Classification of Congruences for Mock Theta Functions and Weakly Holomorphic Modular Forms

Abstract

Abstract. Let f (q) denote Ramanujan's mock theta functionIt is known that there are many linear congruences for the coefficients of f (q) and other mock theta functions. We prove that if the linear congruence a(mn + t) ≡ 0 (mod ℓ) holds for some prime ℓ ≥ 5, then ℓ|m and 24t−1 ℓ = −1 ℓ . We prove analogous results for the mock theta function ω(q) and for a large class of weakly holomorphic modular forms which includes η-quotients. This extends work of Radu [30] in which he proves a conjecture of Ahlgren and Ono for the partition function p(n).

Mock theta functions and weakly holomorphic modular forms modulo 2 and 3

“…Finally, we mention that Radu [29] used similar methods to study the behavior of the partition function modulo primes 5. Andersen [4] has obtained similar results in the present context.…”

Incongruences for modular forms and applications to partition functions

“…This work has inspired a number of results that describe where congruences can (or, rather, cannot) occur for Fourier coefficients of modular forms and mock theta functions [Dew11,And14,AK15,Cho16,L 17]. In this work, we give additional results regarding the non-existence of congruences.…”

“…Let B ∈ Z, k ∈ 1 2 Z, and N ∈ Z + . As in [AK15,And14], we set S(B, k, N, χ) := {η B (τ )F (τ ) : F (τ ) ∈ M ! k (Γ 0 (N), χ)} where M !…”

“…The following theorem can be obtained by applying the q-expansion principle of P. Deligne and M. Rapoport and is very much inspired by the work of Ahlgren, B. Kim, N. Andersen, and S. Löbrich [AK15,And14,L 17].…”

A family of mock theta functions of weights 1/2 and 3/2 and their congruence properties

“…To be more precise, applying certain quadratic twists to mock theta functions, one obtains weakly holomorphic modular forms due to the cancelation of the non-holomorphic parts. In [8] N. Andersen proved that any linear congruence for the coefficients of f (q) and ω(q) must come in this way.…”