Infinite Product Exponents for Modular Forms

TL;DR

This paper analyzes the coefficients of infinite product expansions of modular forms, providing bounds on their growth and exploring cases where these bounds are tight, especially for low-genus modular curves.

Contribution

It offers new bounds on the coefficients of infinite product expansions of modular forms, extending previous descriptions and identifying cases of tight bounds for genus 0 or 1 curves.

Findings

Exponential upper bounds on the growth of coefficients

Lower bounds established for genus 0 or 1 cases

Extension of Choi's description to bounds on coefficients

Abstract

Recently, D. Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over $\Gamma_0(N)$. Using this result, we provide some bounds on these infinite product coefficients for holomorphic modular forms. We give an exponential upper bound for the growth of these coefficients. We show that this bound is also a lower bound in the case that the genus of the associated modular curve $X_0(N)$ is $0$ or $1$.