Weak harmonic Maass forms of weight 5/2 and a mock modular form for the partition function

We construct a natural basis for the space of weak harmonic Maass forms of weight 5/2 on the full modular group. The non-holomorphic part of the first element of this basis encodes the values of the ordinary partition function p(n). We obtain a formula for the coefficients of the mock modular forms of weight 5/2 in terms of regularized inner products of weakly holomorphic modular forms of weight -1/2, and we obtain Hecke-type relations among these mock modular forms.


Introduction
A number of recent works have considered bases for spaces of weak harmonic Maass forms of small weight. Borcherds [3] and Zagier [23] (in their study of infinite product expansions of modular forms, among many other topics) made use of the basis {f d } d>0 defined by f −d = q −d + O(q) for the space of weakly holomorphic modular forms of weight 1/2 in the Kohnen plus space of level 4. Duke, Imamoḡlu and Tóth [14] extended this to a basis {f d } d∈Z for the space of weak harmonic Maass forms of the same weight and level and interpreted the coefficients in terms of cycle integrals of the modular j function. In subsequent work, [13] they constructed a similar basis in the case of weight 2 for the full modular group, and related the coefficients of these forms to regularized inner products of an infinite family of modular functions. To construct these bases requires various types of Maass-Poincaré series. These have played an fundamental role in the theory (see for example [5], [10] among many other works).
Here we will construct a natural basis for the space of weak harmonic Maass forms of weight 5/2 on SL 2 (Z) with a certain multiplier. To develop the necessary notation, let the Dedekind eta-function be defined by η(z) := q 1 24 n≥1 (1 − q n ), q := e 2πiz .
We have the generating function where p(n) is the ordinary partition function. The transformation property η(γz) = ε(γ)(cz + d) SL 2 (Z). Then η −1 (z) is the first element of a natural basis {g m } for this space. To construct the basis, we set g 1 := η −1 , and for each m ≡ 1 mod 24 we define g m := g 1 P (j), where P (j) is a suitable monic polynomial in the Hauptmodul j(z) such that g m = q − m 24 + O(q 23 24 ). We list the first few examples here: (ε) the space of harmonic weak Maass forms of weight 5/2 and character ε on SL 2 (Z). These are real-analytic functions which transform as , which are annihilated by the weight 5/2 hyperbolic Laplacian and which have at most linear exponential growth at ∞ (see the next section for details). If M !

2
(ε) denotes the space of weakly holomorphic modular forms of weight 5/2 and character ε, then we have M !
Here we construct a basis {h m } for the space H ! : (ε) is the standard differential operator. In particular, the non-holomorphic part of h 1 encodes the values of the partition function.
To state the first result it will be convenient to introduce the notation β(y) = Γ − 3 2 , πy 6 , where the incomplete gamma function is given by Γ(s, y) := ∞ y e −t t s−1 dt.
The set {h m } forms a basis for H !
Furthermore, for m = n, the coefficients p + m (n) are real. When m = 1, we have Using Lemma 7 below we obtain the following corollary.
is a weight 5 2 weak harmonic Maass form on SL 2 (Z) with character ε. We will construct the functions h m (z) in Section 3 using Maass-Poincaré series. Unfortunately, the standard construction does not produce any non-holomorphic forms in H ! 5 2 (ε), and we must therefore consider derivatives of these series with respect to the auxiliary parameter. This method was recently used by Duke, Imamoḡlu and Tóth [13] in the case of weight 2 (see also [4] and [19]). The construction provides an exact formula for the coefficients p ± m (n). In particular, when m = 1 we obtain the famous exact formula of Rademacher [21] for p(n) as a corollary (see Section 4 for details).
Work of Bruinier and Ono [11] provides an algebraic formula for the coefficients p − m (n) (and in particular the values of p(n)) as the trace of certain weak Maass forms over CM points. Forthcoming work of the second author investigates the analogous arithmetic and geometric nature of the coefficients p + m (n). In analogy with [14] and [8], the coefficients are interpreted as the real quadratic traces (i.e. sums of cycle integrals) of weak Maass forms.
Remark. When m < 0, we can construct the forms h m directly as in (1.2). We list a few examples here. Together with the family (1.2), these form a "grid" in the sense of Guerzhoy [16] or Zagier [23] (note that the integers appearing as coefficients are the same up to sign as those in (1.2)).
For m > 0, the formula for p + m (n) which results from the construction is an infinite series whose terms are Kloostermann sums multiplied by a derivative of the J-Bessel function in its index. Here we give an alternate interpretation of these coefficients involving the regularized Petersson inner product, in analogy with [13] and [15].
For modular forms f and g of weight k, define provided that this limit exists. Here F (Y ) denotes the usual fundamental domain for SL 2 (Z) truncated at height Y . Then we prove We note that when n = m the integral defining this inner product does not converge.
As an immediate corollary of Theorem 3 we obtain Corollary 4. For positive m, n ≡ 1 (mod 24) we have There are also Hecke relations among the forms h m . Letting T5 2 (ℓ 2 ) denote the Hecke operator of index ℓ 2 on H ! 5 2 (ε) (see Section 6 for definitions), we prove Theorem 5. For any m ≡ 1 (mod 24) and for any prime ℓ ≥ 5 we have Using Theorem 5 it is possible to deduce many relations among the coefficients p + m (n). We record a typical example as a corollary.
Corollary 6. If m, n ≡ 1 (mod 24) and ℓ ≥ 5 is a prime with m ℓ = n ℓ , then Remark. It is possible to derive results analogous to Theorem 5 and Corollary 6 involving the operators T5 2 ℓ 2k for any k (see, for example, [1] or [2]). For brevity, we will not record these statements here.
Example. Using the formula given in Proposition 11 below, we compute In the next section we provide some background material. In Section 3 we adapt the method of [13] to construct the basis described in Theorem 1. The last three sections contain the proofs of the remaining results.

