Effective Congruences for Mock Theta Functions

Let M(q)=\sum c(n) q^n be one of Ramanujan's mock theta functions. We establish the existence of infinitely many linear congruences of the form c(An+B) \equiv 0 (mod \ell^j), where A is a multiple of \ell and an auxiliary prime p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on prior works of Lichtenstein and Treneer.

The coefficients a f (n) of f (q) can be used to determine the number of partitions of n of even rank and of odd rank [6].
The function f (q) is one of Ramanujan's seventeen original mock theta functions, which are strange q-series that often have combinatorial interpretations (see [11] for a comprehensive survey of mock theta functions). These functions have been the source of much recent study. In [3,7,8,9,10,18], congruences for the coefficients of various mock theta functions are established. For example, in their investigation of strongly unimodal sequences, Bryson, Ono, Pitman, and Rhoades [8] prove the existence of congruences for the coefficients of Ramanujan's mock theta function In particular, they establish the congruence a Ψ (11 4 · 5n + 721) ≡ 0 (mod 5). (1.1) In [18], Waldherr shows that Ramanujan's mock theta function Congruences like the examples above have also been proven for other mock theta functions, such as Ramanujan's φ(q) function [7,9]. It is natural to ask if a general theory of such congruences exists. In this paper, we build on the approaches of these previous works to establish the existence of linear congruences for all of Ramanujan's mock theta functions.
If M(q) is one of Ramanujan's mock theta functions, then by work of Zwegers [20] there are relative prime integers δ and τ for which is the holomorphic part of a weight 1/2 harmonic weak Maass form (to be defined in Section 2). We obtain congruences for the coefficients of M(q) as in (1.1) by obtaining them for F (z).
Theorem 1. Let M(q) be one of Ramanujan's mock theta functions with F (z) as in (1.2), let N be the level of F , and let ℓ j be a prime power with (ℓ, N) = 1. Then there is a prime Q and infinitely many primes p such that, for some m, B ∈ N, we have b(p 4 ℓ m Qn + B) ≡ 0 (mod ℓ j ).
Furthermore, the smallest such p satisfies p ≤ C, where C is an effectively computable constant that depends on ℓ j , N, and other computable parameters.
Remark. Theorem 1 is a special case of Theorem 5, a more general result that applies to weight 1/2 harmonic Maass forms whose holomorphic parts have algebraic coefficients and whose nonholomorphic parts are period integrals of weight 3/2 unary theta series.

Nuts and Bolts
The proof of Theorem 5 utilizes several important concepts from the theory of modular forms and harmonic Maass forms. In this section we summarize those topics and results that will be key ingredients in the proof.

Harmonic Maass Forms.
Ramanujan's mock theta functions are essentially the holomorphic parts of certain weight 1/2 harmonic Maass forms. To begin, we define half integral weight harmonic weak Maass forms. Here "harmonic" refers to the fact that these functions vanish under the weight k hyperbolic Laplacian ∆ k , as y → +∞ for some ǫ > 0. Analogous conditions are required at all cusps. The term "weak" refers to the relaxed growth condition at the cusps described by (3). For convenience, we will refer to these harmonic weak Maass forms simply as harmonic Maass forms. We adopt the following notation: if χ is a Dirichlet character modulo N, ) denote the space of cusp forms (resp., holomorphic modular forms, weakly holomorphic modular forms, harmonic Maass forms) of weight k on Γ 0 (N) with Nebentypus χ. For to be the holomorphic part and If M(q) is one of Ramanujan's mock theta functions with F (z) as in (1.2), then by [20], f + = F is the holomorphic part of a weight 1/2 harmonic weak Maass form whose nonholomorphic part f − is a period integral of a weight 3/2 unary theta series. As a consequence, there exist integers δ 1 , . . . , δ h such that the coefficients a − (n) are supported on exponents of the form −δ i m 2 .
As stated in Section 1, Theorem 1 is a special case of Theorem 5, which applies to weight 1/2 harmonic Maass forms with algebraic coefficients whose nonholomorphic parts are period integrals of weight 3/2 unary theta series. Essentially, these congruences are obtained from the annihilation of a cusp form g(z), related to f (z), by the Hecke operators T (p 2 ). The cusp form g(z) is determined by a result of Treener [17]. Moreover, work of Lichtenstein [14] allows us to bound the first prime p such that T (p 2 ) annihilates g(z). The details of the construction of g(z) follow.

