Congruences like Atkin's for the partition function

Scott Ahlgren, Patrick B. Allen, Shiang Tang

Published 2021-12-17, updated 2022-07-18Version 2

Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form $p( Q^3 \ell n+\beta)\equiv0\pmod\ell$ where $\ell$ and $Q$ are prime and $5\leq \ell\leq 31$; these lie in two natural families distinguished by the square class of $1-24\beta\pmod\ell$. In recent decades much work has been done to understand congruences of the form $p(Q^m\ell n+\beta)\equiv 0\pmod\ell$. It is now known that there are many such congruences when $m\geq 4$, that such congruences are scarce (if they exist at all) when $m=1, 2$, and that for $m=0$ such congruences exist only when $\ell=5, 7, 11$. For congruences like Atkin's (when $m=3$), more examples have been found for $5\leq \ell\leq 31$ but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime $\ell\geq 5$, there are infinitely many congruences like Atkin's in the first natural family which he discovered and that for at least $17/24$ of the primes $\ell$ there are infinitely many congruences in the second family.