Modular Forms of Degree 2 and Curves of Genus 2 in Characteristic 2

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Introduction
In the 1960s, Igusa determined the ring of Siegel modular forms of degree 2 using the close connection between the moduli of principally polarized abelian surfaces and the moduli of curves of genus 2, see [9].Since curves of genus 2 in characteristic different from 2 can be described by binary sextics, he could use the invariant theory of binary sextics and establish the link with modular forms using Thomae's formulas that express theta constants in terms of cross ratios of zeros of binary sextics.Recently, in a joint work with Carel Faber, we exhibited in [2] a more direct version of this without the detour of Thomae's formulas and used it for describing vector-valued Siegel modular forms of degree 2.
Modular Forms of Degree 2 and Curves of Genus 2 in Characteristic 2 5205 In positive characteristic, modular forms of degree 2 are described as sections of powers of the determinant bundle L of the Hodge bundle E on the moduli space A 2 ⊗ F p of principally polarized abelian surfaces in characteristic p.It turns out that for p ≥ 5 the situation is very close to the case of characteristic 0. For p ≥ 5, the graded ring has generators of weight 4, 6, 10, 12, and 35 and there is a relation of degree 70 expressing the square of the odd weight generator in terms of the even weight generators, just as in characteristic 0. In fact, Igusa determined the ring R 2 (Z) of integral modular forms and the reduction map R 2 (Z) → R 2 (F p ) is surjective for p ≥ 5. We refer to [1,10,11,15,16].
The situation in characteristics 2 and 3 is different.The ring R 2 (F 3 ) was determined in [7] using the relation with binary sextics.
In characteristic 2, curves of genus 2 can no longer be described by binary sextics.Nevertheless, as Igusa showed in [9], there is still a close relationship with the invariant theory of binary sextics.In this paper we use invariants of curves of genus 2 in characteristic 2 to determine the ring of modular forms of degree 2. The result is as follows.
The modular form ψ 1 is the Hasse invariant vanishing on the locus of nonordinary abelian surfaces; the form χ 10 vanishes doubly on the locus of abelian surfaces that are products of elliptic curves.All the forms ψ 1 , χ 10 , ψ 12 , χ 13 and χ 48 are constructed using invariant theory, see Section 4.
We point out that we find non-zero (regular) vector-valued modular forms of weights not allowed in characteristic 0, like (3, −1) or (2, 0).We thank the referees for helpful remarks.

Modular Forms
We denote by A g the moduli stack of principally polarized abelian varieties of dimension g and by E g the Hodge bundle on A g .It extends to a Faltings-Chai-type toroidal compactification Ãg .We write L = det(E g ).For g > 1 sections of L k over A g extend automatically to Ãg , a property known as the Koecher principle, see [5,Prop. 1.5,p. 140].We write M k ( g ) = H 0 ( Ãg ⊗ F 2 , L k ) for the space of scalar-valued modular forms of weight k.Moreover, we set the ring of scalar-valued modular forms of degree g in characteristic 2. It is a finitely ] with c 1 of weight 1 and a cusp form of weight 12.
We are interested in the case g = 2.In the following we shall simply write E for the Hodge bundle E 2 .
In the Chow group with rational coefficients of codimension 1 cycles of Ã2 the class of the locus A 1,1 of products of elliptic curves satisfies the relation where λ 1 is the 1st Chern class of E and D the divisor added to compactify A 2 , see [14, (8.4)].Therefore, there exists a non-zero modular form χ 10 of weight 10 over F 2 vanishing doubly on the divisor A 1,1 ⊗ F 2 .As the cycle relation shows χ 10 is a cusp form.The form is determined up to a non-zero multiplicative constant and we shall normalize χ 10 later.
Since A 1,1 is the image of a degree 2 morphism A 1 × A 1 → A 1,1 by restriction to the locus A 1,1 ⊗ F 2 we find an exact sequence Another divisor that we will use is the locus V 1 of principally polarized abelian surfaces of 2-rank ≤ 1.The cycle class of this locus is known; it is the vanishing locus of the map L → L ⊗2 induced by Verschiebung on the universal abelian surface and we thus find (cf.[6]) Hence, there exists a modular form ψ 1 of weight 1, determined up to a non-zero multiplicative constant, with divisor V 1 .It is called the Hasse invariant.This is not a cusp form since it restricts to the Hasse invariant c 1 on the boundary component We shall normalize ψ 1 later.The existence of ψ 1 implies dim M k ( 2 ) ≤ dim M k+1 ( 2 ).The exact sequence (2) and the structure of ( In order to estimate dim M k ( 2 ) for larger k we can apply semi-continuity (cf.[8, Thm. 12.8] ) to L defined over Ã2 /Z and deduce dim Using this and the exact sequence (2) we can deduce but we will deduce this in another way later, see (7).
Similarly, we can consider vector-valued modular forms.For each irreducible representation ρ of GL(2, Q) there is a corresponding bundle E ρ .Since we are dealing with g = 2 the ρ in question correspond to pairs (j, k) of integers with j ≥ 0 and for the space of cusp forms.Again, these sections extend over Ã2 ⊗ F 2 by the Koecher principle.But note that ρ ⊗ F 2 might be reducible.
Lemma 2.1.We have dim S 6,8 ( 2 ) ≥ 1 and every element of S 6,8 Proof.We know that over C we have dim S 6,8 ( 2 ) = 1.Hence, by semi-continuity of dimensions there exists a cusp form of weight (6,8) in characteristic 2. Note that the Hodge bundle E pulled back via and this is (0).If we develop a form ϕ ∈ S 6,8 ( 2 ) vanishing on A 1,1 ⊗ F 2 in the normal direction of the divisor A 1,1 ⊗ F 2 we see that the 1st ("non-constant") term in its Taylor expansion lands in and the only non-zero factor here is S 12 ( 1 ) ⊗ S 12 ( 1 ) and the image is symmetric under the interchange of factors.But dim S 12 ( 1 ) = 1.Therefore, if ϕ 1 , ϕ 2 are two linearly independent cusp forms of weight (6,8) there exists a non-trivial linear combination that vanishes with order ≥ 2 along A 1,1 ⊗ F 2 .This yields a modular form divisible by χ 10 , hence a regular section of Sym 6 We thus have at least one non-zero modular form f ∈ S 6,8 ( 2 ) that vanishes with multiplicity 1 on A 1,1 ⊗ F 2 .Thus, we also find a rational modular form f /χ 10 with possible poles along A 1,1 ⊗ F 2 .
We can analyze the order of the poles of the coordinates of f /χ 10 near a generic point of A 1,1 .We consider f near a generic point of A 1,1 ⊗ F 2 and write its Taylor expansion as where t is a local normal coordinate.If for fixed i we have γ i,r = 0 for r < r 0 then where QS denotes quasi-modular cusp forms, see [17,Sec. 5] for the notion of quasimodular form.From the dimensions of S k ( 1 ) we see as in the proof of Lemma 2.1 that ord A 1,1 (γ 3 ) ≥ −1 and we can even get the estimate ord A 1,1 (γ 0 , . . ., γ 6 ) ≥ (2, 1, 0, −1, 0, 1, 2) , but we will not use this.

