The ring of modular forms of degree two in characteristic three

We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3.


Introduction
Let A g be the moduli space of principally polarized abelian varieties of dimension g.It is a Deligne-Mumford stack over Z.It carries a natural vector bundle of rank g, the Hodge bundle E g .We write L for its determinant line bundle.The vector bundle E g extends in a natural way over any compactification Ãg of Faltings-Chai type and we will denote the extension of E g and L again by the same symbols.Sections of L ⊗k over Ãg are called modular forms of weight k.It is known that for g ≥ 2 any section of L k over A g extends to a section of L k over Ãg , a fact usually referred to as the Koecher principle, see [6,Prop. 1.5,p. 140].
If F = Z or Z p or a field one has the graded ring It is known by [6] that it is a finitely generated F-algebra.
In the case of F = C the ring R g (C) is the ring of scalar-valued Siegel modular forms of degree g.It is well-known known that R 1 (C) = C[E 4 , E 6 ] is freely generated over C by the Eisenstein series E 4 and E 6 of weights 4 and 6.In the 1960s Igusa [10] determined the structure of R 2 (C): R 2 (C) = C[ψ 4 , ψ 6 , χ 10 , χ 12 , χ 35 ]/(χ 2 35 − P ), where the indices of the generators indicate the weights and P is a polynomial in ψ 4 , ψ 6 , χ 10 and χ 12 .Moreover, the ideal of cusp forms is generated by χ 10 , χ 12 and χ 35 .For g = 3, Tsuyumine showed in [16] that R 3 (C) is generated by 34 elements; recently the number of generators was reduced to 19 by Lercier and Ritzenthaler [13].
For F = F p , a finite field with p elements, the ring R 1 (F p ) was described by Deligne [4].Besides giving the structure of the ring over Z 1991 Mathematics Subject Classification.11F03,14J15, 14G35, 11G18.
In this paper we consider the case p = 3 and determine the structure of R 2 (F 3 ).We use the close connection between the moduli space A 2 and the moduli space M 2 of curves of genus 2 via the Torelli map M 2 → A 2 and the description of M 2 as a quotient stack for the action of GL(2) on the space of binary sextics.In that way invariant theory can be used to construct modular forms.The relation between invariants and modular forms was already exploited by Igusa in [10], but he used theta functions and Thomae's formula to relate these to cross ratios of the zeros of a binary sextic.Here we use not only invariants but also covariants giving vector-valued modular forms as introduced in [2] to analyze the regularity of scalar-valued modular forms.
Our result is: Theorem 1.1.The subring R ev 2 (F 3 ) of modular forms of even weight is generated by forms of weights 2, 10, 12, 14 and 36 and has the form

The proof of Theorem 1.1.
The moduli stack A 2 ⊗ F 3 has a canonical compactification Ã2 ⊗ F 3 .We will denote the space of sections of L k on Ã2 ⊗ F 3 by M k (Γ 2 ) and we thus have R 2 (F 3 ) = ⊕ k M k (Γ 2 ).We write M k (Γ 1 ) for the space H 0 ( Ã1 ⊗ F 3 , L k ).The Satake compactification is denoted by A * 2 ⊗ F 3 .We denote the first Chern class of L by λ 1 .
The locus V 1 of abelian surfaces with p-rank ≤ 1 is a divisor in A 2 ⊗F p and its closure V 1 in Ã2 ⊗F p has cycle class (p−1)λ 1 , so [V 1 ] = 2λ 1 for p = 3, see [8,5].Therefore there is a modular form ψ 2 of weight 2 whose zero divisor is V 1 .It is determined up to multiplication by a non-zero scalar.We will normalize it later.This form is known as the Hasse invariant.Multiplication by The divisor of products of elliptic curves H 1 := A 1,1 ⊗ F 3 gives rise to a second modular form.(The notation refers to the fact that H 1 is the Humbert surface of discriminant 1.)In the Chow group of codimension 1 of with D the divisor at infinity, hence there exists a modular form of weight 10 vanishing with multiplicity 2 on H 1 .We call this form χ 10 (up to normalization to be determined later).The automorphism group of a generic product of elliptic curves has an extra involution (when compared with the automorphism group of a generic principally polarized abelian surface) and it acts by −1 on L, hence every modular form of even weight vanishes with even multiplicity along H 1 .
