Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo $\ell $

Abstract

Let l be a prime and λ, j ≥ 0 be an integer. Suppose that f(z) = Σ n a(n)q n is a weakly holomorphic modular form of weight λ + 1/2 and that a(0) ≢ 0 (mod l). We prove that if the coefficients of f(z) are not well-distributed modulo l j , then λ = 0 or 1 (mod l―1/2). This implies that, under the additional restriction α(0) ≢ 0 (mod l), the following conjecture of Balog, Darmon and Ono is true: if the coefficients of a modular form of weight λ+1/2 are almost (but not all) divisible by l, then either λ ≡ 0 (mod l―1/2) or λ ≡ 1 (mod l―1/2). We also prove that if λ ≢ 0 and 1 (mod l―1/2), then there does not exist an integer β, 0 < β < l, such that α(ln + β) ≡ 0 (mod l) for every nonnegative integer n. As an application, we study congruences for the values of the overpartition function.