A Note on Congruences for Weakly Holomorphic Modular Forms

2020/07/17 by Dembner, Spencer, Jain, Vanshika
#11F30 #11F33 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2007.09274

Abstract

Let O_L be the ring of integers of a number field L. Write q = e^{2 πi z}, and suppose that f(z) = ∑_{n ≫ - ∞}^∞ a_f(n) qⁿ ∈ M_k^!(SL₂(ℤ)) ∩ O_L[[q]] is 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) + 2 p^t for some r ≥ 0 and t > 0, then a_f(p^t) ≡ 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.

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