Determination of the Fricke families

TL;DR

This paper explicitly constructs generators for the ring of weakly holomorphic modular functions for certain congruence subgroups and classifies all associated Fricke families at levels divisible by 4.

Contribution

It provides explicit generators for the ring ^1_N(5Q) and classifies all Fricke families for levels divisible by 4.

Findings

Explicit generators of ^1_N(5Q) are obtained.

Complete classification of Fricke families at these levels.

The structure of the modular function ring is clarified.

Abstract

For a positive integer $N$ divisible by $4$, 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 $\mathbb{Q}$, from which we are able to classify all Fricke families of such level $N$.