The ring of modular forms of degree two in characteristic three

Abstract

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

1 Introduction

Let ${{\mathcal {A}}}_g$ be the moduli space of principally polarized abelian varieties of dimension g. It is a Deligne-Mumford stack over ${{\mathbb {Z}}}$. It carries a natural vector bundle of rank g, the Hodge bundle ${{\mathbb {E}}}_g$. We write L for its determinant line bundle. The vector bundle ${{\mathbb {E}}}_g$ extends in a natural way over any compactification ${\tilde{{\mathcal {A}}}}_g$ of Faltings-Chai type and we will denote the extension of ${{\mathbb {E}}}_g$ and L again by the same symbols. Sections of $L^{\otimes k}$ over ${\tilde{{\mathcal {A}}}}_g$ are called modular forms of weight k. It is known that for $g\ge 2$ any section of $L^k$ over ${{\mathcal {A}}}_g$ extends to a section of $L^k$ over ${\tilde{{\mathcal {A}}}}_g$, a fact usually referred to as the Koecher principle, see [7, Prop. 1.5, p. 140].

If ${{\mathbb {F}}}={{\mathbb {Z}}}$ or ${{\mathbb {Z}}}_p$ or a field one has the graded ring

$$\begin{aligned} {{\mathcal {R}}}_g({{\mathbb {F}}})= \oplus _k H^0({\tilde{{\mathcal {A}}}}_g \otimes {{\mathbb {F}}},L^k)\, . \end{aligned}$$

It is known by [7] that it is a finitely generated ${{\mathbb {F}}}$-algebra.

In the case of ${{\mathbb {F}}}={{\mathbb {C}}}$ the ring ${{\mathcal {R}}}_g({{\mathbb {C}}})$ is the ring of scalar-valued Siegel modular forms of degree g. It is well-known known that ${{\mathcal {R}}}_1({{\mathbb {C}}})={{\mathbb {C}}}[E_4,E_6]$ is freely generated over ${{\mathbb {C}}}$ by the Eisenstein series $E_4$ and $E_6$ of weights 4 and 6. In the 1960s Igusa [11] determined the structure of ${{\mathcal {R}}}_2({{\mathbb {C}}})$:

$$\begin{aligned} {{\mathcal {R}}}_2({{\mathbb {C}}})={{\mathbb {C}}}[\psi _4,\psi _6,\chi _{10},\chi _{12}, \chi _{35}]/\left( \chi _{35}^2-P\right) , \end{aligned}$$

where the indices of the generators indicate the weights and P is a polynomial in $\psi _4,\psi _6,\chi _{10}$ and $\chi _{12}$. Moreover, the ideal of cusp forms is generated by $\chi _{10},\chi _{12}$ and $\chi _{35}$. For $g=3$, Tsuyumine showed in [20] that ${{\mathcal {R}}}_3({{\mathbb {C}}})$ is generated by 34 elements; recently the number of generators was reduced to 19 by Lercier and Ritzenthaler [14].

For ${{\mathbb {F}}}={{\mathbb {F}}}_p$, a finite field with p elements, the ring $R_1({{\mathbb {F}}}_p)$ was described by Deligne [5]. Besides giving the structure of the ring over ${{\mathbb {Z}}}$

$$\begin{aligned} \mathcal {R}_1({{\mathbb {Z}}})={{\mathbb {Z}}}[c_4,c_6,\Delta ]/\left( c_4^3-c_6^2-1728 \, \Delta \right) , \end{aligned}$$

he showed that

$$\begin{aligned} {{\mathcal {R}}}_1({{\mathbb {F}}}_2)={{\mathbb {F}}}_2[a_1,\Delta ] \quad \text { and} \quad {{\mathcal {R}}}_1({{\mathbb {F}}}_3)={{\mathbb {F}}}_3[b_2,\Delta ]\, , \end{aligned}$$

where $\Delta$ is of weight 12 and $a_1$ (resp $b_2$) is of weight 1 (resp. 2). For $p\ge 5$ we have ${{\mathcal {R}}}_1({{\mathbb {F}}}_p)={{\mathbb {F}}}_p[c_4,c_6]$.

