Congruences on Square-Classes for the Partition Function

TL;DR

This paper advances the understanding of Ramanujan-type congruences for the partition function by linking them to square-classes, introducing a new framework based on modular curves and group representations.

Contribution

It introduces a novel framework connecting partition congruences to square-classes using modular curve models and finite group representations, improving prior results.

Findings

Ramanujan-type congruences are governed by square-classes.

The framework applies to all weakly holomorphic modular forms.

First elucidation of Atkin-O'Brien's family of congruences.

Abstract

We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes. We thus elucidate for the first time an exemplary family of congruences found by Atkin-O'Brien. Our results are based on a novel framework that leverages available results on integral models of modular curves via representations of finite quotients of $\mathrm{SL}_2(\mathbb{Z})$ or $\mathrm{Mp}_1(\mathbb{Z})$. This framework applies to congruences of all weakly holomorphic modular forms.