Congruences for Level $1$ cusp forms of half-integral weight

Abstract

Suppose that $\ell \ge 5$ is prime. For a positive integer $N$ with $4\mid N$, previous works studied properties of half-integral weight modular forms on ${\Gamma }_0(N)$ which are supported on finitely many square classes modulo $\ell$, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan $\Theta$-operator. Here, we study the analogous problem for modular forms of half-integral weight on ${\mathrm{SL}}_2(\mathbb{Z})$. Let $\eta$ be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on ${\mathrm{SL}}_2(\mathbb{Z})$ which is supported on finitely many square classes modulo $\ell$ can be written modulo $\ell$ in terms of ${\eta }^{\ell }$ and an iterated derivative of $\eta$.

Accepted to Proceedings of the AMS