Existence of $\eta$-Quotients For Squarefree Levels

TL;DR

This paper characterizes when eta-quotients can exist within certain modular form spaces, providing necessary and sufficient conditions for squarefree levels coprime to 6, advancing understanding of eta-quotients in modular forms.

Contribution

It establishes necessary and sufficient conditions for eta-quotients to exist in modular form spaces for a broad class of squarefree levels coprime to 6.

Findings

Necessary conditions for eta-quotients in modular form spaces are identified.

Sufficient conditions are proven for a large family of squarefree levels coprime to 6.

The results clarify the structure of eta-quotients in modular forms for specific levels.

Abstract

In this paper, we investigate which modular form spaces can contain $\eta$-quotients, functions of the form $f(\tau) = \prod_{\delta \mid N} \eta(\delta\tau)^{r_\delta}$. For $k\geq 2$ even and $N$ coprime to 6, we give necessary conditions for the space $M_k(\Gamma_1(N))$ to contain $\eta$-quotients. We then show that these conditions are sufficient as well for a large family of squarefree levels $N$ coprime to 6.