule.The S -linear morphism v ⊗ λ λ · v ⊗ v ⊗ · · · ⊗ v ( p copies) induces an S -isomorphism V ( p ) ∼ −→ ST p ( V ) /Sym p ( ⊗ p V ) by [D], I §
10. If we compose theprojection ST p ( V ) → ST p ( V ) /Sym p ( ⊗ p V ) with the inverse of this isomorphism and then with Φ , we obtain a S -linear mor-phism ST p ( V ) → V . It can be considered as an element in V ⊗ S S p ( V ∗ ) , where S p ( V ∗ ) is the symmetric tensor quotient of the p -th tensor product of V ∗ defined as the quotient by the equivalence relation induced from the symmetryrelations ... ⊗ x ⊗ ... ⊗ y ⊗ .. ∼ ... ⊗ y ⊗ ... ⊗ x ⊗ .. .Applied for the Hasse-Witt map Φ this construction yields a sheaf homomorphism ST p ( Lie ( A/S )) → Lie ( A/S ) or equivalently an element A ∈ Γ( Spec ( S ) , S p (Ω A ) ⊗ Ω ∗ A ) . Exterior powers of p -linear maps are p -linear. Hence the exterior powers Λ i ( A ) of A define vector valued modular forms in characteristic p . In particular for i = g this defines the Hasse invariant A as a modular form of weight k = p − in thespace M p − g, F p defined above A ∈ Γ( Spec ( S ) , ω p − ) . The ungraded ring of modular forms . Assume κ to be F p or its algebraic clos-ure F p . Let denote M g,κ the associated ungraded ring of modular forms over κ obtained from the graded ring M • g,κ . This ring is a normal domain. Then Unique Faktorization Theorem . There exists a prime p such that for p > p thenormal domain M g,κ of modular forms over κ is a unique factorization ring forall g ≥ .The lemma is related to the following graded version for modular forms over κ ofsome level N ≥ , where we tacitly assume a root of unity ζ N to be chosen in κ . Theorem . For p > p and g ≥ and N ≥ and p ∤ N the Picard group of themoduli stack A g,κ,N defined over κ is P ic ( A g,κ,N ) ∼ = Z · c ( ω ) ⊕ sp g ( N Z /N Z ) . Remark . In particular this group is of rank one, and generated by the Chern class c ( ω ) modulo a finite torsion subgroup isomorphic to the ‘Lie algebra’ sp g ( Z /N Z ) of the symplectic group over the ring Z /N Z . The corresponding assertion ist falsefor g = 2 . It does not hold for g ≥ and levels of type N = 2 i either. Howeverthere are similar statements in this latter case. Remark . To obtain the results above on Picard groups for F p it is enough to provethis over F p using descend via Hilbert theorem 90 and the vanishing of the Brauergroups of these fields. Fourier expansion . The q -expansion of a modular form is obtained by evaluatingforms at the Mumford families of principally polarized abelian varieties definedby torus quotients. The q -expansion principle [CF] gives injective maps M kg,R ֒ → R [[ q ]] := R [[ q ij ]] , q ij = q ji , ≤ i, j ≤ g defined by f X T a f ( T ) q T with Fourier coefficients a f ( T ) ∈ R attached to half-integral symmetric matrices T (written as symbolic exponents). Notice, the relevant Mumford family definesan abelian scheme over R [[ q ]][ q ij ] . Nevertheless by Koecher’s effect f ∈ R [[ q ]] .The theory of Chai-Faltings [CF] implies that M kg, Z = { f ∈ M kg, C | a f ( T ) ∈ Z } .This relates modular form over Z to classical Siegel modular forms. Example . For R = F p the Hasse invariant A ∈ M p − g, F p has Fourier expansion A = 1 . In other words a f ( T ) = 0 for all T = 0 .We will see below that the last theorem implies Proposition . Assume p > p for p as in the lemma. Then the kernel of the ringhomomorphism ϕ : M g, F p → F p [[ q ]] of ungraded rings induced from the q -expansion is the principal ideal generatedby ( A − for the Hasse invariant A . Let S g, F p ⊆ F p [[ q ]] denote the image of the homomorphism ϕ . It is not hard to seefor p ≥ g + 2 that S g, F p coincides with the ring obtained by reduction mod p fromthe ring of the integral q -expansion of all modular forms in M g, Z . Furthermore onehas the following rather obvious properties: The subring S g, F p of the ring of powerseries F p [[ q ]] is stable under the following operators(1) Hecke operators T ( l ) for primes l different from p (2) The operators U : X T a ( T ) q T X T a ( pT ) q T V : X T a ( T ) q T X T a ( T ) q pT . (3) Φ : F p [ q , .., q gg ]] F p [[ q , .., q g − g − ]] q ∗ g = q g ∗ and q ij q ij for i = g, j = g . Notice Φ( S g, F p ) = S g − , F p and Φ( M g, F p ) ⊆ M g − , F p . For the operator V use that reduction modulo p of the normalized Hecke operator T ( p ) coincides with V for weights k ≥ g + 1 . Of course one can assume k ≥ g + 1 without restriction of generality, since multiplication with the Hasse invariant A commutes with the Hecke operators T ( l ) for all ( l, p ) = 1 . For forms f withcoefficients in F p we have f | V = f p . Proof of the