Generators of the ring of weakly holomorphic modular functions for ${\Gamma }_1(N)$

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.

16 pages