$\ell$-adic properties and congruences of $\ell$-regular partition functions

TL;DR

This paper investigates the $ ext{ell}$-adic properties of $ ext{ell}$-regular partition functions by connecting them to modular forms, establishing congruences, and revealing $ ext{ell}$-adic relations among partition values.

Contribution

It introduces a new modular form sequence encoding $ ext{ell}$-regular partitions and proves congruences and isomorphisms that reveal $ ext{ell}$-adic relations and uniform bounds.

Findings

Established congruences modulo powers of $ ext{ell}$ between partition-related forms and level 1 modular forms.

Proved isomorphisms between modules generated by the sequence and subspaces of level 1 cusp forms.

Derived uniform bounds on the ranks of modules related to $ ext{ell}$-regular partitions.

Abstract

We study $\ell$-regular partitions by defining a sequence of modular forms of level $\ell$ and quadratic character which encode their $\ell$-adic behavior. We show that this sequence is congruent modulo increasing powers of $\ell$ to level $1$ modular forms of increasing weights. We then prove that certain $\mathbb{Z}/\ell^m\mathbb{Z}$-modules generated by our sequence are isomorphic to certain subspaces of level $1$ cusp forms of weight independent of the power of $\ell$, leading to a uniform bound on the ranks of those modules and consequently to $\ell$-adic relations between $\ell$-regular partition values.