Weakly holomorphic modular forms in prime power levels of genus zero

TL;DR

This paper constructs explicit bases for weakly holomorphic modular forms at certain prime power levels, reveals divisibility properties of their Fourier coefficients, and extends known congruences and duality results to these levels.

Contribution

It provides explicit bases for weakly holomorphic modular forms at specific prime power levels and extends key properties like divisibility, duality, and congruences beyond level 1.

Findings

Fourier coefficients divisible by high powers of primes dividing the level

Basis elements satisfy Zagier duality

Extended congruence results to multiple levels

Abstract

Let $M_k^\sharp(N)$ be the space of weight $k$, level $N$ weakly holomorphic modular forms with poles only at the cusp at $\infty$. We explicitly construct a canonical basis for $M_k^\sharp(N)$ for $N\in\{8,9,16,25\}$, and show that many of the Fourier coefficients of the basis elements in $M_0^\sharp(N)$ are divisible by high powers of the prime dividing the level $N$. Additionally, we show that these basis elements satisfy a Zagier duality property, and extend Griffin's results on congruences in level 1 to levels 2, 3, 4, 5, 7, 8, 9, 16, and 25.