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 2πiz , and suppose thatis a weakly holomorphic modular form of even weight k ≤ 2. We answer a question of Ono by showing that if p ≥ 5 is prime and 2 − k = r(p − 1) + 2p t for some r ≥ 0 and t > 0, then a f (p t ) ≡ 0 (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 Theorem 2.5 from [3].