Two-divisibility of the coefficients of certain weakly holomorphic modular forms

TL;DR

This paper investigates the divisibility properties of Fourier coefficients of certain weakly holomorphic modular forms, establishing a connection with the Ramanujan tau-function and demonstrating frequent divisibility by 2.

Contribution

It introduces a canonical basis for specific weights of weakly holomorphic modular forms and links their coefficients to the Ramanujan tau-function, revealing divisibility patterns.

Findings

Fourier coefficients are often highly divisible by 2.

Established a relation between these coefficients and the Ramanujan tau-function.

Provided new insights into divisibility properties of modular form coefficients.

Abstract

We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan tau-function, and use this to prove that these Fourier coefficients are often highly divisible by 2.