proposition . For the elliptic case g = 1 see [Sw]. The general case isreduced to the elliptic case as follows. First notice dim ( S g, F p ) ≥ dim ( M g, F p ) − . In fact, if κ is a field and B = L ∞ i =0 B i is a finitely generated integral (graded) κ -algebra of Krull dimension dim ( B ) , and if ϕ : B → S is a ring homomorphismonto an integral domain S such that ϕ | B i is injective for every i , then dim ( S ) ≥ dim ( B ) − . Since the stack A g, F p is irreducible by a result of Chai, the gradedring of modular forms satisfies the assumptions made on B . Hence this generalfact from commutative algebra can be applied for the graded ring of modularforms. Obviously Ker ( ϕ ) is a prime ideal. Therefore it is a prime ideal of heightone because the kernel is nontrivial, since = A − ∈ Ker ( ϕ ) . Hence Ker ( ϕ ) is a principal ideal by the unique factorization theorem stated above.Since A − is contained in the kernel of ϕ , it is enough to show that A − isirreducible. If not A − F G for F = F + · · · + F s and G = G + · · · + G t and p − r + s, r > , s > and forms F i , G i of weight i . But then Φ g − ( A −
1) = A − (in genus g = 1 ) implies Φ g − ( F s ) = 0 and Φ g − ( G t ) = 0 by degreereasons (notice the Hasse invariant A is nonvanishing of weight p − for genus g = 1 ). But this factorization of A − for genus g = 1 would contradict the resultof Swinnerton-Dyer [Sw]. This proves the proposition. (cid:3) By a similar argument, using the Φ -operator to reduce to the case g = 2 , we havethe following Corollary . For g ≥ the Hasse invariant is an irreducible modular form. Inparticular the locus of singular abelian varieties (the zero locus of the Hasseinvariant A ) is irreducible. For g = 2 this follows from results of Koblitz. Corollary . Suppose g ≥ , p > p and f ∈ M kg, F p . Then the following statementsare equivalent1) f is totally p -singular, i.e. f = P T a ( T ) q T and a ( T ) = 0 unless p | T .2) f = A r h p for some r ∈ N and some h ∈ M k ′ g, F p .In particular f = 0 for k < p − , if one of these two equivalent conditions holds.Proof . To show that 1) implies 2) we may assume, that A does not divide f . If f hasweight k , we choose l such that K = k + l ( p − ≥ g + 1 . Then A l f has weight K ≥ g + 1 , hence also ( A l f ) | U . Then (( A l f ) | U ) | V = (( A l f ) | U ) p has weight pK , and it has the same Fourier expansion as f . Therefore (( A l f ) | U ) p = A m f for m = k + lp by the q -expansion principle, since both sides have the sameweight and the same q -expansion. So at least the power A lp divides (( A l f ) | U ) p .Hence A l divides ( A l f ) | U and h = A − l ( A l f ) | U ) is a modular form of weight k such that h p = A k f . Therefore the irreducible form A divides h . In other words h = Ah , hence A p h p = A k f . Since A does not divide f by assumption, weconclude h p = A k − p f and either k = p , or again h = Ah holds for somemodular form h . If we continue, this must terminate after finitely many steps.Hence k is divisible by p and there exists a modular form h = h ν such that h p = f . (cid:3) Question . Is S g, F p ⊆ F q [[ q ]] preserved by the differential operator P T a ( T ) · q T P T det ( T ) a ( T ) · q T ? Similarly is P T T · a ( T ) q T a vector valued modular form? Proof of the theorem . For the proof we can extend the base field to become κ = F p .Use Hilbert theorem 90 and vanishing of the Brauer group for finite fields, andlater for fields of fractions of Henselian discrete valuation rings with algebraicallyclosed residue field. We can also replace N by a suitably large integer not divisibleby p , so that X is a smooth variety without restriction of generality.1) Change of level . Assume we are over a base scheme S over Z [ N , ζ N ] for N ≥ .Consider the natural morphism A g,S,N → A g,S , which is equivariant with respectto the group G N = Sp (2 g, Z /N Z ) . Pullback defines a natural homomorphism P ic ( A g,S ) → P ic ( A g,S,N ) . A line bundle on A g,S is given by its pullback L on A g,S,N together with descent data ϕ σ : σ ∗ ( L ) ∼ = L for σ ∈ G N satisfying theobvious cocycle conditions. Since H ( A g,S,N , O ∗ ) = O ∗ S ( S ) and the group G N is perfect, the cocycle datum is uniquely determined, once it exists. Hence thereexists an exact sequence → P ic ( A g,S ) → P ic ( A g,S,N ) G N → H ( G N , O ∗ S ( S )) For odd N and g ≥ we claim H ( G N , C ) = 0 for C = O ∗ S ( S ) . We may replace C by an arbitrary finite trivial G N -module. For perfect groups G the K¨unneth theo-rem and the universal coefficient theorem imply H ( G, C ) =
