The ring of Siegel modular forms of degree 2 for $\Gamma(2)$ (awaiting review)

We define the following subgroups of the symplectic group $\Sp(4,\Z)$: $$ \Gamma[2]=\{ M\in \Sp(4,\Z): M\equiv 1_4 \bmod 2 \} \, , \qquad \Gamma_1[2]= \{ M\in \Sp(4,\Z): M\equiv \left(\begin{matrix} 1_2 & * \\ 0 & 1_2\end{matrix} \right) \bmod 2\} \, , \qquad \Gamma_0[2]= \{ M\in \Sp(4,\Z): M\equiv \left(\begin{matrix} * & * \\ 0 & * \end{matrix} \right) \bmod 2 \}. $$ The successive quotients can be identified as follows $$\Gamma_1[2]/ \Gamma[2] \simeq (\Z/2\Z)^3, \quad \Gamma_0[2]/ \Gamma[2] \simeq \Z/2\Z \times S_4, \quad \Gamma_0[2]/ \Gamma_1[2] \simeq S_3, \quad \Gamma/ \Gamma[2] \simeq S_6\, $$ where $S_n$ is the symmetric group on $n$ letters.

Recall that the irreducible representations of the symmetric group $S_n$ correspond bijectively to the partitions $(\lambda)$ of $n$; the representation corresponding to the partition $\lambda$ will be denoted $s[\lambda]$. Repetions of a digit in a partition will be abreviated by a power, e.g. $[a,a,b,\ldots]=[a^2,b,\ldots]$. For convenience, we give the following: $$ \text{Irreducible representations of }S_6\\ \begin{matrix}\lambda\colon & [6] & [5,1] & [4,2] & [4,1^2] & [3^2] & [3,2,1] & [3,1^3] & [2^3] & [2^2,1^2] & [2,1^4] & [1^6]\\ \dim\colon & 1 & 5 & 9 & 10 & 5 & 16 & 10 & 5 & 9 & 5 & 1\end{matrix} $$

$$ \text{Irreducible representations of }S_3\\ \begin{matrix}\lambda\colon & [3] & [2,1] & [1^3]\\ \dim\colon & 1 & 2 & 1\end{matrix} $$

It turns out that the vector space of vector-valued modular forms (resp. cusp forms) of weight $(k,j)$ on $\Gamma[2]$, denoted $M_{k,j}(\Gamma[2])$ (resp. $S_{k,j}(\Gamma[2])$), is a $S_6$-representation space. We have the following splitting: $$ M_{k,j}(\Gamma[2])=S_{k,j}(\Gamma[2])\oplus E_{k,j}(\Gamma[2]), $$ where $E_{k,j}(\Gamma[2])$ denotes the space of Eisenstein series of weight $(k,j)$. We give the isotypic decomposition of these spaces as follows. For fixed $(k,j)$, the isotypic decompositions - $ M_{k,j}(\Gamma[2])=m_{s[6]}s[6]+m_{s[5,1]}s[5,1]+\ldots +m_{s[1^6]}s[1^6]$, - $ E_{k,j}(\Gamma[2])=n_{s[6]}s[6]+n_{s[5,1]}s[5,1]+\ldots +n_{s[1^6]}s[1^6]$, - $ S_{k,j}(\Gamma[2])=l_{s[6]}s[6]+l_{s[5,1]}s[5,1]+\ldots +l_{s[1^6]}s[1^6]$, will be denoted by $$ \begin{matrix} \lambda\colon & [6] & \ldots & [1^6]\\ [M_{k,j}(\Gamma[2]),E_{k,j}(\Gamma[2]),S_{k,j}(\Gamma[2])] & [m_{s[6]},n_{s[6]},l_{s[6]}] & \ldots & [m_{s[1^6]},n_{s[1^6]},l_{s[1^6]}] \end{matrix} $$ Since $\Gamma_0[2]/ \Gamma_1[2] \simeq S_3$, the space of vector-valued modular forms on $\Gamma_1[2]$ is a $S_3$-representation space.