For $g=2$, Igusa determined in [13] also the ring of modular forms over ${{\mathbb {Z}}}$; it is generated by elements of weight

$$\begin{aligned} 4, 6, 10, 12, 12, 16, 18, 24, 28, 30, 35, 36, 40, 42, 48\, . \end{aligned}$$

For finite fields the structure of ${{\mathcal {R}}}_2({{\mathbb {F}}}_p)$ is known for $p\ge 5$. For this we refer to Ichikawa’s paper [10]. For $p\ge 5$ the ring is just as in characteristic zero generated by modular forms $\psi _4$, $\psi _6$, $\chi _{10}$, $\chi _{12}$ and $\chi _{35}$ with $\chi _{35}$ satisfying a relation $\chi _{35}^2=P(\psi _4,\psi _6,\chi _{10},\chi _{12})$. Moreover for $p\ge 5$ the reduction map ${{\mathcal {R}}}_2({{\mathbb {Z}}}_p) \rightarrow {{\mathcal {R}}}_2({{\mathbb {F}}}_p)$ is surjective. Nagaoka studied the image of the reduction map in [17, 18], see also [1].

In this paper we consider the case $p=3$ and determine the structure of ${{\mathcal {R}}}_2({{\mathbb {F}}}_3)$. We use the close connection between the moduli space ${{\mathcal {A}}}_2$ and the moduli space ${{\mathcal {M}}}_2$ of curves of genus 2 via the Torelli map ${{\mathcal {M}}}_2 \hookrightarrow {{\mathcal {A}}}_2$ and the description of ${{\mathcal {M}}}_2$ as a quotient stack for the action of $\mathrm{GL}(2)$ 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 [11], 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 ${{\mathcal {R}}}_2^{\mathrm{ev}}({{\mathbb {F}}}_3)$ of modular forms of even weight is generated by forms of weights 2, 10, 12, 14 and 36 and has the form

$$\begin{aligned} {{\mathcal {R}}}_2^{\mathrm{ev}}({{\mathbb {F}}}_3)= {{\mathbb {F}}}_3[\psi _2, \chi _{10}, \psi _{12}, \chi _{14},\chi _{36}]/J \end{aligned}$$

with J the ideal generated by the relation

$$\begin{aligned} \psi _2^3\chi _{36}-\chi _{10}^3\psi _{12}-\psi _2^2\chi _{10}\chi _{14}^2+\chi _{14}^3\, . \end{aligned}$$

Moreover, ${{\mathcal {R}}}_2({{\mathbb {F}}}_3)={{\mathcal {R}}}_2^{\mathrm{ev}}({{\mathbb {F}}}_3)[\chi _{35}]/(\chi _{35}^2-P)$ with P a polynomial in $\psi _2$, $\chi _{10}$, $\psi _{12},\chi _{14}$ and $\chi _{36}$. The ideal of cusp forms is generated by $\chi _{10}, \chi _{14},\chi _{35}, \chi _{36}$.

The generator $\psi _2$ is the Hasse invariant that vanishes on the locus of non-ordinary abelian surfaces and $\chi _{10}$ is a form that vanishes on the locus of products of elliptic curves. The ring of modular forms of degree 2 in characteristic 2 is described in [4].

2 The proof of Theorem 1.1

Since for $g=2$ the moduli stack ${{\mathcal {A}}}_g\otimes {{\mathbb {F}}}_3$ has a canonical compactification due to Igusa we will use this compactification ${\tilde{{\mathcal {A}}}}_2 \otimes {{\mathbb {F}}}_3$. We will denote the space of sections of $L^k$ on ${\tilde{{\mathcal {A}}}}_2 \otimes {{\mathbb {F}}}_3$ by $M_k(\Gamma _2)$ and we thus have ${{\mathcal {R}}}_2({{\mathbb {F}}}_3)=\oplus _k M_k(\Gamma _2)$. We write $M_k(\Gamma _1)$ for the space $H^0({\tilde{{\mathcal {A}}}}_1\otimes {{\mathbb {F}}}_3,L^k)$. The Satake compactification is denoted by ${{\mathcal {A}}}_2^{*}\otimes {{\mathbb {F}}}_3$. We denote the first Chern class of L by $\lambda _1$.

We begin by constructing generators of weight 2 and 10. The locus $V_1$ of abelian surfaces with p-rank $\le 1$ is a divisor in ${{\mathcal {A}}}_{2}\otimes {{\mathbb {F}}}_p$ and its closure ${\overline{V}}_1$ in ${\tilde{{\mathcal {A}}}}_2\otimes {{\mathbb {F}}}_p$ has cycle class $(p-1)\lambda _1$ in the Chow group with ${{\mathbb {Q}}}$-coefficients, so $[{\overline{V}}_1]=2\lambda _1$ for $p=3$, see [6, 22]. Therefore the effective divisor ${\overline{V}}_1$ is the divisor of a section of $L^{\otimes 2}$ and there is a modular form $\psi _2$ of weight 2 whose zero divisor is ${\overline{V}}_1$. 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 $\psi _2$ implies that $\dim M_k(\Gamma _2) \le \dim M_{k+2}(\Gamma _2)$.

