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

Let $\ell$ be a prime and $\lambda,j\geq 0$ be an integer. Suppose that $f(z)=\sum_{n}a(n)q^n$ is a weakly holomorphic modular form of weight $\lambda+\frac{1}{2}$ and that $a(0)\not \equiv 0 \pmod{\ell}$. We prove that if the coefficients of $f(z)$ are not ``well-distributed'' modulo $\ell^j$, then

$$\lambda=0$$     or  $$1 \pmod{\frac{\ell-1}{2}}.$$

This implies that, under the additional restriction $a(0)\not \equiv 0 \pmod{\ell}$, the following conjecture of Balog, Darmon and Ono is true: if the coefficients of a modular form of weight $\lambda+\frac{1}{2}$ are almost (but not all) divisible by $\ell$, then either $\lambda\equiv 0\pmod{\frac{\ell-1}{2}}$ or $\lambda\equiv 1 \pmod{\frac{\ell-1}{2}}$. We also prove that if $\lambda \not \equiv 0$    and $1 \pmod{\frac{\ell-1}{2}},$ then there does not exist an integer $\beta$, $0\leq \beta <\ell$, such that $a(\ell n+ \beta)\equiv 0 \pmod{\ell}$ for every nonnegative integer $n$. As an application, we study congruences for the values of the overpartition function.