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

Abstract:

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 \[ \lambda =0 \text { or } 1 \pmod {\frac {\ell -1}{2}}.\] 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 $\lambda \not \equiv 0 \text { and } 1 \pmod {\frac {\ell -1}{2}},$ 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.