ℓ-adic properties of smallest parts functions

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 ℓ-adic properties of certain modular forms and mock modular forms of weight 3/2 with respect to the Hecke operators T(ℓ2m).