Hom ( H ( G, Z ) , C ) .So it is enough to show vanishing of the Schur multiplier M ( G ) = H ( G, Q / Z ) ∼ = H ( G, Z ) D . Since the Schur multiplier is additive for products of perfect groups,one can reduce to the case where N is a prime power. According to the groupAtlas, see also[FP], this Schur multiplier vanishes in the case where N is a primedifferent from two. Using results of [FP] one reduces this to show H ( G N , Z /l Z ) vanishes. For ( l, N ) = 1 see [FP]. So we can assume C = F p . For N = p r for r ≥ and p odd now we proceed by induction on r using G p r /Lie =: G = G p r − with respect to the abelian group Lie = sp g ( Z /p Z ) . The Hochschild-Serre spec-tral sequence and the induction assumption together with H ( G, H ( Lie )) = 0 and H ( G, Lie D ) = 0 then prove the induction step. For p = 2 one has an exactsequence → Lie D → H ( Lie, F p ) → Λ ( Lie D ) → , hence the first state-ment H ( G, H ( Lie )) = 0 follows from the obvious facts ( Lie D ) G = 0 and Λ ( Lie D ) G = 0 . The second assertion H ( G, Lie D ) = 0 is less obvious. It fol-lows by induction on the genus g using restriction to parabolic subgroups andstandard techniques of group cohomology. Hence under our assumptions on N the pullback defines an isomorphism P ic ( A g,S ) ∼ = P ic ( A g,S,N ) G N . By the methods above one also obtains the vanishing of cohomology groups ofthe finite group G N = Sp g ( Z /N Z ) for odd N in the following special cases H i (cid:0) G N , C ∗ (cid:1) = 0 , g ≥ , i = 1 , H i (cid:0) G N , ( sp g ( Z /N Z ) , Ad ) (cid:1) = 0 , g ≥ , i = 0 , and ∤ NH (cid:0) G N , Z ) l − torsion = 0 , for g ≥ and ∤ l . In the proof these vanishing theorems can be used to allow the passage from level N = 1 to level N > and vice versa.2) Analytic results over C . For g ≥ we have a) For Γ g = Sp g ( Z ) P ic (Γ g ) := H ( H g / Γ g , O ∗ ( H g )) ∼ = H (Γ g , Z ) = Z · c ( ω ) , and b) For all principal congruence subgroups Γ = Γ g ( N ) for N ≥ H (Γ , Z ) ∼ = P ic an ( H g / Γ) ∼ = P ic ( A g, C ,N ) , and c) P ic (Γ) tors ∼ = (Γ ab ) D . See [F], p. 257 and Hilfssatz 4. Finally d), that
P ic (Γ) / Z c ( ω ) is a finite group for g ≥ and arbitrary subgroups Γ of finite index in Γ g . This follows from theoremsof A.Borel and vanishing theorems (resp. L -vanishing theorems for g = 3 ) forLie algebra cohomology.Recall G N = Γ g / Γ g ( N ) . By step 1 the E -term of the Hochschild-Serre spec-tral sequence H i ( G N , H j (Γ g ( N ) , Z )) = ⇒ H i + j (Γ g ) for i + j = 2 has only H ( G N , H (Γ g ( N ) , Z )) as nontrivial summand, and degenerates in this degreeat the E -term, i.e. the differential d induces a canonical exact sequence → H (Γ g , Z ) → H (Γ g ( N ) , Z ) Γ g → H ( G N , Z ) . Since H ( G N , Z ) ∼ = H ( G N , Q / Z ) the vanishing of Schur multiplier explainedin step 1 implies H ( G N , Z ) = 0 for odd N and g ≥ . Hence the restriction mapinduces a canonical isomorphism res : H (Γ g , Z ) ∼ = H (Γ g ( N ) , Z ) Γ g for g ≥ and odd integers N . Hence both for the stack setting as well as for theanalytic setting the descend from level N to level one only amounts to consider G N -invariants. This is used in step 4) below.A first application of this descend principle: For N ≥ and g ≥ the long exactsequence of G N -invariants for the exact sequence → (Γ ab ) D → P ic ( A g, C ,N ) p → Z → from assertion 2c) and 2d) and the vanishing of the cohomology groups H i ( G N , (Γ ab ) D ) = 0 for i = 0 , implies that p induces an isomorphism p : P ic ( A G N g, C ,N ) ∼ = Z . Here we used Γ g ( N ) ab = Γ g ( N ) / Γ g (2 N ) ∼ = sp g ( Z /N Z ) and the vanishing theorems of step 1, in particular that the perfect group G N actstrivially on the cyclic quotient group Z . The descend principle above thus givesthe canonical isomorphism p : P ic ( A g, C ) ∼ = Z . Since c ( ω ) generates P ic ( A g, C ) ∼ = H (Γ g , Z ) by 2b) we conclude that the pro-jection map p induces an isomorphism P ic ( A g, C ,N ) ∼ = Z · c ( ω ) ⊕ P ic ( A g, C ,N ) tors for odd N and g ≥ . We would like to carry this over to the Picard groups of A g, F p ,N via smooth and proper base change for etale l -adic cohomology for all l = p . For this consider N ≥ and compactifications.3) Passage to compactification . There exists an exact G N -equivariant sequence → Z r → P ic ( A g,κ,N ) → P ic ( A g,κ,N ) → for suitable toroidal compactification A g,κ,N of A g,κ,N in the sense of Chai-Faltings.Indeed the subgroup generated by the boundary divisors is torsion-free by Koe-cher’s principle, hence isomorphic to Z r for a suitable integer r depending on N, g and the compactification. The same sequences exist over Z [ N , ζ N ] and C .4) Base change . The strategy of proof is to reduce