2.2.
Elements of the Proof. To prove Theorem 5, we first obtain a weakly holomorphic modular form f (z) by applying quadratic twists to annihilate the nonholomorphic part of f (z). If Q is an odd prime, define ψ Q := • Q and G :=

Remarks.
The definition of f ⊗ψ Q given in [12, III, Proposition 17] applies to modular forms, but this definition also makes sense for f ∈ H 2−k (N, χ) since the transformation z → z −λ/Q only affects the real part of z (the Γ-factor in f − remains unchanged). As in the modular case (see [ . Then . For each λ with 0 ≤ λ < M, let λ ′ denote the smallest nonnegative integer satisfying λ ′ a ≡ λd (mod M). Then we have The lemma now follows from a standard argument (see [12, Proposition III.17(b)]).
In the proof of Theorem 5, we require a cusp form g(z) with the property that f (z) ≡ g(z) (mod ℓ j ). The existence of such a cusp form is guaranteed by work of Treneer [17]. We first fix some notation. For f ∈ M ! k (N, χ) and a prime ℓ, define α = α(f, ℓ) and β = β(f, ℓ) to be the smallest nonnegative integers satisfying − ℓ α < 4 min where a c runs over a set of representatives for the cusps of Γ 0 (N). Theorem 3 below follows from Theorems 1.1 and 3.1 of [17], along with the proof of Theorem 3.1 of [17].
Further, a positive proportion of primes p ≡ −1 (mod 4Nℓ j ) satisfy for all n coprime to ℓp.
To state Lichtenstein's result, we require the following notation. Let E be the smallest number field containing the coefficients of g(z) and let ℓO E factor as ℓO E = m λ em m , where the λ m are prime ideals of O E . Let S := S k (Γ 0 (2N)) and set d := dim C S. Let  There is an effectively computable constant A 1 (defined in [13]) such that for some prime we have g | T (p 2 ) ≡ 0 (mod ℓ j ). Assuming the Generalized Riemann Hypothesis, the prime p satisfies p ≤ 280B 2 (log B + log L) 2 .
Remark. The quantity B defined in Theorem 4 arises from the elementary bound which, in certain cases, is easy to compute and is much smaller (see [14,Example 4.3]).

Statement of the General Theorem and its Proof
Here we state our general result of which Theorem 1 is a special case. For ease of notation, we state it for harmonic Maass forms with holomorphic parts whose coefficients lie in Q, but an analogous result holds for such forms with algebraic coefficients.
Further, a positive proportion of the primes p ≡ −1 (mod 4NQ 3 ℓ j ) have for all n coprime to ℓp. (iii ) Define S := S ℓ β (ℓ 2 −1) (Γ 0 (2NQ 3 ℓ 2 )), and let B = B(S, ℓ j ) and L = L(S, ℓ) as given in Theorem 4 above. Then the smallest prime p for which (3.1) holds satisfies Assuming GRH, this prime p satisfies Proof of Theorem 5. To prove (i), we use Q-quadratic twists to annihilate the nonholomorphic part of f . We have Since the nonholomorphic part of f is supported on exponents of the form −δ j Q 2 n 2 , the harmonic Maass form f : The nth coefficient of f is That is, a(n) is a + (n) if −n Q = −1 and is 0 otherwise.
To prove (ii), apply Theorem 3 to the weakly holomorphic modular form f . Statement (iii) is immediate from Theorem 4.
Theorem 1 now follows quickly from Theorem 5.
Proof of Theorem 1. Using Theorem 5 and taking m = α we obtain congruences of the form a(p 3 ℓ m n) ≡ 0 (mod ℓ j ) for all n coprime to ℓp. We may choose Q in Theorem 5 to also be coprime to ℓ. Then since a(k) = b(n) when −k Q = −1, we can take any integer A satisfying (A, pℓ) = 1 and Remark. For the case where α = 0, Theorem 1 can be improved. Namely, we can have Q = ℓ and, with A chosen as above, we can similarly obtain congruences b(p 4 ℓn + B) ≡ 0 (mod ℓ j ) where B = p 3 A.