l-adic properties of partition functions

TL;DR

This paper extends the modular forms approach to analyze powers of the partition function and the spt-function, providing a unified framework for understanding their congruences modulo powers of primes.

Contribution

It generalizes previous work to include powers of the partition function and spt-function, establishing a modular forms-based framework for their congruences.

Findings

Explains exceptional congruences of $p_r$ observed by Boylan.

Provides new congruences for spt modulo 5, 7, and 13.

Shows generating functions lie in a small space of modular form reductions.

Abstract

Folsom, Kent, and Ono used the theory of modular forms modulo $\ell$ to establish remarkable ``self-similarity'' properties of the partition function and give an overarching explanation of many partition congruences. We generalize their work to analyze powers $p_r$ of the partition function as well as Andrews's spt-function. By showing that certain generating functions reside in a small space made up of reductions of modular forms, we set up a general framework for congruences for $p_r$ and spt on arithmetic progressions of the form $\ell^mn+\delta$ modulo powers of $\ell$. Our work gives a conceptual explanation of the exceptional congruences of $p_r$ observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.