the assertion of the theorem tothe case of level N = 1 using step 1). For N = 1 we have to show P ic ( A g,κ ) = Z · c ( ω ) . For this choose some auxiliary prime l different from p , and first show P ic ( A g,κ ) ⊗ Z l = Z l · c ( ω ) . If the stack A g were a smooth and proper scheme,this could be easily deduced from the analytic facts stated in 2) by proper andsmooth base change. Since it is not one has to pass to some larger level N andconsider a toroidal compactification for which it is possible to apply the smooth0and proper base change theorem. Using step 1) and 3) one can pass back to derive P ic ( A g,κ ) ⊗ Z l = Z l · c ( ω ) . For varying l this implies that P ic ( A g,κ ) / ( Z · c ( ω )) is a finite p -primary abelian group.To make this outline more precise choose an integer l prime to p . One can assume N ≥ . First one replaces F p by its algebraic closure F p . Apply the snake lemmato the exact sequences of step 3 with respect to multiplication by l . This gives adiagram / / P ic ( X/κ )[ l ] (cid:15) (cid:15) / / P ic ( A g,κ,N )[ l ] (cid:15) (cid:15) / / ( Z /l Z ) r / / P ic ( X/κ ) /l (cid:15) (cid:15) / / P ic ( X/ C )[ l ] / / P ic ( A g, C ,N )[ l ] / / ( Z /l Z ) r / / P ic ( X/ C ) /l for toroidal compactifications X as in step 3. Notice P ic ( X/κ ) ∼ = H ( X/κ, µ l ) for κ = F p and κ = C . Hence the left vertical arrow is an isomorphism by theproper smooth etale base change theorem. This implies that the second verticalspecialization map is injective. Next notice that the abelian variety P ic ( X/κ ) vanishes. Look at m -torsion points for large auxiliary integers m prime to p . Then P ic ( X/κ ) is a group of type ( Z /m Z ) d . As a subgroup of P ic ( X/κ ) this forces d = 0 . To show this use base change for the first etale cohomology to redu-ce this to the case κ = C , where this immediately follows from step 2). Thevanishing of P ic ( X/κ ) implies P ic ( X/κ ) =
N S ( X/κ ) . By the Kummer se-quence one has inclusions P ic ( X/κ ) /l ֒ → H et ( X/κ, µ l ) respectively canoni-cal inclusions P ic ( X/ F p ) ⊗ Z l ֒ → H et ( X/ C , Z l ) for prime l . By base change H et ( X/ F p , µ l ) ∼ = H et ( X/ C , µ l ) respectively H et ( X/ F p , Z l ) ∼ = H et ( X/ C , Z l ) .Furthermore H et ( X/ C , Z l ) ∼ = H et ( X/ C , Z ) ⊗ Z l . Hence the smooth and properbase change theorem implies that the fourth vertical map of the diagram aboveis injective. This implies by the 5-lemma, that the second vertical map also issurjective. Hence the second vertical map is an isomorphism P ic ( A g, F p ,N )[ l ] ∼ = P ic ( A g, C ,N )[ l ] . The argument also gives an injective map
P ic ( A g, F p ,N ) /l ֒ → P ic ( A g, C ,N ) /l . Together this implies the analog of assertion 2d) over F p . Hence P ic ( A g, F p ,N ) /P ic ( A g, F p ,N ) tors ∼ = Z . P ic ( A g, F p ,N ) tors is (Γ g ( N ) ab ) D up to p -power torsion as shown above, bytaking G N -invariants the argument at the end of step 2 give the following result P ic ( A g, C ,N ) ⊗ Z l ∼ = ( Z l · c ( ω )) ⊕ ( Z l ⊗ P ic ( A g, C ,N ) tors ) . for all primes l different from p . Warning: Usually taking invariants does not com-mutes with the tensor product with Z l . Here it does by a simple inspection.5) p -index . To control the cotorsion of the subgroup generated by c ( ω ) in thegroup P ic ( A g, F p ,N ) for N = 1 one suitably embeds A , F p ֒ → A g, F p such that ω pulls back to ω . Suppose L e = ω for a line bundle L on A g, F p . Then of coursethe same holds for the pullback of L to A , F p . So this reduces us to consider thecase g = 2 for this question. Now Igusa [I] has given an explicit description ofthe coarse moduli space A coarse of the stack A . In fact for p = 2 , the space A coarse , F p is a quotient U/G , where U is the open complement of some closed subsetof codimension 2 in the projective space P , and where G is the finite group G = Z / Z × Z / Z × Z / Z × Z / Z acting on U . The same holds over κ = F p . Consider A , F p , { { $ $ HHHHHHHHH A , F p A coarse , F p . A suitable power of ω r descends to A coarse , F p . Since the Picard group of U is cyclic,this provides an a priori bound e for the integer e independent from g and p . Sinceon the other hand e must be a power of p by step 4) we find that for large enough p > p necessarily e = 1 holds. This proves P ic ( A g, F p ) = Z c ( ω ) ⊕ T for a finite p -primary torsion group T .6) p -torsion . To show that the torsion T vanishes it is enough to control the p -torsion P ic ( A g,κ )[ p ] . For this we may pass to a suitably large level N not divisibleby p . Then the covering space X = A g,κ,N is smooth. It is enough to show