Generators of graded rings of modular forms

Abstract

We study graded rings of modular forms over congruence subgroups, with coefficients in a subring $A$ of $\mathbb{C}$, and specifically the highest weight needed to generate these rings as $A$-algebras. In particular, we determine upper bounds, independent of $N$, for the highest needed weight that generates the $\mathbb{C}$-algebras of modular forms over $\Gamma (N)$, ${\Gamma }_1(N)$ and ${\Gamma }_0(N)$ with some conditions on $N$. For $N\ge 5$, we prove that the $\mathbb{Z}[1/N]$-algebra of modular forms over ${\Gamma }_1(N)$ with coefficients in $\mathbb{Z}[1/N]$ is generated in weight at most 3. We give an algorithm that computes the generators, and supply some computations that allow us to state two conjectures concerning the situation over ${\Gamma }_0(N)$.