论文标题
一致的注释较弱
A Note on Congruences for Weakly Holomorphic Modular Forms
论文作者
论文摘要
令$ o_l $为数字字段$ l $的整数环。写入$ q = e^{2πiz} $，并假设$$ f（z）= \ sum_ {n \ gg- \ gg - \ infty}^{\ infty} a_f（n）q^n \ in m_ {k} o_l [[q]] $$是一种弱的全态模块化形式，均为$ k \ leq 2 $。我们通过证明$ p \ geq 5 $为prime，而$ 2-k = r（p-1） + 2 p^t $对于某些$ r \ geq 0 $和$ t> 0 $，则回答一个Ono的问题，然后$ A_F（p^t）\ equiv 0 \ pmod p $。对于$ p = 2,3，$我们显示出相同的结果，条件是$ 2 -k -2 p^t $均匀且至少$ 4 $。这代表了Jin，MA和Ono证明的定理的“缺失案例”。
Let $O_L$ be the ring of integers of a number field $L$. Write $q = e^{2 πi z}$, and suppose that $$f(z) = \sum_{n \gg - \infty}^{\infty} a_f(n) q^n \in M_{k}^{!}(\operatorname{SL}_2(\mathbb{Z})) \cap O_L[[q]]$$ 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 a theorem proved by Jin, Ma, and Ono.
