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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中文翻译：

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关于弱全纯模形式的同余的注解

摘要：令$O_L$为数域$L$的整数环。写出 $q = e^{2 \pi iz}$，并假设 \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*} 是偶权重 $k \leq 2$ 的弱全纯模形式. 我们通过证明如果 $p \geq 5$ 是素数并且 $2-k = r(p-1) + 2 p^t$ 对于某些 $r \geq 0$ 和 $t > 0$ 来回答小野的问题，然后 $a_f(p^t) \equiv 0 \pmod p$。对于 $p = 2,3,$ 我们显示相同的结果，条件是 $2 - k - 2 p^t$ 是偶数且至少为 $4$。这代表了 [Proc. 2.5] 中定理 2.5 的“缺失情况”。阿米尔。数学。社会。144 (2016)，第 4591-4597 页]。

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