The divisor of products of elliptic curves $H_1:={{\mathcal {A}}}_{1,1}\otimes {{\mathbb {F}}}_3$ 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 ${\tilde{{\mathcal {A}}}}_2\otimes {{\mathbb {F}}}_3$ (resp. ${{\mathcal {A}}}_2^*\otimes {{\mathbb {F}}}_3$) we have the relation (cf. e.g. [16, p. 317])

$$\begin{aligned} 2[{\overline{H}}_1]+[D]= 10\, \lambda _1 \quad (\mathrm{resp.\ } \quad 2[{\overline{H}}_{1}]= 10\, \lambda _1)\, , \end{aligned}$$

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 $\chi _{10}$ (up to a 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

$$\begin{aligned} 0 \rightarrow H^0(\mathcal {A}_2\otimes {{\mathbb {F}}}_3,L^k\otimes O(-2H_1))\rightarrow H^0(\mathcal {A}_2\otimes {{\mathbb {F}}}_3, L^k) \rightarrow H^0\left( H_1,L^k_{|H_1}\right) \end{aligned}$$

and in view of the degree 2 morphism $\mathcal {A}_1\times \mathcal {A}_1 \rightarrow \mathcal {A}_{1,1}$ induced by interchanging the two factors, we can identify this with

$$\begin{aligned} 0 \rightarrow M_{k-10}(\Gamma _2) \rightarrow M_k(\Gamma _2) \rightarrow \mathrm{Sym}^2(M_k(\Gamma _1))\, , \end{aligned}$$

(2.1)

where the second arrow is multiplication by $\chi _{10}$. Moreover $M_{k-10}(\Gamma _2)=(0)$ for $k <8$ since L is ample on ${{\mathcal {A}}}_2^*\otimes {{\mathbb {F}}}_3$. The exact sequence (2.1) and the fact that we know $M_k(\Gamma _1)$ implies that $\dim M_k(\Gamma _2)=1$ for $k=2,4,6,8$ and $\dim M_{10}(\Gamma _2)=2$ and $M_{10}(\Gamma _2)$ is generated by $\psi _2^5$ and $\chi _{10}$.

We now turn to the construction of the other generators. We use the ideas of [2]. The Torelli map defines an embedding ${{\mathcal {M}}}_2\otimes {{\mathbb {F}}}_3 \rightarrow {{\mathcal {A}}}_2\otimes {{\mathbb {F}}}_3$. A smooth projective curve of genus 2 can be given by an equation

$$\begin{aligned} y^2=f(x) \quad \text { with }f=\sum _{i=0}^6 a_i \, x^{6-i} \, . \end{aligned}$$

(2.2)

We let $V=\langle x_1,x_2 \rangle$ be the ${{\mathbb {F}}}_3$-vector space generated by $x_1,x_2$ and write f as a homogeneous polynomial $\sum _{i=0}^6 a_i x_1^{6-i}x_2^i$. Note that a curve as in (2.2) comes with a basis of the space of regular differentials, viz. $dx/y, \, xdx/y$.

We have a description of ${{\mathcal {M}}}_2\otimes {{\mathbb {F}}}_3$ as the stack quotient $[\mathcal {X}^0/\mathrm{GL}(V)]$ with $\mathcal {X}^0 \subset \mathcal {X}=\mathrm{Sym}^6(V) \otimes \det (V)^{-2}$ the locus given by the non-vanishing of the discriminant, see [3, Section 3, p. 3].

The pullback to ${{\mathcal {X}}}^0$ of the Hodge bundle under the composition of ${{\mathcal {X}}}^0 \rightarrow {{\mathcal {M}}}_2$ with the Torelli map ${{\mathcal {M}}}_2 \hookrightarrow {{\mathcal {A}}}_2$ is the equivariant bundle V on ${{\mathcal {X}}}^0$ as the basis $dx/y, \, xdx/y$ of the space of regular differentials on the curve $y^2=f(x)$ shows. The pullback of L is $\det (V)$. As a consequence pulling back defines a homomorphism

$$\begin{aligned} \mu : {{\mathcal {R}}}_2({{\mathbb {F}}}_3) \rightarrow I \end{aligned}$$

