On spaces of modular forms spanned by eta-quotients

Abstract

An eta-quotient of level $N$ is a modular form of the shape $f(z)={\prod }_{\delta |N}\eta (\delta z{)}^{r_{\delta }}$. We study the problem of determining levels $N$ for which the graded ring of holomorphic modular forms for ${\Gamma }_0(N)$ is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level $N$. In addition, we prove that if $f(z)$ is a holomorphic modular form that is non-vanishing on the upper half plane and has integer Fourier coefficients at infinity, then $f(z)$ is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of $J_0(2^k)(\mathbb{Q})$.

arXiv comment23 pages