# Generators of level $2$ modular forms

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 $$\mathfrak{H}$$.

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)

• 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)?
– Meow
Dec 16 '20 at 17:14
• 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.
– m.s
Dec 16 '20 at 17:38