CLASSIFICATION OF CONGRUENCES FOR MOCK THETA FUNCTIONS AND WEAKLY HOLOMORPHIC MODULAR FORMS

Let f(q) denote Ramanujan's mock theta function It is known that there are many linear congruences for the coefficients of f(q) and other mock theta functions. We prove that if the linear congruence a(mn+t) ≡ 0 (mod ℓ) holds for some prime ℓ ≥ 5, then ℓ|m and ((24t − 1)/ℓ) ≠ (−1/ℓ). We prove analogous results for the mock theta function ω(q) and for a large class of weakly holomorphic modular forms which includes η-quotients. This extends work of Radu, in which he proves a conjecture of Ahlgren and Ono for the partition function p(n).