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The ring of modular forms for $\Gamma_1=\text{SL}(2,\mathbf{Z})$ $$M(\Gamma_1)\ =\ \bigoplus_{k\ge 0} M(\Gamma_1)_k\ =\ k[E_4,E_6]$$ is a free ring, with a generator in weight $4$ and weight $6$. (The reason is that $\mathfrak{H}/\Gamma_1=\mathbf{P}(4,6)$ is a weighted projective space, and the above is $\bigoplus_{k\ge 0}H^0(\mathbf{P}(4,6),\mathcal{O}(k))$). The generators are Eisenstein series.

Question: what is the ring of modular forms of level $2$ $$M(\Gamma_2)\ =\ \bigoplus_{k\ge 0} M(\Gamma_2)_k\ ?$$ Since $\mathfrak{H}/\Gamma_2=\mathbf{P}(2,2)$ I believe it should be free on two generators of weight $2$, but I don't know what they are explicitly as functions on $\mathfrak{H}$.

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Yes, the ring of holomorphic modular forms on $\Gamma(2) = \Gamma_2 \triangleleft \mathrm{SL}_2(\mathbb{Z})$ is equal to $\mathbb{C}[X,Y]$ where $X = \Theta_2^4$ and $Y= \Theta_3^4$ and where

$$ \Theta_2(z) = \sum_{n \in \mathbb{Z}}{e^{\pi i (n+1/2)^2 z}}, \quad \Theta_3(z) = \sum_{n \in \mathbb{Z}}{e^{\pi i n^2 z}}, \qquad \Theta_4(z) = \sum_{n \in \mathbb{Z}}{(-1)^ne^{\pi i n^2 z}} $$

are the classical theta constants (sometimes also denoted $\Theta_2 =\theta_{10}, \Theta_3 = \theta_{00}, \Theta_4 =\theta_{01}$)

Thus, for example $$ M_2(\Gamma_2)= \mathbb{C} \Theta_2^4 \oplus \mathbb{C} \Theta_3^4, \quad M_4(\Gamma_2) = \mathbb{C} \Theta_2^8 \oplus \mathbb{C} \Theta_2^4\Theta_3^4 \oplus \mathbb{C}\Theta_3^8. $$

You can prove this description using the following valency formula (which is in some sense simpler than the one for level 1 forms and also be proved via contour integration, as is done (in level 1) in Serre's course of arithmetic, for example)

$$ \nu_{[1]}(f) + \nu_{[0]}(f) + \nu_{[\infty]}(f) + \sum_{[p] \in \Gamma_2 \backslash \mathbb{H}}{\nu_{[p]}(f)}= \frac{k}{2} $$

valid for any nonzero meromorphic modular form $f$ of even integral weight $k$. It implies first that none of the theta constants $\Theta_2, \Theta_3, \Theta_4$ has a zero on the upper half plane. Another input is that multiplication by $\Theta_3^4$ defines an isomorphism from $M_k(\Gamma_2)$ onto $S_{k+2}^{[1]}(\Gamma_2) \subset M_{k+2}(\Gamma_2)$ where $S_{k+2}^{[1]}(\Gamma_2)$ denotes the subspace of forms vanishing at the cusp 1. (This can be seen as an analogue of multiplication by $\Delta \in S_{12}(\Gamma_1)$, shifting the weight by 12)

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  • $\begingroup$ Thank you! There's a slight typo- $Y=\Theta_3^4$, and theta. Also, is $\Theta_4^4$ not a level 2 modular form, or is it just expressible in terms of $\Theta_2^4$ and $\Theta_3^4$ (in which case, what is the expression)? $\endgroup$
    – Pulcinella
    Dec 16, 2020 at 17:14
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    $\begingroup$ Thanks, I fixed some of the typos. $\Theta_4^4 = \Theta_3^4- \Theta_2^4$ is a modular form of level 2 and weight 2. The latter identity is also attributed to Jacobi. $\endgroup$
    – m.s
    Dec 16, 2020 at 17:38

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