On spaces of modular forms spanned by eta-quotients
An eta-quotient of level $N$ is a modular form of the shape $f(z) = \prod_{δ| N} η(δz)^{r_δ}$. We study the problem of determining levels $N$ for which the graded ring of holomorphic modular forms for $Γ_{0}(N)$ 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 $J_{0}(2^{k})(\mathbb{Q})$.
math.NT↗