# Congruences for a mock modular form on $\operatorname{SL}_2(\mathbb{Z})$ and the smallest parts function

## TL;DR

This paper investigates the congruence properties of a mock modular form related to the smallest parts function, demonstrating the rarity of such congruences modulo primes and reestablishing known results.

## Contribution

It introduces a new approach using Zagier's mock modular forms to analyze congruences of the smallest parts function and proves their infrequency.

## Key findings

- Congruences modulo primes  are rare for the smallest parts function.
- Reproves Garvan's theorem on the function's properties modulo .
- Establishes a connection between mock modular forms and partition congruences.

## Abstract

Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight $3/2$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo primes $\ell\geq 5$. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo $\ell$. As another application, we show that congruences modulo $\ell$ for the smallest parts function are rare in a precise sense.