Generators of the ring of weakly holomorphic modular functions for $\Gamma_1(N)$

TL;DR

This paper explicitly constructs generators for the ring of weakly holomorphic modular functions for certain congruence subgroups, using modular units with infinite product expansions, for specific divisibility conditions on N.

Contribution

It provides explicit generators for the ring of weakly holomorphic modular functions for (N) with rational Fourier coefficients, expanding understanding of modular function rings.

Findings

Explicit generators for (N) rings are given for N divisible by 4,5,6,7, or 9.

Uses modular units with infinite product expansions to construct generators.

Enhances the algebraic understanding of modular functions for these subgroups.

Abstract

For a positive integer $N$ divisible by $4,5,6,7$ or $9$, let $\mathcal{O}_{1,N}(\mathbb{Q})$ be the ring of weakly holomorphic modular functions for the congruence subgroup $\Gamma_1(N)$ with rational Fourier coefficients. We present explicit generators of the ring $\mathcal{O}_{1,N}(\mathbb{Q})$ over $\mathbb{Q}$ by making use of modular units which have infinite product expansions.