Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo ℓ

Let ℓ be a prime and λ,j ≥ 0 be an integer. Suppose that f(z) = Σ_n a(n)qⁿ is a weakly holomorphic modular form of weight λ + ¹₂ 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 a(0) ≢ 0 (mod ℓ), the following conjecture of Balog, Darmon and Ono is true: if the coefficients of a modular form of weight λ+ ¹₂ 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 a(ℓn+ β) = 0 (mod ℓ) for every nonnegative integer n. As an application, we study congruences for the values of the overpartition function.