A note on congruences for weakly holomorphic modular forms

Abstract

Let O L O_L be the ring of integers of a number field L L . Write q = e 2 π i z q = e^{2 \pi i z} , and suppose that f ( z ) = ∑ n ≫ − ∞ a f ( n ) q n ∈ M k ! ( SL 2 ⁡ ( Z ) ) ∩ O L [ [ q ] ] \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 ≤ 2 k \leq 2 . We answer a question of Ono by showing that if p ≥ 5 p \geq 5 is prime and 2 − k = r ( p − 1 ) + 2 p t 2-k = r(p-1) + 2 p^t for some r ≥ 0 r \geq 0 and t > 0 t > 0 , then a f ( p t ) ≡ 0 ( mod p ) a_f(p^t) \equiv 0 \pmod p . For p = 2 , 3 , p = 2,3, we show the same result, under the condition that 2 − k − 2 p t 2 - k - 2 p^t is even and at least 4 4 . This represents the “missing case” of Theorem 2.5 from [Proc. Amer. Math. Soc. 144 (2016), pp. 4591–4597].