P ic ( X )[ p ] = 0 . In general, for smooth X over κ one can control P ic ( X )[ p ] = 0 by the closed 1-forms on X . In fact, if C denotes the Cartier operator, then onehas a surjection (cid:26) η ∈ H ( X, Ω X ) | dη = 0 , Cη = η (cid:27) ∼ = H (cid:0) X flat , µ p (cid:1) ։ P ic ( X )[ p ] . P ic ( X )[ p ] vanishes under the assumptions made, it is enoughto proof the following Vanishing Theorem . Suppose g ≥ and p = 2 and let κ be F p [ ζ N ] or F p . Thenfor X = A g,κ,N and arbitrary level N not divisible by p the following holds H ( X, Ω X ) = 0 . Remark . The proof of this vanishing theorem can be extended to the case of regularalternating differential forms Ω iX on X of degree i for i < g and i < p − . Remark . We could compute p from step 5 of the above proof of the theorem andthe assumption p = 2 . Proof of the vanishing theorem . The proof proceeds by induction on g . In somesense the case g = 2 is the most complicated case. For the proof we can choosesome embedding Z [ ζ N ] to F p and extend the base field κ to become F p . We canalso replace N by a suitably large integer not divisible by p , so that X is a smoothvariety without restriction of generality.a) q -expansion . As a first step in the proof use that any regular 1-form η on A g, F p ,N automatically extends to a 1-form η ∈ Ω X ( log ) with log-poles on thetoroidal compactification X = A g,κ,N (Koecher’s principle). Furthermore noticethe Kodaira-Spencer isomorphism in the sense of [CF], p.107 Ω X ( log ) ∼ = S (Ω A ) . In other words, one can expand η in the following form using q -expansion η = X T ≥ T race (cid:0) a ( T ) · dlog ( q ) (cid:1) · q T , where a ( T ) is a symmetric g by g matrix with coefficients in κ and where T runsover the halfintegral semidefinite g by g matrices (notation is chosen to hide N ).E.g. for g = 2 dlog ( q ) = dq q dq q dq q dq q ! . Equivariance . For unimodular U ∈ Gl ( g, Z ) congruent to id modulo N withimage U in Gl ( g, Z /p Z ) the Fourier coefficients a ( T ) of the 1-form η satisfy ( ∗ ) a ( t U T U ) = t U a ( T ) U .
In particular a (0) = 0 .c) Cartier operator . Suppose there exists a 1-form η = 0 on X . Without restrictionof generality one can choose η such that there exists T with a ( T ) = 0 and p ∤ T .Notice otherwise the form is closed, so one can apply the Cartier operator C :Ω X,closed → Ω X ( p ) . Since C ( P T a ( T ) q T dlog ( q )) = P T ˜ a ( T ) q T/p dlog ( q ) , thisprocess terminates after finitely many steps and gives the desired form η on X ( p s ) .We may replace X ( p s ) by X , since this amounts just to another choice of ζ N ∈ κ .By b) we can in addition assume T = ( ∗ ∗∗ ν ) with last entry T gg = ν mod p .d) A boundary chart . For the sake of simplicity temporarily assume N = 1 . Let π : A g − → A g − be the universal abelian scheme with polarization λ : A g − → A ∨ g − . On thepullback Ξ = Ξ ξ = j ∗ ( P − ) of the Poincare G m -torsor P over A g − × A g − A ∨ g − under j = id × λ the group G m acts such that the canonical map u : Ξ → A g − induces an isomorphism Ξ / G m ∼ = A g − . A formal boundary chart S = S ξ ofthe toroidal compactification is now obtained by the formal completion of Ξ =(Ξ × A ) / G m along the zero section. Indeed u : Ξ → A g − is a line bundle over A g − . The complement of the zero section is the G m -bundle Ξ = (Ξ × G m ) / G m we started from. See [CH], p.104 ff for X ξ = Z , E ξ = G m and E ξ = A .e) Fourier-Jacobi expansion . Then η , considered as a differential form on S withlog-poles, can be expanded u ∗ ( η | S ) = P ν u ∗ ( η ) χ ν q ν with respect to the cha-racters χ ν ( t ) = t ν of G m . Here q ∈ Γ(Ξ , L − ) is the tautological section for L = ( id × λ ) ∗ ( P ) . Notice q = τ ( χ , χ ) in the notation of [CF], p.106. Torelate this to the Fourier-Jacobi expansion one needs to consider the semiabe-lian principally polarized scheme ♥ G defined over S by Mumford’s construc-tion. It contains a split torus T over S of rank one with the abelian scheme4quotient ♥ G/T ∼ = A g − . Notice Ω T ∼ = O S via dqq . The exact sequence → Ω A g − → Ω ♥ G → Ω T → of relative differential forms over S and theKodaira-Spencer isomorphism Symm (Ω ♥ G ) ∼ = Ω ( log ) give rise to an isomor-phism Ω S = u ∗ ( E ) for a vector bundle E on A g − , which is pinched into an exactsequence → Ω A g − → E r → O A g − → . The coefficients u ∗ ( η ) χ are global sections of u ∗ ( u ∗ ( E )) χ = E ⊗ O A g − u ∗ ( O S ) χ . Separability . Notice u ∗ ( O S ) χ = L ν , where L = ( id × λ ) ∗ ( P ) is a totally symme-tric line bundle on A g − normalized along the zero section. Recall p = 2 . Hencefor ν p the line bundle L = L ν is separable over the base A g − ,κ of characteristic p . Thisallows to relate the sections of u ∗ ( O ) χ to theta functions. This will be explained inthe next section. Indeed on some etale extension S of the base (e.g. S = A g − ,κ, ν )the line bundle L is of type δ = (2 ν, .., ν ) in the sense of Mumford (see also thenext section). By further increasing the level we will see that then ( A g − , π, L ) admits a δ -marking in the following sense.f) Theta functions . For an abelian scheme