(2.3)

with I the ring of invariants of the action of $\mathrm{GL}(V)$ on $\mathrm{Sym}^6(V)$. Here an invariant is a polynomial in $a_0,\ldots ,a_6$, the coefficients of f that is invariant under $\mathrm{SL}(V)$. Since the image of ${{\mathcal {M}}}_2$ in ${{\mathcal {A}}}_2$ 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 ${{\mathcal {A}}}_2$ when multiplied with a sufficiently high power of $\chi _{10}$. This provides us with homomorphisms

$$\begin{aligned} {{\mathcal {R}}}_2({{\mathbb {F}}}_3) {\buildrel \mu \over \longrightarrow } I {\buildrel \nu \over \longrightarrow } {{\mathcal {R}}}_2({{\mathbb {F}}}_3)_{\chi _{10}}\, , \end{aligned}$$

where ${{\mathcal {R}}}_2({{\mathbb {F}}}_3)_{\chi _{10}}$ is obtained from ${{\mathcal {R}}}_2({{\mathbb {F}}}_3)$ by allowing powers of $\chi _{10}$ in the denominator. We have $\nu \circ \mu =\mathrm{id}$.

This generalizes as follows to vector-valued modular forms. For each finite dimensional irreducible representation $\rho$ of $\mathrm{GL}(2)$ there is a vector bundle ${{\mathbb {E}}}_2^{\rho }$ obtained from ${{\mathbb {E}}}_2$ by applying a Schur functor. Such a $\rho$ is of the form $\mathrm{Sym}^j(\mathrm{St}) \otimes \det ^k(\mathrm{St})$ with $\mathrm{St}$ the standard representation of $\mathrm{GL}(V)$. A section of $\mathrm{Sym}^j ({{\mathbb {E}}}_2) \otimes \det ({{\mathbb {E}}}_2)^k$ over ${{\mathcal {A}}}_2$ is called a modular form of degree 2 and weight (j, k). The Koecher principle also applies to these modular forms: sections of ${{\mathbb {E}}}_2^{\rho }$ over ${{\mathcal {A}}}_2$ extend over ${\tilde{{\mathcal {A}}}}_2$, see [7, Prop. 1.5, p. 140]. We write

$$\begin{aligned} M_{j,k}(\Gamma _2)=H^0({\tilde{{\mathcal {A}}}}_2\otimes {{\mathbb {F}}}_3, \mathrm{Sym}^j({{\mathbb {E}}}_2) \otimes \det ({{\mathbb {E}}}_2)^k) \end{aligned}$$

and we consider the ${{\mathcal {R}}}_2({{\mathbb {F}}}_3)$-module

$$\begin{aligned} M= \oplus _{j,k} M_{j,k}(\Gamma _2)\, . \end{aligned}$$

It is even a ring. The map (2.3) 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 $\mathrm{GL}(V)$ on $V \oplus \mathrm{Sym}^6(V)$. Alternatively, covariants can be obtained by taking an equivariant embedding of an irreducible $\mathrm{GL}(V)$-representation $U \rightarrow \mathrm{Sym}^d(\mathrm{Sym}^6(V))$, or equivalently, an equivariant map

$$\begin{aligned} \varphi : {{\mathbb {F}}}_3 \rightarrow \mathrm{Sym}^d(\mathrm{Sym}^6(V))\otimes U^{\vee } \end{aligned}$$

and then $\Phi =\varphi (1)$ is a covariant. If U is an irreducible representation of highest weight $(w_1,w_2)$ then one may view $\Phi$ as a homogeneous form in $a_0,\ldots ,a_6$ of degree d and in $x_1, x_2$ of degree $w_1-w_2$, see [2, 9, 19]. For example, taking $U=\mathrm{Sym}^6(V)$ and $d=1$ yields the covariant $\Phi =f$, the universal binary sextic. Covariants form a ring ${{\mathcal {C}}}$ that was much studied in the 19th and early 20th century. Grace and Young determined generators of this ring in [9].

The maps ${{\mathcal {R}}}_2({{\mathbb {F}}}_3) \rightarrow I \rightarrow {{\mathcal {R}}}_2({{\mathbb {F}}}_3)_{\chi _{10}}$ now extend to

$$\begin{aligned} M {\buildrel \mu \over \longrightarrow }\, {{\mathcal {C}}} {\buildrel \nu \over \longrightarrow } M_{\chi _{10}}\, , \end{aligned}$$

where $M_{\chi _{10}}$ is obtained from M by admitting powers of $\chi _{10}$ as denominators. We have $\nu \circ \mu = \mathrm{id}_M$.