Curves of Genus 2 in Characteristic 2
Let C be a smooth projective curve of genus 2 over a perfect field k of characteristic 2. Let This corresponds to the 2-rank of Jac(C) being equal to 2, 1 or 0.
We point out that a curve of genus 2 defined by an equation ( 5) comes with a basis dx/a, x dx/a of H 0 (C, O(K)).
The choices we made for arriving at equation ( 5) are a basis of H 0 (C, O(K)) and an element of η in H 0 (C, O(3K)).Clearly, another choice of η is of the form η + θ with a unit in k and θ homogeneous of degree 3 in ξ 0 , ξ 1 , or y → uy of degree ≤ 3.
The induced action on the pair (a, b) is by Another choice of basis of H 0 (C, O(K)) is given by an element of GL(V) with V = ξ 0 , ξ 1 .
The action on x, y is by Let Y be the algebraic stack of triples (π , α, β) with π : C → S a curve of genus 2, . We may view it as the stack of curves of genus 2 with a framed Hodge bundle and a section of ω ⊗3 π satisfying the additional condition that it yields an equation as (5).There is an obvious action of GL( 2) and an action of Sym 3 (O ⊕2 S ).We thus consider the stack and we can identify it with the moduli space M 2 ⊗ F 2 of curves of genus 2 in characteristic 2.
We now describe a concrete form of this stack.Let V be a 2-dimensional k-vector space generated by x 1 , x 2 .Consider the subspace of Sym We write V j,l for Sym j (V) ⊗ det(V) ⊗l .We let the semi-direct product GL(V) V 3,−1 act on V 3,−1 × V 6,−2 via twisting the two actions of GL(V) and Sym 3 (V).Without twisting an element M ∈ GL(V) acts by and v ∈ Sym  We thus consider where X 0 is the open substack given by the condition that y 2 + ay = b defines a smooth projective curve of genus 2.
The moduli space M 2 ⊗ F 2 can be identified with the stack quotient Note that by our choice of twisting the stabilizer of a pair (a, b) contains (id V , a) as it should, since the generic curve has an automorphism group of order 2. The pullback of the Hodge bundle E on M 2 ⊗ F 2 is the equivariant bundle V.

