$\ell$-adic properties of smallest parts functions

TL;DR

This paper establishes explicit congruences modulo prime powers for various smallest parts functions related to partitions, using $ ext{ell}$-adic properties of modular and mock modular forms of weight 3/2.

Contribution

It introduces new explicit congruences for smallest parts functions based on $ ext{ell}$-adic analysis of modular forms and mock modular forms of weight 3/2.

Findings

Proves congruences for partition-related smallest parts functions.

Uses $ ext{ell}$-adic properties of modular forms and mock modular forms.

Applies Hecke operators to establish these congruences.

Abstract

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on $\ell$-adic properties of certain modular forms and mock modular forms of weight $3/2$ with respect to the Hecke operators $T(\ell^{2m})$.