The image under $\nu$ of the covariant f, the universal binary sextic, is a rational modular form $\chi _{6,-2}$, that is, a rational section of $\mathrm{Sym}^6({{\mathbb {E}}}_2) \otimes {\det }({{\mathbb {E}}}_2)^{-2}$ that is regular after multiplication by an appropriate power of $\chi _{10}$. The power $-2$ comes from the twisting used in the description of the stack quotient $[\mathcal {X}^0/\mathrm{GL}(V)]$, where $\mathcal {X}^0 \subset \mathrm{Sym}^6(V) \otimes \det (V)^{-2}$, see [3, Section 3, p. 3].

This construction was given in [2] in characteristic zero and yields a meromorphic modular form, here denoted $\varphi _{6,-2}$, that becomes holomorphic after multiplication by $\chi _{10}$. The reduction of the characteristic zero rational modular form $\varphi _{6,-2}$ yields a rational modular form in characteristic 3. This implies that $\chi _{6,-2}$ becomes regular after multiplication by $\chi _{10}$. We can write the form $\chi _{6,-2}$ locally on ${{\mathcal {A}}}_2\otimes {{\mathbb {F}}}_3$ symbolically as

$$\begin{aligned} \chi _{6,-2}= \sum _{i=0}^6 \alpha _i X_1^{6-i} X_2^i\, , \end{aligned}$$

(2.4)

where the monomials $X_1^{6-i} X_2^i$ are dummies to indicate the coordinates in the fibres of $\mathrm{Sym}^6({{\mathbb {E}}}_2) \otimes {\det }({{\mathbb {E}}}_2)^{-2}$. Here we view $\alpha _i$ locally as a rational function on ${{\mathcal {A}}}_2 \otimes {{\mathbb {F}}}_3$. Using the local expression (2.4) one can give the image $\nu (T)$ of an invariant $T=T(a_0,\ldots ,a_6)$ locally by $T(\alpha _0,\ldots ,\alpha _6)$.

We note that interchanging $X_1$ and $X_2$ induces an involution replacing $\alpha _i$ by $\alpha _{6-i}$.

Comparing with the characteristic 0 case and using semi-continuity we see that the orders of the rational functions $\alpha _i$ 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 $\alpha _i$ along $H_1$ in characteristic 3:

$$\begin{aligned} \mathrm{ord}_{H_{1}}(\alpha _0,\ldots ,\alpha _6)= (\ge 2, \ge 1, \ge 0, \ge -1, \ge 0, \ge 1, \ge 2)\, . \end{aligned}$$

(2.5)

Moreover, the symmetry that interchanges $x_1$ and $x_2$ implies that the orders of $\alpha _i$ and $\alpha _{6-i}$ along $H_1$ are equal. Another way to see the estimates for the orders is by developing $\chi _{6,8}= \chi _{6,-2} \chi _{10}$ along the locus ${{\mathcal {A}}}_{1,1} \otimes {{\mathbb {F}}}_3 \subset {{\mathcal {A}}}_2\otimes {{\mathbb {F}}}_3$. Since the pullback of the Hodge bundle ${{\mathbb {E}}}_2$ to ${{\mathcal {A}}}_1 \times {{\mathcal {A}}}_1$ via ${{\mathcal {A}}}_1^2 \rightarrow {{\mathcal {A}}}_{1,1} \subset {{\mathcal {A}}}_2$ is $\oplus _{i=0}^6 p_1^*({{\mathbb {E}}}_1)^i \otimes p_2^*({{\mathbb {E}}}_1)^{6-i}$ the restriction of $\alpha _i \chi _{10}$ lies in $S_{14-i}(\Gamma _1)\otimes S_{8+i}(\Gamma _1)$ and this is zero. The next Taylor term in the Taylor development along ${{\mathcal {A}}}_{1,1}$ lies in $S_{15-i}(\Gamma _1) \otimes S_{9+i}(\Gamma _1)$ and this is zero for $i\ne 3$.

The ring of invariants I for the action of $\mathrm{GL}(V)$ on $\mathrm{Sym}^6(V)$ in characteristic 3 is generated by invariants A, B, C, D and E of degree 2, 4, 6, 10 and 15, see e.g. [11] or [8]. The invariants A, B, C, D that we use here can be expressed in the reductions modulo 3 of the invariants $J_2,J_4,J_6$ et $J_{10}$ given in [15]: $A=-J_2 (\bmod 3)$, $B= -J_4 (\bmod 3)$, $C=-J_6-A^3 (\bmod 3)$, $D= J_{10} (\bmod 3)$. The invariant E can be found in [12, p. 848].

The invariant A has the form $A=a_1a_5-a_2a_4$. We know of the existence of a modular form $\psi _2$ of weight 2. Under the map $\mu$ it must map to a non-zero multiple of A. We fix $\psi _2$ by requiring $\mu (\psi _2)=A$. The restriction to $H_1$ of the Hasse invariant $\psi _2$ is a non-zero multiple of $\mathrm{Sym}^2(b_2)$, with $b_2$ the Hasse invariant for $g=1$, hence $\psi _2$ does not vanish identically on $H_{1}$.

