The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(4)$ (awaiting review)

By results of Aoki and Ibukiyama [MR:2130626, 10.1142/S0129167X05002837], and of Hayashida and Ibukiyama [MR:1840071], the ring $M_{*}(\Gamma_0(4))$ of Siegel modular forms of degree 2 with respect to the group $\Gamma_0(4)$ is generated by the following functions, which are specified in terms of theta constants:

• $X$, a form of weight 2, with formula $X = ((\theta_{0000})^4+(\theta_{0001})^4+(\theta_{0010})^4+(\theta_{0011})^4)/4.$
• $X(2\Omega)$, a form of weight 2.
• $f_2(2\Omega)$, a form of weight 4, with formula $f_2 = (\theta_{0000})^4.$
• $K(2\Omega)$, a form of of weight 6, with formula $$K = \theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111})^2/4096.$$
• $f_{11}(2\Omega)$, a cusp form of weight 11, with formula $f_{11} = f_6\chi_5,$ where $$ \chi_5=\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011}\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111},\qquad f_6 = ((\theta_{0001})^{4}-(\theta_{0010})^{4}) ((\theta_{0001})^{4}-(\theta_{0011})^{4})((\theta_{0010})^{4}-(\theta_{0011})^{4}). $$

Note that we write $F(2\Omega)$ to mean "apply $F$ after doubling the input".

The generators $X, X(2\Omega),f_2(2\Omega), K(2\Omega)$ are algebraically independent. Let $B={\Bbb C}[X, X(2\Omega),f_2(2\Omega), K(2\Omega)]$. The ring of modular forms is $$ M(\Gamma_0(4)) = B + Y(2\Omega) B + f_{11}(2\Omega)(B + Y(2\Omega) B),$$ where $Y = (\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011})^2$.