Divisibility properties of coefficients of modular functions in genus zero levels

TL;DR

This paper establishes divisibility properties and duality relations for Fourier coefficients of modular forms at specific genus zero levels, providing new formulas for their generating functions.

Contribution

It introduces new divisibility results, proves Zagier duality for these coefficients, and derives a general formula for generating functions across all genus zero levels.

Findings

Fourier coefficients exhibit specific divisibility properties.

Fourier coefficients satisfy Zagier duality across all weights.

Provides a general formula for generating functions of canonical bases.

Abstract

We prove divisibility results for the Fourier coefficients of canonical basis elements for the spaces of weakly holomorphic modular forms of weight $0$ and levels $6, 10, 12, 18$ with poles only at the cusp at infinity. In addition, we show that these Fourier coefficients satisfy Zagier duality in all weights, and give a general formula for the generating functions of such canonical bases for all genus zero levels.