A → S of relative dimension g and atotally symmetric relative ample line bundle L on A normalized along the zerosection Mumford defined the theta group scheme G ( L ) over S ([M2], prop 1).For a type δ = ( d , .., d g ) of even integers with d i | d i +1 for all i one also has theHeisenberg group scheme G ( δ ) over S as defined in [M2], p.77. Here, as in [M2],we will suppose a separability condition, namely that d = Q gi =1 d i is invertibleon the base scheme S . Then by definition a symmetric ϑ -structure, or δ -marking,for ( A , π, L ) is an isomorphism G ( L ) ∼ = G ( δ ) with some additional compatibilityproperties as defined in [M2], p.79. Let V ( δ ) be the free Z [ d − ] -module generatedof functions K ( δ ) = L i ( Z /d i Z ) → Z [ d − ] . Then [M2] prop. 2 and the discus-sion in loc. cit p.81 implies for a given δ -marking that there exists an equivariantisomorphism of O S -sheaves α : V δ ⊗ Z K ∼ = π ∗ ( L ) for some invertible sheaf K on S , unique up to multiplication with some unit in O ∗ ( S ) . Furthermore [CF], theorem 5.1 gives K d ⊗ ω d ∼ = O S , since det ( π ∗ ( L )) = K d for d = (2 ν ) g . Hence K ⊗ ω is a torsion line bundle on S killed by d . We5notice, that Mumford’s theory of theta functions also gives an injections K ֒ → O S . Additional remark . Suppose g ≥ . By suitably extending the level N to large N ′ , but still with ( N ′ , p ) = 1 , all torsion line bundles in P ic ( S )[4 d ] becometrivial over A g − ,κ,N ′ . For g ≥ this is first shown over C , but then follows by themethods used for the proof of the theorem. Alternative argument (for all g ≥ ).Use that K is defined over Z [ ν , ζ N ′ ] . Since the characteristic p fiber of A g,κ,N ′ isirreducible for ( N ′ , p ) = 1 , hence a principal divisor, one can reduce the statementto characteristic zero, where this can be explicitly computed by cocycles.g) Existence of δ -markings . In the case of base fields existence of δ -markingsfor a given symplectic isomorphism g : H ( L ) ∼ = H (2 δ ) is shown in [M] § g exists over S = A g − ,κ,N by extending the level N . It is enough that ν divides the level N . For a generalbase S , different to the case of a base field considered in loc. cit., there existsan obstruction to lifting g to a symmetric isomorphism f : G ( L ) → G (2 δ ) .The proof of loc. cit. carries over, if this obstruction vanishes. The obstruction isdefined by the map δ : H ( L ) → P ic ( S )[8 ν ] given by δ ( s ) = t s ∗ ( L ) ⊗ L − for sections s : S → H ( L ) . This map is quadraticby the theorem of cube and ‘pointe de degree 2’ in the sense of [MB], p.12. Indeed δ ( rs ) = r δ ( s ) for all r ∈ N , since L is symmetric. If m annihilates H ( L ) thevalues of δ are m -torsion (resp m -torsion if | m ) line bundles on S by [MB],lemma 5.6. In our case this applies for m = 4 ν . If we extend the base to level m , then every s ∈ H ( L ) can be written in the form s = 2 my . But δ ( s ) = δ (2 my ) = 4 m δ ( y ) . Since δ ( y ) is m -torsion, we get δ ( s ) = 0 . Hence a δ -marking exists after a suitable choice of level, i.e. after an etale extension of ourbase scheme S = A g − ,κ,N .h) The induction step . Suppose g ≥ . To show Ω ( X ) = 0 it suffices to prove u ∗ ( η ) χ = 0 for all χ ( t ) = t ν with ν ≥ and ν mod p by step e) and f).Indeed the Fourier-Jacobi expansion corresponds to the expansion with respect tothe parameter q = q gg . So it is enough to show Γ( A g − , E ⊗ L ) for the separableinvertible sheaves L = L ν . Using the exact sequence for E and the sequence → Ω X → Ω A g − → Ω A → for A = A g − over S it suffices to show for X = A g − ,κ,N the vanishing Γ( X, K ) = 0 , Γ( X, Ω A ⊗ K ) = 0 and Γ( X, Ω X ⊗ K ) = 0 .By K ֒ → O X and by the Kodaira-Spencer isomorphism the last two statementsare reduced to the induction assumption Γ( X, Ω X ) = 0 in genus g − . This provesthe induction step, since obviously Γ( X, K ) = 0 .i)
