A note on congruences for weakly holomorphic modular forms

Dembner, Spencer; Jain, Vanshika

doi:10.1090/proc/15501

A note on congruences for weakly holomorphic modular forms
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by Spencer Dembner and Vanshika Jain PDF

Proc. Amer. Math. Soc. 149 (2021), 3683-3686

Abstract:

Let $O_L$ be the ring of integers of a number field $L$. Write $q = e^{2 \pi i z}$, and suppose that \begin{equation*} f(z) = \sum _{n \gg - \infty } a_f(n) q^n \in M_{k}^{!}(\operatorname {SL}_2(\mathbb {Z})) \cap O_L[[q]] \end{equation*} is a weakly holomorphic modular form of even weight $k \leq 2$. We answer a question of Ono by showing that if $p \geq 5$ is prime and $2-k = r(p-1) + 2 p^t$ for some $r \geq 0$ and $t > 0$, then $a_f(p^t) \equiv 0 \pmod p$. For $p = 2,3,$ we show the same result, under the condition that $2 - k - 2 p^t$ is even and at least $4$. This represents the “missing case” of Theorem 2.5 from [Proc. Amer. Math. Soc. 144 (2016), pp. 4591–4597].

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