By the inequalities (2.5) and the expression for A we see that $\mathrm{ord}_{{H}_{1}}(\alpha _2)=0= \mathrm{ord}_{{H}_{1}}(\alpha _4)$ and

$$\begin{aligned} \mathrm{ord}_{H_{1}}(\alpha _0,\ldots ,\alpha _6)= (\ge 2, \ge 1, 0, \ge -1, 0, \ge 1, \ge 2)\, . \end{aligned}$$

In degree 4 we find another invariant B, not a multiple of $A^2$:

$$\begin{aligned} \begin{aligned} B&=2\, a_0a_1a_5a_6+a_0a_2a_4a_6+2\, a_0a_2a_5^2+2\, a_0a_4^3+ 2\, a_1^2a_4a_6+2\, a_1a_2a_4a_5 \\&\quad +a_1a_3^2a_5+a_1a_3a_4^2+2\,a_2^3a_6+a_2^2a_3a_5+a_2^2\, a_4^2+2\, a_2a_3^2a_4. \\\end{aligned} \end{aligned}$$

Since we know $\dim M_4(\Gamma _2)=1$ there cannot be a regular modular form in weight 4 that is not a multiple of $\psi _2^2$. This implies that $\mathrm{ord}_{H_1}(\alpha _3)<0$ and hence $\mathrm{ord}_{H_1}(\alpha _3)=-1$. Thus $B=(a_1a_5-a_2a_4) a_3^2+ (a_1a_4^2+a_2^2a_5) a_3+ \cdots$ defines a rational modular form $\chi _B=\nu (B)$ of weight 4 with order $-2$ along $H_1$. Since $\chi _{10}$ vanishes with multiplicity 2 along $H_1$ we thus find that

$$\begin{aligned} \chi _{14}:=\chi _B \chi _{10} \end{aligned}$$

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

$$\begin{aligned} C= 2 \, a_3^6 + A \, a_3^4+ 2(a_1a_4^2+a_2^2a_5)a_3^3+ \cdots \end{aligned}$$

and we see that $\chi _C=\nu (C)$ has order $-6$ along $H_1$. In degree 10 there is a new invariant

$$\begin{aligned} D=(a_1a_5)^3a_3^4+(a_0a_2^3a_5^3+a_1^3a_4^3a_6+ 2 a_1^3a_4^2a_5^2+2a_1^2a_2^2a_5^3)a_3^3+\cdots \end{aligned}$$

yielding a modular form that vanishes with multiplicity $\ge 2$ on ${H}_{1}$. Indeed, since $\alpha _1\alpha _5$ vanishes with multiplicity $\ge 2$ the first term $(\alpha _1\alpha _5)^3 \alpha _3^4$ vanishes with order $\ge 2$; the next terms also vanish with order $\ge 2$ as one easily checks. Therefore $\chi _D$ is regular and vanishes with multiplicity $\ge 2$. Since $\chi _D$ is not zero, it must be a multiple of $\chi _{10}$ and then vanishes on $H_1$ with multiplicity 2. This implies that the order of vanishing of $\alpha _1$ and $\alpha _5$ along $H_1$ is 1.

Corollary 1

We have $\mathrm{ord}_{H_1}(\alpha _0,\ldots ,\alpha _6)=(\ge 2, 1, 0, -1, 0, 1, \ge 2)$.

We fix $\chi _{10}$ by setting it equal to $\chi _D=\nu (D)$. This fixes $\chi _{14}$ too.

In a similar manner one checks that the rational modular form $\psi _S=\nu (S)$ with S equal to

$$\begin{aligned} S= B^3+A^3C-A^2B^2=(a_1a_4^2+a_2^2a_5)^3a_3^3+ \cdots \end{aligned}$$

is regular too. We put $\psi _{12}=\psi _S$. We thus find a 3-dimensional subspace of $M_{12}(\Gamma _2)$ generated by $\psi _2^6, \psi _2 \chi _{10}$ and $\psi _{12}$. From the fact that B and D are not divisible by A we see that $\chi _{14}$ does not lie in $\psi _2 M_{12}(\Gamma _2)$. Therefore $\dim M_{12}(\Gamma _2) < \dim M_{14}(\Gamma _2)$. Since we know by (2.1) that $\dim M_{14}(\Gamma _2)\le 4$ we conclude that $\dim M_{12}(\Gamma _2)=3$.

