Graded rings of paramodular forms of levels 5 and 7

Abstract

We compute generators and relations for the graded rings of paramodular forms of degree two and levels 5 and 7. The generators are expressed as quotients of Gritsenko lifts and Borcherds products. The computation is made possible by a characterization of modular forms on the Humbert surfaces of discriminant 4 that arise from paramodular forms by restriction.

Introduction

Paramodular forms (of degree 2, level $N\in \mathbb{N}$, and weight k) are holomorphic functions f on the Siegel upper half-space ${\mathbb{H}}_2$ which transform under the action of the paramodular group

$$K(N)=\{M\in {\mathrm{Sp}}_4(\mathbb{Q}):{\sigma }_N^{-1}M{\sigma }_N\in {\mathbb{Z}}^{4\times 4}\},{\sigma }_N=\mathrm{diag}(1,1,1,N)$$

by $f(M\cdot \tau )=\det {(c\tau +d)}^{k}f(\tau )$ for $M=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\in K(N)$ and $\tau \in {\mathbb{H}}_2$.

For a fixed level N, paramodular forms of all integral weights form a finitely generated graded ring $M_{⁎}(K(N))$ and a natural question is to ask for the structure of this ring. This yields information about the geometry of $X_{K(N)}=\overline{K(N)\backslash {\mathbb{H}}_2}$ (a moduli space for abelian surfaces with a polarization of type $(1,N)$) since the Baily-Borel isomorphism identifies $X_{K(N)}$ with the projective cone $\mathrm{Proj}{M}_{⁎}(K(N))$. Unfortunately these rings are difficult to compute. Besides Igusa's celebrated result for $N=1$ [21], the ring structure is only understood for levels $N=2$ (by Ibukiyama and Onodera [20]), $N=3$ (by Dern [8]) and $N=4$ (the group $K(4)$ is conjugate to a congruence subgroup of ${\mathrm{Sp}}_4(\mathbb{Z})$ for which this is implicit in the work of Igusa [21]). Substantial progress in levels $N=5,7$ was made by Marschner [23] and Gehre [13] respectively but the problem has remained open for all levels $N\ge 5$.

A general approach to these problems is to use pullback maps to lower-dimensional modular varieties and the existence of modular forms with special divisors. The levels $N=1,2,3,4$ admit a paramodular form which vanishes only on the $K(N)$-orbit of the diagonal (by [14]). Any other paramodular form f can be evaluated along the diagonal through the Witt operator, which we denote

$$P_1:M_{⁎}(K(N))⟶M_{⁎}({\mathrm{SL}}_2(\mathbb{Z})\times {\mathrm{SL}}_2(\mathbb{Z})),P_{1}f({\tau }_1,{\tau }_2)=f(\left(\begin{array}{cc}\hfill {\tau }_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\tau }_2/N\hfill \end{array}\right)).$$

One constructs a family of paramodular forms whose images under $P_1$ generate the ring of modular forms for ${\mathrm{SL}}_2(\mathbb{Z})\times {\mathrm{SL}}_2(\mathbb{Z})$ with appropriate characters. Any paramodular form can then be reduced against this family to yield a form which vanishes on the diagonal and is therefore divisible by the distinguished form with a simple zero by the Koecher principle. In this way the graded ring can be computed by induction on the weight.

Unfortunately in higher levels $N\ge 5$ it is never possible to find a paramodular form which vanishes only along the diagonal (by Proposition 1.1 of [14]) so this argument fails. (In fact, allowing congruence subgroups hardly improves the situation; see the classification in [7]. Some related computations of graded rings were given by Aoki and Ibukiyama in [1].) One might instead try to reduce against paramodular forms whose divisor consists not only of the diagonal but also Humbert surfaces of larger discriminant $D>1$ (which correspond to polarized abelian surfaces with special endomorphisms; see e.g. the lecture notes [10] for an introduction), the diagonal being a Humbert surface of discriminant one. Such paramodular forms can be realized as Borcherds products. There are instances in the literature where this approach has succeeded (e.g. [9]). However the pullbacks (generalizations of the Witt operator) to Humbert surfaces other than the diagonal are more complicated to work with explicitly and are usually not surjective, with the image being rather difficult to determine in general.

In this note we take a closer look at the pullback $P_4$ to the Humbert surfaces of discriminant four for odd prime levels N. In particular we list 5 candidate modular forms which one might expect to generate the image of symmetric paramodular forms under $P_4$. They do generate it in levels $N=5,7$ and this reduces the computation of the graded ring $M_{⁎}(K(N))$ to a logical puzzle of constructing paramodular forms which vanish to varying orders along certain Humbert surfaces. (A similar argument is outlined by Marschner and Gehre in [23] and [13] respectively, although we do not follow their suggestion to reduce along the Humbert surface of discriminant 9. Instead we use Humbert surfaces of discriminants $1,4,5$ and 8.)

We can prove the following theorems. Let ${\mathcal{E}}_k$ be the paramodular Eisenstein series of weight k.

Theorem 1

In level $N=5$, there are Borcherds products $b_5,b_8,b_{12},b_{14}$, Gritsenko lifts $g_6,g_7,g_8,g_{10}$, and holomorphic quotient expressions $h_9,h_{10},h_{11},h_{12},h_{16}$ in them such that the graded ring $M_{⁎}(K(5))$ is minimally presented by the generators

$${\mathcal{E}}_4,b_5,{\mathcal{E}}_6,g_6,g_7,g_8,b_8,h_9,g_{10},h_{10},h_{11},b_{12},h_{12},b_{14},h_{16}$$

of weights $4,5,6,6,7,8,8,9,10,10,11,12,12,14,16$ and by 59 relations in weights 13 through 32.

Theorem 2

In level $N=7$, there are Borcherds products $b_4,b_6,b_7,b_9,b_{10},b_{12}^{sym},b_{12}^{anti},b_{13}$, Gritsenko lifts $g_5,g_6,g_7,g_8,g_{10}$, and holomorphic quotient expressions $h_8,h_9,h_{11},h_{14},h_{15},h_{16}$ in them such that the graded ring $M_{⁎}(K(7))$ is minimally presented by the generators

$${\mathcal{E}}_4,b_4,g_5,{\mathcal{E}}_6,b_6,g_6,b_7,g_7,g_8,h_8,b_9,h_9,b_{10},g_{10},h_{11},b_{12}^{sym},b_{12}^{anti},b_{13},h_{14},h_{15},h_{16}$$

of weights $4,4,5,6,6,6,7,7,8,8,9,9,10,10,11,12,12,13,14,15,16$ and by 144 relations in weights 10 through 32.

The definitions of the forms $b_i,g_i,h_i$ are given in sections 5 and 6 below. The relations are listed in the ancillary files on arXiv. Fourier coefficients of the generators are available on the author's university webpage.