Generators of graded rings of modular forms

TL;DR

This paper establishes uniform upper bounds on the weights needed to generate graded rings of modular forms over various congruence subgroups, provides algorithms for computing generators, and proposes conjectures for further cases.

Contribution

It determines bounds independent of level N for generators of modular form rings and introduces algorithms for explicit computation.

Findings

Bound of weight at most 3 for generators over (N) with coefficients in /N

Algorithm developed for computing generators of modular form rings

Conjectures proposed for the structure over (N)

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 $N \geq 5$, 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 $\Gamma_0(N)$.