An $S_6$-representation $ m_{s[6]}s[6]+m_{s[5,1]}s[5,1]+\ldots +m_{s[1^6]}s[1^6]$ contributes a $S_3$-representation $$ (m_{s[6]}+m_{s[4,2]}+m_{s[2^3]}s[3]+(m_{s[5,1]}+m_{s[4,2]}+m_{s[3,2,1]})s[2,1]+(m_{s[4,1^2]}+m_{s[3^2]})s[1^3] $$ to $M_{k,j}(\Gamma_1[2])$ and a contribution $m_{s[6]}+m_{s[4,2]}+m_{s[2^3]}$ to the dimension of $M_{k,j}(\Gamma_0[2])$. We use the same notations as for $\Gamma[2]$ for the isotypic decomposition (under $S_3$) of the spaces $M_{k,j}(\Gamma_1[2])$.

The structures of the graded rings $R= \oplus_k M_{k,0}(\Gamma_[2])$ and $R^{\rm ev}=\oplus_k M_{2k,0}(\Gamma[2])$ of scalar-valued modular forms on $\Gamma[2]$ has been determined by Jun-Ichi Igusa [MR:0141613, 10.2307/2372812]. In order to describe these structures, we need to introduce theta series with characteristics.

We use the following (lexicographic) notation for the ten even theta characteristics $$ \begin{aligned} & n_1=\left[\begin{matrix} 0 & 0 \cr 0 & 0 \cr\end{matrix}\right], n_2=\left[\begin{matrix} 0 & 0 \cr 0 & 1 \cr\end{matrix}\right], n_3=\left[\begin{matrix} 0 & 0 \cr 1 & 0 \cr\end{matrix}\right], n_4=\left[\begin{matrix} 0 & 0 \cr 1 & 1 \cr\end{matrix}\right], n_5=\left[\begin{matrix} 0 & 1 \cr 0 & 0 \cr\end{matrix}\right], \\ & n_6=\left[\begin{matrix} 0 & 1 \cr 1 & 0 \cr\end{matrix}\right], n_7=\left[\begin{matrix} 1 & 0 \cr 0 & 0 \cr\end{matrix}\right], n_8=\left[\begin{matrix} 1 & 0 \cr 0 & 1 \cr\end{matrix}\right], n_9=\left[\begin{matrix} 1 & 1 \cr 0 & 0 \cr\end{matrix}\right], n_{10}=\left[\begin{matrix} 1 & 1 \cr 1 & 1 \cr\end{matrix}\right] \\ \end{aligned} $$ and define, for $\tau \in \mathbb{H}_2$ and $$\begin{bmatrix} \mu \\ \nu \\ \end{bmatrix} = \begin{bmatrix} \mu_1 & \mu_2 \\ \nu_1 & \nu_2 \\ \end{bmatrix} $$ with $\mu =(\mu_1,\mu_2)$ and $\nu =(\nu_1,\nu_2)$ in ${\Z}^2$, the standard theta constants with characteristics $$ \vartheta_{\begin{bmatrix}\mu \\ \nu \end{bmatrix}}(\tau)= \sum_{n=(n_1,n_2) \in \Z^2} e^{\pi i \left((n+\mu /2)\left(\tau(n+\mu/2)^t+\nu^t\right)\right)}. $$ We denote $\vartheta_{n_i}=\vartheta_i$ and $\vartheta_i^4=x_i$. The results of Jun-Ichi Igusa can be read as follows: $$ R^{\rm ev} \cong {\C}[x_1,\ldots,x_5]/(f) $$ with $f$ a homogeneous polynomial of degree $4$ in the $x_i$, the so-called Igusa quartic. The full ring $R$ is a degree $2$ extension $R^{\rm ev}[\chi_5]/(\chi_5^2+2^{14}\chi_{10})$ generated by the modular form $\chi_5$ of weight $5$ $$ \chi_5= \prod_{i=1}^{10} \vartheta_{i}\, . $$

Also see F. Cléry, G. van der Geer, S. Grushevsky, [arXiv:1306.6018] and J. Bergström, C. Faber, G. van der Geer [MR:2439544, 10.1093/imrn/rnn100].