A further generator is

$$\begin{aligned} \chi _{36}=\nu (C D^3)=\chi _{C}\chi _{10}^3\, . \end{aligned}$$

Since the orders of $\chi _C$ and $\chi _{10}$ along $H_1$ are $-6$ and 2 the modular form $\chi _{36}$ is regular and does not vanish identically on $H_1$. The modular form $\chi _{36}$ is not contained in the subring generated by $\psi _2$,$\chi _{10}$, $\psi _{12}$ and $\chi _{14}$ as one sees by looking at the invariants. We have the identity

$$\begin{aligned} (B^3+A^3C-A^2B^2) D^3= B^3 D^3 + A^3C D^3 -A^2DB^2D^2 \end{aligned}$$

by which we can express $\psi _{12}\chi _{10}^3$ in the other generators:

$$\begin{aligned} \psi _{12}\chi _{10}^3=\chi _{14}^3+\psi _2^3\chi _{36}-\psi _2^2\chi _{10} \chi _{14}^2\, . \end{aligned}$$

(2.6)

Since A, B, C, D are generators of the ring of invariants and are algebraically independent the forms $\psi _2, \chi _{10}, \psi _{12}, \chi _{14}$ are algebraically independent. The form $\chi _{36}$ then satisfies the algebraic relation (2.6) and since there is no non-trivial relation of lower weight involving $\chi _{36}$ it implies that this relation generates the ideal of relations between the generators $\psi _2$, $\chi _{10}$, $\psi _{12}$, $\chi _{14}$ and $\chi _{36}$.

The forms $\psi _2$, $\chi _{10}$, $\psi _{12}$, $\chi _{14}$ and $\chi _{36}$ generate a subring $R^{\mathrm{ev}}$ of the ring ${{\mathcal {R}}}_2^{\mathrm{ev}}({{\mathbb {F}}}_3)$ with generating function

$$\begin{aligned} G=\frac{(1-t^{42})}{(1-t^2)(1-t^{10})(1-t^{12})(1-t^{14})(1-t^{36})}\, . \end{aligned}$$

and by the Riemann-Roch theorem we have $\dim M_{k}(\Gamma _2)=k^3/1080 + O(k^2)$ for even k. Note that

$$\begin{aligned} \frac{42}{2\cdot 10 \cdot 12 \cdot 14\cdot 36}= \frac{1}{2880}\, . \end{aligned}$$

On the other hand we have $c_1(L)^3=1/2880$, see [22, p. 74]. We can use the degree of $\mathrm{Proj}({{\mathcal {R}}}_2^{\mathrm{ev}}({{\mathbb {F}}}_3))$ to show that there cannot be more generators of ${{\mathcal {R}}}_2^{\mathrm{ev}}({{\mathbb {F}}}_3)$, but one can see this also in a more elementary way as follows.

Let $d(k)=\dim M_{k}(\Gamma _2)$ and $r(k)=\dim R_k$ where $R_k= R^{\mathrm{ev}} \cap M_k(\Gamma _2)$.

Proposition 1

We have $d(k)=r(k)$ for even $k\ge 0$.

Proof

We know that $d(k) \ge r(k)$ for even k and $d(k)=r(k)$ for even $0\le k \le 14$. Suppose by induction that $d(k) =r(k)$ for even $k \le m$. The exact sequence (2.1) gives the upper bound $d(k) \le r(k-10)+ c(k)(c(k)+1)/2$ for $k\le m+10$, where $c(k)=\dim M_k(\Gamma _1)= \lfloor k/12\rfloor +1$. Using the generating function G one sees that $r(k)-r(k-10)= c(k)(c(k)+1)/2$ for $k\not \equiv 0 \, (\bmod \, 12)$ and $k \not \equiv 2 \, (\bmod \, 12)$. Hence $d(k)=r(k)$ for even $k\le m+10$ with $k \not \equiv 0, 2 (\bmod 12)$. But we have

$$\begin{aligned} d(k+2)-d(k) \ge r(k+2)-r(k)\, , \end{aligned}$$

as we show in the next lemma. This proves $d(k)=r(k)$ for even $k \le m+10$. Therefore we conclude the proof by induction. $\square$

Lemma 1

We have $d(k+2)-d(k) \ge r(k+2)-r(k)$ for even $k\ge 0$.

