$p$-adic properties of coefficients of weakly holomorphic modular forms

TL;DR

This paper investigates the divisibility properties of Fourier coefficients of weakly holomorphic modular forms in certain weights, revealing frequent high divisibility by small primes 2, 3, and 5.

Contribution

It provides new insights into the prime divisibility patterns of Fourier coefficients in a canonical basis for specific weights of weakly holomorphic modular forms.

Findings

Coefficients are often highly divisible by 2, 3, and 5.

Divisibility patterns are analyzed across weights 4, 6, 8, 10, and 14.

The results suggest underlying arithmetic structures in modular form coefficients.

Abstract

We examine the Fourier coefficients of modular forms in a canonical basis for the spaces of weakly holomorphic modular forms of weights 4, 6, 8, 10, and 14, and show that these coefficients are often highly divisible by the primes 2, 3, and 5.