Restriction to H 1 yields for even k an exact sequence We now turn to the construction of the other generators.We use the ideas of [2].The Torelli map defines an embedding M 2 ⊗ F 3 → A 2 ⊗ F 3 .A smooth projective curve of genus 2 can be given by an equation We let V = x 1 , x 2 be the F 3 -vector space generated by x 1 , x 2 and write f as a homogeneous polynomial 6 i=0 a i x 6−i 1 x i 2 .Note that a curve as in (3) comes with a basis of the space of regular differentials, viz.dx/y, xdx/y.
The pull back to X 0 of the Hodge bundle under the composition of X 0 → M 2 with the Torelli map M 2 → A 2 is the equivariant bundle V on X 0 .The pullback of L is det(V ).As a consequence pulling back defines a homomorphism µ : R 2 (F 3 ) → I with I the ring of invariants of the action of GL(V ) on Sym 6 (V ).Here an invariant is a polynomial in a 0 , . . ., a 6 , the coefficients of f that is invariant under SL(V ).Since the image of M 2 in A 2 is a Zariski open part with complement H 1 , not every invariant corresponds to a modular form; but every invariant corresponds to a rational modular form that is regular outside H 1 .In particular, it becomes regular on all of A 2 when multiplied with a sufficiently high power of χ 10 .This provides us with homomorphisms where R 2 (F 3 ) χ 10 is obtained from R 2 (F 3 ) by allowing powers of χ 10 in the denominator.We have ν • µ = id.This generalizes as follows to vector-valued modular forms.For each finite dimensional irreducible representation ρ of GL(2) there is a vector bundle E ρ 2 obtained from E 2 by applying a Schur functor.Such a ρ is of the form Sym j (St)⊗det k (St) with St the standard representation of GL(V ).A section of Sym j (E 2 ) ⊗ det(E 2 ) k over A 2 is called a modular form of degree 2 and weight (j, k).The Koecher principle also applies to these modular forms: sections of E ρ 2 over A 2 extend over Ã2 .We write and we consider the R 2 (F 3 )-module It is even a ring.The map (4) can be extended to a map from M to the ring of covariants.Here a covariant can be described as an invariant for the action of GL(V ) on V ⊕ Sym 6 (V ).Alternatively, covariants can be obtained by taking an equivariant embedding of an irreducible GL(V )-representation U → Sym d (Sym 6 (V )), or equivalently, an equivariant map and then Φ = ϕ(1) is a covariant.If U is an irreducible representation of highest weight (w 1 , w 2 ) then one may view Φ as a homogeneous form in a 0 , . . ., a 6 of degree d and of degree w 1 − w 2 in x 1 , x 2 .For example, taking U = Sym 6 (V ) and d = 1 yields the covariant Φ = f , the universal binary sextic.Covariants form a ring C that was much studied in the 19th and early 20th century.Grace and Young determined generators of this ring.The maps where M χ 10 is obtained from M by admitting powers of χ 10 as denominators.
We have ν The image under ν of the covariant f , the universal binary sextic, is a rational modular form χ 6,−2 , that is, a rational section of Sym 6 (E 2 ) ⊗ det(E 2 ) −2 that is regular after multiplication by an appropriate power of χ 10 .
This construction was given in [2] in characteristic zero and yields a meromorphic modular form, here denoted ϕ 6,−2 , that becomes holomorphic after multiplication by χ 10 .The reduction of the characteristic zero rational modular form ϕ 6,−2 yields a rational modular form in characteristic 3.This implies that χ 6,−2 becomes regular after multiplication by χ 10 .We can write the form χ 6,−2 locally on A 2 ⊗ F 3 symbolically as where the monomials X 6−i 1 X i 2 are dummies to indicate the coordinates in the fibres of Sym 6 (E 2 ) ⊗ det(E 2 ) −2 .Here we view α i locally as a rational function on A 2 ⊗ F 3 .Using the local expression ( 5) one can give the image ν(T ) of an invariant T = T (a 0 , . . ., a 6 ) locally by T (α 0 , . . ., α 6 ).
We note that interchanging X 1 and X 2 induces an involution replacing α i by α 6−i .
Comparing with the characteristic 0 case and using semi-continuity we see that the orders of the rational functions α i along the divisor H 1 are at least equal to the orders of their complex analogues along H 1 .The Fourier expansion in characteristic 0 given in [2, page 1658] implies the following inequalities for the orders of α i along H 1 in characteristic 3: Moreover, the symmetry that interchanges x 1 and x 2 implies that the orders of α i and α 6−i along H 1 are equal.
The ring of invariants I for the action of GL(V ) on Sym 6 (V ) in characteristic 3 is generated by invariants A, B, C, D and E of degree 2, 4, 6, 10 and 15, see e.g.[10] or [9].
