A Note on Congruences for Weakly Holomorphic Modular Forms

Abstract

Let $O_L$ be the ring of integers of a number field $L$. Write $q=e^{2\pi iz}$, and suppose that $$f(z)=\sum _{n\gg -\infty }^{\infty }{a}_f(n)q^n\in M_k^{!}({\mathrm{SL}}_2(\mathbb{Z}))\cap O_L[[q]]$$ is a weakly holomorphic modular form of even weight $k\le 2$. We answer a question of Ono by showing that if $p\ge 5$ is prime and $2-k=r(p-1)+2p^t$ for some $r\ge 0$ and $t>0$, then $a_f(p^t)\equiv 0(\mathrm{mod}p)$. For $p=2,3,$ we show the same result, under the condition that $2-k-2p^t$ is even and at least $4$. This represents the "missing case" of a theorem proved by Jin, Ma, and Ono.

4 pages; to appear in Proceedings of the American Mathematical Society