The partition function modulo prime powers

Abstract:

Let $\ell \geq 5$ be prime, let $m\geq 1$ be an integer, and let $p(n)$ denote the partition function. Folsom, Kent, and Ono recently proved that there exists a positive integer $b_{\ell }(m)$ of size roughly $m^2$ such that the module formed from the $\mathbb {Z}/\ell ^m\mathbb {Z}$-span of generating functions for $p\left (\frac {\ell ^bn + 1}{24}\right )$ with odd $b\geq b_{\ell }(m)$ has finite rank. The same result holds with “odd” $b$ replaced by “even” $b$. Furthermore, they proved an upper bound on the ranks of these modules. This upper bound is independent of $m$; it is $\left \lfloor \frac {\ell + 12}{24}\right \rfloor$.

In this paper, we prove, with a mild condition on $\ell$, that $b_{\ell }(m)\leq 2m - 1$. Our bound is sharp in all computed cases with $\ell \geq 29$. To deduce it, we prove structure theorems for the relevant $\mathbb {Z}/\ell ^m\mathbb {Z}$-modules of modular forms. This work sheds further light on a question of Mazur posed to Folsom, Kent, and Ono.