The invariant A has the form A = a 1 a 5 − a 2 a 4 .We know of the existence of a modular form ψ 2 of weight 2. Under the map µ it must map to a nonzero multiple of A. We fix ψ 2 by requiring µ(ψ 2 ) = A. The restriction to H 1 of the Hasse invariant ψ 2 is a non-zero multiple of Sym 2 (b 2 ), with b 2 the Hasse invariant for g = 1, hence ψ 2 does not vanish identically on H 1 .
By the inequalities (6) and the expression for A we see that ord In degree 4 we find another invariant B, not a multiple of A 2 : Since we know dim M 4 (Γ 2 ) = 1 there cannot be a regular modular form in weight 4 that is not a multiple of ψ 2 2 .This implies that ord H 1 (α 3 ) < 0 and hence ord H 1 (α 3 ) = −1.Thus B = (a 1 a 5 − a 2 a 4 )a 2  3 + (a 1 a 2 4 + a 2 2 a 5 )a 3 + • • • defines a rational modular form χ B = ν(B) of weight 4 with order −2 along H 1 .Since χ 10 vanishes with multiplicity 2 along H 1 we thus find that χ 14 := χ B χ 10 is a regular modular form of weight 14.
The vector space of invariants of degree 6 is generated by A 3 , AB and an invariant C C = 2 a 6 3 + A a 4 3 + 2(a 1 a 2 2 + a 2 2 a 5 )a 3 3 + • • • and we see that χ C = ν(C) has order −6 along H 1 .In degree 10 there is a new invariant • yielding a modular form that vanishes with multiplicity ≥ 2 on H 1 .Indeed, since α 1 α 5 vanishes with multiplicity 2 the first term (α 1 α 5 ) 3 α 4 3 vanishes with order 2; the next terms also vanish with order ≥ 2 as one easily checks.Therefore χ D is regular and vanishes with multiplicity ≥ 2. Since χ D is not zero, it must be a multiple of χ 10 and then vanishes on H 1 with multiplicity 2. We fix χ 10 by setting it equal to χ D = ν(D).This fixes χ 14 too.
A further generator is . Since the orders of χ C and χ 10 along H 1 are −6 and 2 the modular form χ 36 is regular and does not vanish identically on H 1 .The modular form χ 36 is not contained in the subring generated by ψ 2 ,χ 10 , ψ 12 and χ 14 .We have the identity by which we can express ψ 12 χ 3 10 in the other generators: Since A, B, C, D are generators of the ring of invariants and are algebraically independent the ideal of relations between the generators ψ 2 , χ 10 , ψ 12 , χ 14 and χ 36 is generated by the relation (6).The forms ψ 2 , χ 10 , ψ 12 , χ 14 and χ 36 generate a subring of the ring R ev 2 (F 3 ) with generating function .
The leading coefficient of the corresponding Hilbert polynomial is 42 2 .
On the other hand we have c 1 (L) 35 is of even weight, hence can be expressed as a polynomial in ψ 2 , χ 10 , ψ 12 and χ 36 .If ψ is an odd weight modular form then it must vanish on H 1 and H 4 , hence it will be divisible by χ 35 .The invariant µ(χ) of any cusp form χ is divisible by D in I.The invariants that correspond to χ 10 , χ 14 are divisible by D, hence also these are cusp forms.From the form of the generators one easily sees that χ 10 , χ 14 , χ 36 and χ 35 generate the ideal of cusp forms.This completes the proof.
3= 1/2880, see[8, p. 74].By the Riemann-Roch theorem we have dim M k (Γ 2 ) = k 3 /2880 + O(k 2 ) for even k.Therefore there cannot be more generators of R ev 2 (F 3 ).The invariant E of degree 15 is of the form has order −3 along H 1 .Thereforeχ 35 := ν(ED2 ) is a regular modular form.It vanishes on H 1 and on the Humbert surface H 4 of discriminant 4, both with multiplicity 1.The surfaces H 1 and H 4 parametrize abelian surfaces that possess an extra involution.Locally near H 4 the extra automorphism corresponds to the symmetry that interchanges x 1 and x 2 .We know that the cycle class of 2 H 4 on A * 2 ⊗ F 3 is 60λ 1 , see [7, Prop.3.3, p. 217].Therefore the divisor of χ 35 is H 1 + H 4 and since the closure of H 1 contains the 1-dimensional cusp χ 35 is a cusp form.Then χ 2