The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(3)$ with character $\psi_3$ (awaiting review)

The character $\psi_3:\Gamma_0(3)\to\{\pm1\}$ is defined by $$\psi_3\left(\begin{pmatrix}A&B\\C&D\end{pmatrix}\right) =\left(\frac{-3}{\det D}\right).$$ By a result of Tomoyoshi Ibukiyama [MR:1082834, 10.1142/S0129167X9100003X], the module $\oplus_k M_{k}(\Gamma_0(3),\psi_3)$ of Siegel modular forms of degree 2 with respect to the group $\Gamma_0(3)$ and the character $\psi_3$ is generated by five functions that can be specified in terms of theta series.

Let $$ A_2\begin{pmatrix}2&1\\1&2\end{pmatrix},\qquad E_6\begin{pmatrix} 2&-1&0&0&0&0\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&0&-1\\ 0&0&-1&2&-1&0\\ 0&0&0&-1&2&0\\ 0&0&-1&0&0&2 \end{pmatrix},\qquad E_6^* = 3E6^{-1},\qquad S\begin{pmatrix} 1&0&3/2&0\\ 0&1&0&3/2\\ 3/2&0&3&0\\ 0&3/2&0&3 \end{pmatrix}, $$ and let $P:{\Bbb C}^4\times{\Bbb C}^4\to{\Bbb C}$ be a pluriharmonic polynomial defined by $$ P(x,y)=(x_1y_3-x_3y_1)+x(_2y_4-x_4y_2)^2-(x_1y_4-x_4y_1 + x_3y_2-x_2y_3 + x_1y_2-x_2y_1 )^2. $$

The five generators are

• $\alpha_1$, a form of weight 1 in $S_1(\Gamma_0(3),\psi_3)$, with formula $\alpha_1 = \theta_{A_2}.$
• $\beta_3$, a form of weight 3 in $S_3(\Gamma_0(3),\psi_3)$, with formula $\beta_3 = \theta_{E_6}-10 \theta_{A_2}^3 + 9 \theta_{E_6^*}.$
• $\delta_3$, a form of weight 3 in $S_3(\Gamma_0(3),\psi_3)$, with formula $\delta_3 = \theta_{E_6}- 9 \theta_{E_6^*}.$
• $\gamma_4$, a form of weight 4 in $S_4(\Gamma_0(3))$, with formula $\gamma_4 = \theta_{S,P}.$
• $\chi_{14}$, a cusp form of weight 14 in $S_{14}(\Gamma_0(3),\psi_3)$, with formula $$ \chi_{14} = \frac1{2^9 3^{10}} \cdot \frac1{(2\pi i)^3} \begin{vmatrix} \alpha_1&3\beta_3&4\gamma_4/2&3\delta_4\\ \frac{\partial\alpha_1}{\partial\tau}& \frac{\partial\beta_3}{\partial\tau}& \frac{\partial\gamma_4}{\partial\tau}& \frac{\partial\delta_3}{\partial\tau}\\ \frac{\partial\alpha_1}{\partial z}& \frac{\partial\beta_3}{\partial z}& \frac{\partial\gamma_4}{\partial z}& \frac{\partial\delta_3}{\partial z}\\ \frac{\partial\alpha_1}{\partial\omega}& \frac{\partial\beta_3}{\partial\omega}& \frac{\partial\gamma_4}{\partial\omega}& \frac{\partial\delta_3}{\partial\omega} \end{vmatrix} . $$

The generators $\alpha_1, \beta_3, \gamma_4, \delta_3$ are algebraically independent. Let $$B = {\Bbb C}[\alpha_1, \beta_3, \gamma_4, \delta_3],\qquad C = {\Bbb C}[\alpha_1^2, \beta_3^2, \gamma_4, \delta_3^2].$$ We then have $$\oplus_k M_k(\Gamma_0(3),\psi_3) = B^{\rm{odd}} \oplus B^{\rm{even}}\chi_{14}.$$