Proof

We can write $R_{k+2}=\psi _2 R_k \oplus N_{k+2}$ with $N_{k+2}$ the subspace with basis the forms $\chi _{10}^a \psi _{12}^b \chi _{14}^c \chi _{36}^d$ with $a,b,c,d \ge 0$ and $c\le 2$ in view of the relation (2.6). Then we have $\dim N_{k+2}=r(k+2)-r(k)$. The inequality $d(k+2)-d(k)\ge \dim N_{k+2}$ follows from the fact that $N_{k+2}\cap \psi _2 M_k(\Gamma _2) =(0)$. To see this fact, suppose that $f \in M_{k}(\Gamma _2)$ such that $f \not \in R_k$ and $\psi _2 f \in R_{k+2}$. Then $\psi _2 f =P$ with P a sum of monomials $\chi _{10}^a \psi _{12}^b \chi _{14}^c \chi _{36}^d$ with $c \le 2$. Then $P=\nu (Q)$ with Q a polynomial in

$$\begin{aligned} D, B^3+A^3C-A^2B^2, BD, CD^3 \, . \end{aligned}$$

Since $P=\psi _2 f$ this polynomial must be divisible by A. But this implies that if $Q\ne 0$ then it must have at least one monomial with $c\ge 3$, but we excluded this. $\square$

The invariant E of degree 15 is of the form

$$\begin{aligned} E= (a_1a_4^2-a_2^2a_5)^3 a_3^6+\cdots \end{aligned}$$

and $\nu (E)$ has order $-3$ along $H_1$. Therefore

$$\begin{aligned} \chi _{35}:=\nu (E D^2) \end{aligned}$$

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 ${{\mathcal {A}}}_2^*\otimes {{\mathbb {F}}}_3$ is $60 \lambda _1$, see [21, Prop. 3.3, p. 217]. Therefore the divisor of $\chi _{35}$ is $H_1+H_4$ and since the closure of $H_1$ contains the 1-dimensional cusp $\chi _{35}$ is a cusp form. Then $\chi _{35}^2$ is of even weight, hence can be expressed as a polynomial in $\psi _2,\chi _{10},\psi _{12}, \chi _{14}$ and $\chi _{36}$. If $\psi$ is an odd weight modular form then it must vanish on $H_1$ and $H_4$, hence it will be divisible by $\chi _{35}$.

The relation between the space of binary sextics and the moduli space $\overline{\mathcal {M}}_2$ (see for example [3, Section 4]) implies that a modular form $\chi$ is a cusp form if and only if the invariant $\mu (\chi )$ is divisible by the discriminant D in I. From the form of the generators one easily sees that $\chi _{10},\chi _{14}, \chi _{36}$ and $\chi _{35}$ generate the ideal of cusp forms. This completes the proof.

Remark 1

One can use the knowledge of the dimensions of $M_k(\Gamma _2)$ to deduce non-vanishing of $H^1({\tilde{{\mathcal {A}}}}_2\otimes {{\mathbb {F}}}_3, L^k)$ for certain values of k. The short exact sequence of sheaves on ${\tilde{{\mathcal {A}}}}_2\otimes {{\mathbb {F}}}_3$

$$\begin{aligned} 0 \rightarrow L^k\otimes {{\mathcal {O}}}(-{\overline{V}}_1) \rightarrow L^k \rightarrow L^k_{|{\overline{V}}_1} \rightarrow 0 \end{aligned}$$

gives rise to a long exact sequence which can be identified with

$$\begin{aligned} 0 \rightarrow M_{k-2}(\Gamma _2) \rightarrow M_k(\Gamma _2) \rightarrow H^0({\overline{V}}_1,L^k) \rightarrow H^1({\tilde{{\mathcal {A}}}}_2\otimes {{\mathbb {F}}}_3,L^{k-2}) \rightarrow \cdots \end{aligned}$$

For example, if $\dim M_{k-2}(\Gamma _2)=\dim M_k(\Gamma _2)$ we get an injection $H^0({\overline{V}}_1,L^k) \rightarrow H^1({\tilde{{\mathcal {A}}}}_2\otimes {{\mathbb {F}}}_3,L^{k-2})$ and if $k\equiv 0 (\bmod \, 4)$ and $k\ge 0$ one can show that $H^0({\overline{V}}_1,L^k)\ne 0$ by showing that $H^0({\overline{V}}_1[2],L^k)^{{\mathfrak {S}}_6}\ne (0)$, the space of invariants under the symmetric group ${\mathfrak {S}}_6$ acting on $H^0({\overline{V}}_1[2],L^k)$ with $V_1[2]$ the 3-rank $\le 1$ locus in the level 2 moduli space ${\tilde{{\mathcal {A}}}}_2[2]$. Thus for example, $H^1({\tilde{{\mathcal {A}}}}_2\otimes {{\mathbb {F}}}_3,L^{14})\ne (0)$.