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

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 $a(mn+t)\equiv 0(\mathrm{mod}\ell )$ holds for some prime $\ell \ge 5$, then $\ell |m$ and $(\frac{24t-1}{\ell })\ne (\frac{-1}{\ell })$. 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 $p(n)$.