Classification of congruences for mock theta functions and weakly holomorphic modular forms

TL;DR

This paper classifies linear congruences for coefficients of Ramanujan's mock theta functions and related weakly holomorphic modular forms, extending previous results on partition functions and modular forms.

Contribution

It establishes conditions under which linear congruences hold for mock theta coefficients and extends these results to a broad class of modular forms, including eta-quotients.

Findings

Linear congruences imply divisibility conditions on parameters.

Conditions involve Legendre symbols and divisibility of m.

Results generalize previous work on partition functions.

Abstract

Let $f(q)$ denote Ramanujan's mock theta function \[f(q) = \sum_{n=0}^{\infty} a(n) q^{n} := 1+\sum_{n=1}^{\infty} \frac{q^{n^{2}}}{(1+q)^{2}(1+q^{2})^{2}\cdots(1+q^{n})^{2}}.\] It is known that there are many linear congruences for the coefficients of $f(q)$ and other mock theta functions. We prove that if the linear congruence $a(mn+t) \equiv 0 \pmod{\ell}$ holds for some prime $\ell \geq 5$, then $\ell | m$ and $(\frac{24t-1}{\ell}) \neq (\frac{-1}{\ell})$. We prove analogous results for the mock theta function $\omega(q)$ and for a large class of weakly holomorphic modular forms which includes $\eta$-quotients. This extends work of Radu in which he proves a conjecture of Ahlgren and Ono for the partition function $p(n)$.