WEAKLY HOLOMORPHIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT WITH NONVANISHING CONSTANT TERMS MODULO l

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