The ring of Siegel modular forms of degree 2 for $\Gamma_0(2)$ (awaiting review)

By a result of Tomoyoshi Ibukiyama [MR:1082834, 10.1142/S0129167X9100003X], the ring $M_{*}(\Gamma_0(2))$ of Siegel modular forms of degree 2 with respect to the group $\Gamma_0(2)$ is generated by the following five generators, which are specified in terms of theta constants.

• $X$, a form of weight 2, with formula $X = ((\theta_{0000})^4+(\theta_{0001})^4+(\theta_{0010})^4+(\theta_{0011})^4)/4.$
• $Y$, a form of weight 4, with formula $Y = (\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011})^2.$
• $Z$, a form of weight 4, with formula $Z = ((\theta_{0100})^4-(\theta_{0110})^4)^2)/16384.$
• $K$, a form of weight 6, with formula $K = (\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111})^2/4096.$
• $\chi_{19}$, a cusp form of weight 19, with formula $\chi_{19} = \theta \theta'(8YZ-X^2T +YT +1024ZT+96T^2 - 8XK)/32,$ where $$\theta=\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011}\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111},\qquad \theta' = ((\theta_{1000})^{12}+(\theta_{1001})^{12}+(\theta_{1100})^{12}+(\theta_{1111})^{12})/1536,\qquad T = (\theta_{0100}\theta_{0110})^{4}/256.$$

The generators $X, Y, Z, K$ are algebraically independent. The ring of modular forms is

$$M(\Gamma_0(2)) = \C[X,Y,Z,K] + \chi_{19}\,\C[X,Y,Z,K].$$