Start of induction . We now consider the case g = 2 . Here the argument of steph) fails, since the space of 1-forms in genus one usually is nontrivial. However inthe situation as in h) the argument implies that under the projection r : E ⊗ L → L r ( u ∗ ( η ) χ ) ∈ Γ( A g − , L ) vanishes. Indeed, this is a section in Γ( A g − , L ) ∼ = Γ( X, π ∗ ( L )) . Let us considerits Fourier expansion. On can easily reduce to consider the standard cusp at infinityby replacing η without changing the condition obtained by c). By the Koechereffect applied for η this expansion is holomorphic at the cusp. Hence r ( u ∗ ( η ) χ ) vanishes, since a power of it is a modular form of negative weight. Now thisimplies that all f ( q , q ) q ν · dq q terms in the Fourier expansion of η vanish for ν mod p . In other words ( ∗∗ ) t vT v = 0 for v ∈ F gp = ⇒ t va ( T ) v = 0 . In fact by step b) it is enough to have this for t v = (0 , , since in general t v =(0 , t U for some unimodular matrix U congruent to id mod N .j) Suppose det ( T ) = 0 mod p . Then the reduction T of T mod p is a nondegeneratebinary quadratic form over F p . Also S = a ( T ) is a binary quadratic form, and itis easy to see that (**) then implies S = 0 . Also a (0) = 0 by step b), hence a ( T ) vanishes unless the reduction T of T mod p has rank 1 with coefficient ν = T = 0 in F p ⊆ κ . Then necessarily T = (cid:18) x ν xνxν ν (cid:19) = (cid:18) x (cid:19) (cid:18) ν (cid:19) (cid:18) x (cid:19) for some x ∈ F p ⊆ κ .k) Fourier-Jacobi expansion revisited . We now compare the Fourier expansion of η ∈ Ω ( X ) with its Fourier-Jacobi expansion for q = q . To unburden notationlet us assume level N = 1 for simplicity. Then η = ∞ X ν =0 η ν ( q , q ) q ν , η ν = P A ( t t t ν ) q t q t with summation over semidefinite matrices ( t t t ν ) with t ∈ Z and t ∈ Z . Using the transformation property b) for U = ( g ) with g ∈ Z we can also write η ν ( q , q ) = X a (cid:0)X t ϑ A ( t ) a · q t − a ν (cid:1) where now a runs over the finitely many cosets a ∈ Z g − /ν Z g − , and where t runs over all integers t ≥ a ν . Notice that A ( t ) = ( t aa ν ) depends on t . Here weused for an integral matrix × -matrix A = ( a a a a ) (or its reduction mod p ) thenotation ϑ Aa = X g ∈ Z (cid:18) g (cid:19) A (cid:18) g (cid:19) · q ν ( g + aν ) q ν ( g + aν )12 = X g ∈ Z (cid:18) a + 2 ga + g a a + ga a + ga a (cid:19) · q ν ( g + aν ) q ν ( g + aν )12 . = (cid:18) a a a a (cid:19) f a + (cid:18) a a a (cid:19) f (1) a + (cid:18) a
00 0 (cid:19) f (2) a for the derivatives f ( i ) a = P g ∈ Z g i q ν ( g + aν ) q ν ( g + aν )12 of the theta functions f a ( q , q ) .Summation over t therefore expresses η ν ( q , q ) as a sum over a of terms of theform f a · (cid:0) g ,a dq q + 2 g ,a dq q + g ,a dq q (cid:1) + f (1) a · (cid:0) g ,a dq q + 2 g ,a dq q (cid:1) + f (2) a · g ,a dq q for power series g i,a = g i,a ( q ) in the variable q . Linear independency of thetheta functions f a (Mumford’s theory) and step i) imply g ,a ( q ) = 0 by looking at the coefficient at dqq = dq q . Hence the terms simplify to f a · (cid:0) g ,a dq q + 2 g ,a dq q (cid:1) + f (1) a · g ,a dq q . The coefficients g ,a are sections in Γ( A ,κ,n , Ω A ,κ,n ⊗ K − ) and the coefficients g ,a are sections in Γ( A ,κ,n , Ω A ,κ,n ⊗ K − ) for suitably levels n as in step h). For8the choice of n see step f) and g). Morally speaking g ,a , g ,a are modular formsof genus one of level n and weight respectively , if this is suitably defined viatheta line bundles. By the way they arise it is easy to see, that they are holomorphicat the cusps.l) p -singularity . Notice the Fourier coefficients of ϑ A ( t ) q ν are all of the form T ′ = (cid:18) νy νyνy ν (cid:19) where y ∈ M Z for some integer M with ( M, p ) = 1 . Notice M also takes care ofthe level N , which was suppressed so far. Looking at the Fourier coefficients of dq q in the q -expansion of η only those T contribute with T = T ′ + ( l
00 0 ) , where l comes from the Fourier expansion of the section g ,a ( q ) . Reading this modulo p by step j) gives (cid:18) νx νxνx ν (cid:19) ≡ (cid:18) νy νyνy ν (cid:19) + (cid:18) l
00 0 (cid:19) modulo p , since ν mod p . Hence x and y are congruent modulo p , and there-fore l ≡ p . This implies that the q -expansions of the sections g ,a are p -singular for all a .Using this consider the dq g term. From the linear independency of the f a then thesame also follows for the g ,a . In other words g i,a ( q ) = X n ≥ c i,a,n · q n/M , c i,a,n = 0 if p ∤ n . m) Vanishing of the g i,a . Since ( g i,a ) are modular forms of weight 1 resp. 3 of ge-nus one with suitable level n in the sense of Katz [K2] by step k), the p -singularityshown in step l) contradicts corollary (2) of the theorem of loc. cit, p.55 unless ( g i,a ) = 0 since 1 resp. 3 is < p − by our assumption p = 2 , . This theorem ofKatz is a version in the elliptic case with level structure of part 2) of our secondcorollary above. We conclude g i,a = 0 for all i , hence η = 0 . This proves thevanishing theorem for p = 2 , . For p = 3 we only have g ,a = 0 . But this im-plies, that all Fourier coefficients A ( T ) for matrices T with T mod havecoefficients A ( T ) ij = 0 unless i = 1 , j = 1 . But this implies A ( T ) = 0 for allsuch Fourier coefficients by the transformation property (*). By j) it is enough to9suppose T = ( ) . Then otherwise up to a constant A ( T ) = ( ) . Now apply U with U = ( −
