Journal of the
Korean Mathematical Society JKMS

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 $\mathbb{Q}$ in terms of both Fricke functions and Siegel functions, from which we are able to classify all Fricke families of such level $N$.

Keywords: Fricke families, modular functions, modular units

MSC numbers: Primary 11F03; Secondary 11F11