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

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 ${\mathrm{SL}}_2(\mathbb{Z})$ modulo primes $\ell \ge 5$. 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.

7 pages