11 1 ) . Then we get T = ( ) and A ( T ) = ( ) from (*). Acontradiction to (**). This proves the theorem for p = 3 . (cid:3) Proof of the unique factorization theorem . We may suppose g ≥ , since for g = 2 the assertion follows from [I]. Let R N = M g,κ,N denote the ungradedring of modular form of level g with level- N -structure. We have to show that theclass group Cl ( R N ) of the Krull domain R N = M g,κ,N vanishes for level N = 1 .For suitably high level N (divisible by 8) there exists a projective embedding φ : A ∗ g,κ,N ֒ → P of the minimal compactification X ∗ = A ∗ g,κ,N of X = A g,κ,N into projective space by the theory of theta functions, such that j ∗ ( O P (1)) = ω .See [CF], p.157-159 and [CF], V.5. (It is preferable to use theta-level-structures ( N, N ) , but we skip a discussion of this).By definition Spec ( R N ) = C ( X ∗ ) is the affine cone over the projective subvariety j : X ∗ = A ∗ g,κ,N ֒ → P , since it is the ungraded ring of the corresponding gradedcoordinate ring of X ∗ . Therefore Cl ( R N ) = Cl ( C ( X ∗ )) is obtained from Cl ( X ∗ ) by Cl ( C ( X ∗ )) = Cl ( X ∗ ) (cid:14) Z · H where H is the class corresponding to a hyperplane section H , which is a sectionof j ∗ ( O P (1)) = ω . The cone C ( X ∗ ) is normal and regular outside the isolatedcusp and the singular lines over X ∗ s = X ∗ \ X ∗ reg . Since g ≥ , the locus X ∗ s has codimension ≥ . Hence there exist canonical G N -equivariant isomorphisms Cl ( C ( X ∗ )) ∼ = P ic ( C ( X ∗ )) and Cl ( X ∗ ) = P ic ( X ∗ ) and Cl ( X ) ∼ = P ic ( X ) .The complement of X s = X \ X reg in X ∗ s also has codimension 2. Hence also Cl ( X ) ∼ = Cl ( X ∗ ) , and therefore P ic ( X ∗ ) = P ic ( X ) . The class of H thereforecorresponds to the class of ω in P ic ( X ) . Hilbert theorem 90 implies Cl ( R ) ֒ → Cl ( R N ) G N . The composed map δ : Cl ( R N ) G N → H ( G N , Quot ( R N ) ∗ /κ ∗ ) → H ( G N , κ ∗ ) is trivial on the image of Cl ( R ) . A class x in Cl ( R ) defines a class y in Cl ( R N ) with vanishing obstruction in H ( G N , κ ∗ ) (notice for | N this group may benontrivial). Via the isomorphisms above there exists a corresponding class y ∗ in P ic ( X ) / Z · c ( ω ) ∼ = P ic ( X ∗ ) / Z · c ( ω ) ∼ = Cl ( X ∗ ) / Z · j ∗ ( O (1)) ∼ = Cl ( C ( X ∗ )) ,whose obstruction in H ( G N , κ ∗ ) vanishes in the sense of step 1 of the proof ofthe theorem. Hence by descend any representative y ∈ P ic ( A g,κ,N ) of y ∗ is thepullback of a class from P ic ( A g,κ ) . However P ic ( A g,κ ) = Z · c ( ω ) as shown in0the theorem for p > p . Therefore y ∗ = 0 , hence y = 0 and x = 0 . This proves Cl ( R ) = 0 for p > p , hence the unique factorization theorem. (cid:3) References [C] Chai C.L., Compactification of Siegel moduli schemes, London Math. Soc.Lecture Notes Series 107, Cambridge university press, (1985)[CF] Chai C.L.-Faltings G., Degeneration of Abelian Varieties, Ergebnisse derMathematik und ihrer Grenzgebiete, vol. 22, Springer Verlag (1990)[D] Demazure M., Lectures on p -Divisible Groups, SLN 302, Springer Verlag(1972)[I] Igusa J.I., On the ring of modular forms of degree two over Z , Jorn. of Math.101, 1-3 (1979), 149 - 183[FD] Fiederowisz Z.-Priddy S., Homology of Classical Groups over a Finite Field,in Algebraic K-theory (Evanston 1976), edited by M.R.Stein, SLN 551, SpringerVerlag, p. 269 - 282[F] Freitag E., Die Irreduzibilit¨at der Schottkyrelation (Bemerkung zu einem Satzvon J.Igusa), Archiv Math. vol. 40 (1983), 255 - 259[K] Katz N.M. p -adic properties of modular schemes and modular forms, ModularFunctions of One Variable III, SLN vol. 350, Springer Verlag[K2] Katz N.M. A result of modular forms in characteristic p , Modular Functionsof One Variable V, SLN vol. 601, Springer Verlag[MB] Moret-Bailly, L., Pinceaux de varietes abeliennes, Asterisque 129, (1985)[M] Mumford D., On the equations defining abelian varieties I, Invent. Math. 1(1966), 287 - 354129 (1985)[M2] Mumford D., On the equations defining abelian varieties II, Invent. Math. 3(1967), 75 - 135[S] J.P.Serre, Congruences et formes modulaires, Seminaire Bourbaki 1971/72,exp. 416, SLN 317[Sw] Swinnerton-Dyer H.P.F., On ll