Congruences for a mock modular form on SL2(Z) and the smallest parts function

Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight 3/2 on SL2(Z) modulo primes ℓ≥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 ℓ. As another application, we show that congruences modulo ℓ for the smallest parts function are rare in a precise sense.