Weak harmonic Maass forms
We require some preliminaries on weak harmonic Maass forms (see for example [9], [20], or [24] for further details). For convenience, we set Γ := SL 2 (Z). We say that ν is a multiplier system for Γ if ν is a character on Γ of absolute value 1. If k ∈ 1 2 Z then we say that f has weight k and character ν for Γ if for every γ ∈ Γ. Here k denotes the slash operator defined for γ = ( a b c d ) ∈ GL + 2 (Q) by We choose the argument of each non-zero τ ∈ C in (−π, π], and we define τ k using the principal branch of the logarithm. For any k ∈ 1 2 Z, let ξ k denote the Maass-type differential operator which acts on differentiable functions f on H by In this paper, we are interested in the multiplier system ε which is attached to the Dedekind eta function. An explicit description of ε is given, for instance, in [ In what follows, we assume this consistency condition so that the forms in question are not identically zero. Suppose that f : H → C is real analytic and satisfies for all γ ∈ Γ. Then f has a Fourier expansion at ∞ which is supported on exponents of the form n/24 with n ≡ 1 (mod 24). If, in addition, f satisfies ∆ k f = 0, (2.8) then by the discussion in Section 3.2 below we have the Fourier expansion Let H ! k (ε) denote the space of functions satisfying (2.7) and (2.8) with the additional property that only finitely many of the a ± (n) with n ≤ 0 in (2.9) are nonzero. We call elements of H ! k (ε) weak harmonic Maass forms of weight k and character ε. Note that these forms are allowed to have poles in the non-holomorphic part.
denote the subspaces of cusp forms, modular forms, and weakly holomorphic modular forms, respectively.
The next lemma follows from a computation. Care must be taken with the two cases m > 0 and m < 0.

Construction of harmonic Maass forms and proof of Theorem 1
In order to construct the basis described in Theorem 1, we first construct Poincaré series attached to the usual Whittaker functions. It turns out that for positive m these series are identically zero. So we must differentiate with respect to an auxiliary parameter s in order to obtain nontrivial forms in this case. The construction is carried out in several subsections and is summarized in Proposition 11 below.
Here c * indicates that the sum is restricted to residue classes coprime to c, and d denotes the inverse of d modulo c. The second equality comes from writing γz = a c − 1 c 2 (z+d/c) and from making the change of variable x → x − d/c. The last equality comes from writing d = d ′ + ℓc with 0 ≤ d ′ < c and gluing together the integrals for each ℓ.
If we write m = 24m ′ + 1, then by (2.4) we obtain 3.2. Whittaker functions and nonholomorphic Maass-Poincaré series. The Poincaré series P m (z) clearly has the desired transformation behavior; in order to construct harmonic forms, we specialize ϕ 0 (y) to be a function which has the desired behavior under ∆ k . The Whittaker functions M µ,ν (y) and W µ,ν (y) are linearly independent solutions of Whittaker's differential equation (3.5) and are defined in terms of confluent hypergeometric functions (see [22, p. 190] for definitions and properties). Using (3.5) we see after a computation that are linearly independent solutions of the differential equation At the special value s = k 2 , we have and Here J α (x) and I α (x) denote the usual Bessel functions and K(m ′ , n ′ ; c) is defined in (3.2).

Then (3.3) becomes
Using (3.4) we find that The integral in (3.10) can be written as Combining this with the expression for m < 0 from [7, p. 33], we obtain Using this with (3.10) and (3.11) we find that 3.3. Derivatives of nonholomorphic Maass-Poincaré series in weight 5 2 . We specialize to the situation k = 5 2 and s = k 2 = 5 4 . Using (3.7) and (3.8) and noting that g m,n (s) = 0 for n < 0, we obtain We must evaluate the first two terms which appear in the expansion of Q m (z). Proof. By (13.1.34) of [22], we have For the first term, we have . For the second term, we use (13.1.32) and (13.6.10) of [22] to obtain Therefore, the quantity on the left in (3.16) reduces to The lemma follows after using the reflection formula Γ(z)Γ(1 − z) = π csc πz, and its logarithmic derivative

3.4.
Proof of Theorem 1. The next Proposition, which follows directly from Proposition 9 and Lemma 10, summarizes the two constructions.

Proof of Theorem 3
We proceed as in [13] and [15]. Let F (Y ) denote the usual fundamental domain for SL 2 (Z), truncated at height Y . We have the following (see, for example, [3, Section 9]).
The result follows from Stokes' theorem.
To prove Theorem 3, we compute From these estimates we conclude that the expression in (5.1) can be integrated term by term. Also note that for each pair j, k with j ≡ 23 (mod 24), k ≡ 1 (mod 24), and j = −k, we have Since n = m we obtain a(ℓ 2 n) + ℓ −2 −3n ℓ a(n) + ℓ −3 a(n/ℓ 2 ) q n/24 .
There are also Hecke operators on the space H !