TITLE: Generators of the graded ring of modular forms
QUESTION [19 upvotes]: Let $\Gamma$ be a finite-index subgroup of $\operatorname{SL}_2(\mathbb{Z})$. I've seen it stated (in a comment in the code of a computer program) that the graded ring
$$ M(\Gamma, \mathbb{C}) = \bigoplus_{k \ge 0} M_k(\Gamma, \mathbb{C}),$$
where $M_k(\Gamma, \mathbb{C})$ is the space of modular forms of weight $k$ and level $\Gamma$, 
is always generated as a $\mathbb{C}$-algebra by forms of weight $\le 12$.
Why is this true? Moreover, can one improve on the bound of 12? (For the subgroups $\Gamma_0(N)$, weight $\le 6$ always seems to be sufficient.)

REPLY [3 votes]: In the special case of the principal congruence subgroup $\Gamma=\Gamma(N)$ with $N \geq 3$, Khuri-Makdisi has proved that the graded algebra of modular forms on $\Gamma$ is generated by the modular forms of weight 1. In fact, the algebra generated by the Eisenstein series of weight 1 contains all modular forms in weight $\geq 2$, so misses only the cusp forms of weight 1. This is proved in his article Moduli interpretation of Eisenstein series.
In the case $N=2$, then $M_*(\Gamma(2))$ is freely generated by the weight 2 Eisenstein series $e_1,e_2$ obtained by putting the universal elliptic curve in the form $y^2=(x-e_1)(x-e_2)(x-e_3)$.