Invariants and Modular Forms
We write elements a ∈ Sym 3 (V) and b ∈ Sym 6 (V) as Following standard usage, by an invariant for the action of GL(V) Sym 3 (V) we mean a polynomial in the coordinates a 0 , . . ., a 3 and b 0 , . . ., b 6 of the pair (a, b) in Sym 3 (V) × Sym 6 (V) that is invariant under SL(V) Sym 3 (V).We define K as the ring of SL(V) Sym 3 (V)-invariants under the action on X .
The simplest example of such an invariant is the expression the square root of the discriminant of a.
Since the pullback to X 0 of the Hodge bundle E under the Torelli map becomes regular after multiplication by a power of χ 10 .Therefore, we can extend the map μ to such that ν •μ = id; here, the index χ 10 indicates localization at the multiplicative system generated by χ 10 .
Since we know the existence of a modular form ψ 1 ∈ M 1 ( 2 ) we see that μ(ψ 1 ) must equal K 1 as K 1 is the only invariant of weighted degree 3. (Here the weight of a i (resp.b j ) is i (resp.j).)This implies in particular that ν(K 1 ) is a regular modular form of weight 1.
Similarly for covariants.By a covariant under the action of GL(V) Sym 3 (V), we mean a polynomial in the coefficients a i and b j and x 1 , x 2 , which is an invariant for the action of SL(V) Sym 3 (V).
The simplest example of a covariant is given by the polynomial a.It defines an a priori rational modular form of weight (3, −1).Proof.The pullback of the Sym 3 (E) to (A 1 × A 1 ) ⊗ F 2 decomposes as and the coefficients of a correspond to the factors.In view of the symmetry we have ).As we saw above the form ν(K 1 ) = ν(a 0 a 3 + a 1 a 2 ) is regular.By coordinate changes we may assume that a 0 = 0 or a 1 = 0, so it easily follows that ord A 1,1 (a i ) ≥ 0 and the form is regular.Moreover, since its pullback lands in

Corollary 4.2. We have ord
Proof.We note that ψ 1 does not vanish on A 1,1 ⊗ F 2 .

Modular Forms from Invariants
Igusa used in [9] invariants of binary sextics to construct modular forms and to construct a coarse moduli space of curves of genus 2. The Z-algebra of even degree invariants of binary sextics describes a coarse moduli space for curves of genus 2 over Z.
To get characteristic 2 invariants (or covariants) one lifts the curve given by y 2 + ay = b to the Witt ring, say defined by ( ã, b), and takes an invariant (or covariant) of the binary sextic defined by ã2 + 4 b, divides these by an appropriate power of 2 and reduces these modulo 2. This defines invariants (or covariants).
If f = 6 i=0 c i x 6−i is the (universal) binary sextic then the invariant gives by the substitution For the formula for J 2 and the formulas of the invariants J 2i (i = 2, 3, 5) used below we refer to [13, pp. 139-140] and [12, p. 204].
Similarly, the invariant J 4 given by 2 −7 (2640 c 2 0 c 2 6 − 880 c 0 c 1 c 5 c 6 + . ..) yields via I 4 = J 2 2 −24 J 4 an invariant K 4 in characteristic 2 that turns out to be reducible and divisible by K 1 .We thus set K 3 = K 4 /K 1 and get an invariant Similarly, J 6 yields the invariant K 2 3 in characteristic 2. The invariant J 8 yields the invariant and similarly from J 10 we obtain Finally, the invariant I 15 defines an invariant K 15 , but K 15 can be expressed in terms of the invariants already obtained:  Using this lemma we obtain modular forms and as the form of K 3 shows that ν(K 3 ) has order −2 along A 1,1 since K 1 does not vanish on A 1,1 .We see that χ Remark 5.2.From the fact that χ 10 (resp.ψ 12 ) vanishes with multiplicity 2 (resp.0) along A 1,1 we see that the order of a 0 and a 3 along A 1,1 is 1.

there must be a linear relation η 2 +
c 3 η + c 6 = 0 with c 3 (resp.c 6 ) homogeneous of degree 3 (resp.6) in ξ 0 , ξ 1 .Setting x = ξ 1 /ξ 0 and y = η/ξ 3 0 we can write the equation as y 2 + a y = b (5) with a ∈ k[x] non-zero of degree ≤ 3 and b ∈ k[x] of degree ≤ 6.The hyperelliptic involution of the curve C is given by y → y+a.The fixed points are given by the equation a = 0 (or c 3 = 0 in projective coordinates); hence, there are 3, 2 or 1 ramification points.
3 (V) × Sym 6 (V) of pairs (a, b) satisfying the condition that the pair defines a non-singular curve; in the affine version this amounts to a and (a ) 2 b + (b ) 2 (with a and b the derivative) having no root in common.
3 (V) by (a, b) → (a, b + v 2 + va) .After twisting c • Id V acts via c on V 3,−1 and by c 2 on V 6,−2 and this is compatible with the equation y 2 + ay = b if we let c • Id V act on y by y → c y.It is also compatible with b → b + v 2 + av for v ∈ V 3,−1 .

7 )Consider K 12 = K 8 3 + a 0 a 1 a 2 a 3 + a 0 a 3 2 + a 3 1 a 3 ) 4 b 4 3 +
13 is not divisible by ψ 1 , hence dim M 12 ( 2 ) < dim M 13 ( 2 ).In view of the estimates on the dimensions of M k ( 2 ) implied by (2) we conclude that dim M 12 ( 2 ) = 3 and dim M 13 ( 2 ) = 4 .(By inspecting the expression of the invariant K 10 in terms of the a i and b j we see that ν(K 10 ) is a modular form of weight 10 that vanishes with multiplicity 2 along A 1,1 .Therefore, by the cycle class of A 1,1 given in (1) it is a multiple of χ 10 and we normalize χ 10 by setting χ 10 = ν(K 10 ) . . . .and one can check that it has order 0 along A 1,1 .This defines a modular form ψ 12 = ν(K 12 ) .