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Weak Harmonic Maass Forms of Weight 5/2 and a Mock Modular Form for the Partition Function

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Weak Harmonic Maass Forms of Weight 5/2 and a Mock Modular Form for the Partition Function Ahlgren and Andersen Research in Number Theory (2015) 1:10 DOI 10.1007/s40993-015-0011-9 RESEARCH ARTICLE Open Access Weak harmonic Maass forms of weight 5/2 and a mock modular form for the partition function Scott Ahlgren* andNickolasAndersen *Correspondence: [email protected] Abstract Department of Mathematics, We construct a natural basis for the space of weak harmonic Maass forms of weight University of Illinois, Urbana IL, USA 5/2 on the full modular group. The nonholomorphic 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. 1 Introduction AnumberofrecentworkshaveconsideredbasesforspacesofweakharmonicMaass forms of small weight. Borcherds [4] and Zagier [25] (in their study of infinite product expansions of modular forms, among many other topics) made use of the basis {fd}d>0 −d defined by f−d = q +O(q) for the space of weakly holomorphic modular forms of weight 1/2 in the Kohnen plus space of level 4. Duke, Imamoglu¯ and Tóth [14] extended this to abasis{fd}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, [16] 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 a fundamental role in the theory of weak harmonic Maass forms (see for example [6, 10] among many other works). Here, we will construct a natural basis for the space of weak harmonic Maass forms of weight 5/2onSL2(Z) with a certain multiplier. To develop the necessary notation, let the Dedekind eta-function be defined by 1 n 2πiz η(z) := q 24 (1 − q ), q := e , z = x + iy ∈ H, n≥1 where H denotes the complex upper half-plane. We have the generating function −1 n− 1 η (z) = p(n)q 24 , n≥0 where p(n) is the ordinary partition function. The transformation property 1 ab η(γz) = ε(γ )(cz + d) 2 η(z), γ = ∈ SL (Z) (1.1) cd 2 © 2015 Ahlgren and Andersen. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Ahlgren and Andersen Research in Number Theory (2015) 1:10 Page 2 of 16 defines a multiplier system ε of weight 1/2onSL2(Z) which takes values in the 24-th roots ! (ε) of unity. Let M− 1 denote the space of weakly holomorphic modular forms of weight 2 −1 −1/2 and multiplier ε on SL2(Z).Thenη (z) is the first element of a natural basis {gm} −1 forthisspace.Toconstructthebasis,wesetg1 := η ,andforeachm ≡ 1mod24we define gm := g1P(j),where P(j) is a suitable monic polynomial in the Hauptmodul j(z) − m 23 such that gm = q 24 + O q 24 .Welistthefirstfewexampleshere: − g = η 1 1 − 1 23 47 71 95 119 143 167 191 = q 24 + q 24 + 2q 24 + 3q 24 + 5q 24 + 7q 24 + 11q 24 + 15q 24 + O q 24 , − g = η 1( j − 745) 25 − 25 23 47 71 95 = q 24 + 196885q 24 + 21690645q 24 + 886187500q 24 + O q 24 , (1.2) − g = η 1( j2 − 1489j + 160511) 49 − 49 23 47 71 95 = q 24 + 42790636q 24 + 40513206272q 24 + 8543738297129q 24 + O q 24 . ! (ε) / Denote by H 5 the space of weak harmonic Maass forms of weight 5 2 and multiplier 2 ε on SL2(Z). These are real analytic functions which transform as 5 ab F(γ z) = ε(γ )(cz + d) 2 F(z), γ = ∈ SL (Z), cd 2 which are annihilated by the weight 5/2 hyperbolic Laplacian ∂2 ∂2 ∂ ∂ 2 5 5 :=−y + + iy + i , 2 ∂x2 ∂y2 2 ∂x ∂y and which have at most linear exponential growth at ∞ (see the next section for details). ! (ε) / If M 5 denotes the space of weakly holomorphic modular forms of weight 5 2and 2 ε ! (ε) ⊆ ! (ε) multiplier ,thenwehaveM 5 H 5 . 2 2 { } ! (ε) ∈ Here we construct a basis hm for the space H 5 . For negative m we have hm 2 3 ! (ε) ξ ( ) = 6 2 ξ ! (ε) → ! (ε) M 5 , while for positive m we have 5 hm πm gm,where 5 : H 5 M− 1 2 2 2 2 2 is the ξ-operator defined in (2.2) below. In particular, the nonholomorphic part of h1 encodes the values of the partition function. To state the first result it will be convenient to introduce the notation β( ) = − 3 πy y 2 , 6 , ( ) = ∞ −t s−1 where the incomplete gamma function is given by s, y : y e t dt. ≡ ( ) ∈ ! (ε) Theorem 1. For each m 1 mod 24 ,thereisauniquehm H 5 with Fourier 2 expansion of the form ( ) = m + +( ) n < hm z q 24 pm n q 24 if m 0, 0<n≡1(24) and ( ) = +( ) n + β(− ) m + −( )β( ) − n > hm z pm n q 24 i my q 24 pm n ny q 24 if m 0. 0<n≡1 (24) 0<n≡−1 (24) Ahlgren and Andersen Research in Number Theory (2015) 1:10 Page 3 of 16 { } ! (ε) < ∈ ! (ε) > The set hm forms a basis for H 5 .Form 0 we have hm M 5 ,whileform 0 we 2 2 have 3 ξ ( ) = 6 2 · ∈ ! (ε) 5 hm πm gm M− 1 . 2 2 = +( ) Furthermore, for m n, the coefficients pm n are real. When m = 1, we have 3 3 2 2 + 6 −1 6 n 1 n ξ 5 (h1) = η (z) = p q 24 . 2 π π 24 n≥−1 Using Lemma 7 below we obtain the following corollary. Corollary 2. The function P(z) defined by + n + 3 − n ( ) = ( ) 24 − n 1 2 β( ) 24 P z : p1 n q p 24 n ny q 0<n≡1 (24) −1≤n≡23 (24) 5 (Z) ε is a weight 2 weak harmonic Maass form on SL2 with multiplier . The coefficients can be computed using the formula in Proposition 11 below. We have √ 1 25 49 73 ( ) = − ... + 4 π 24 + ... 24 + ... 24 + ... 24 +··· P z 3.96 i 3 q 111.40 q 254.26 q 86.52 q 1 3 − 23 3 − 47 +iβ(−y)q 24 − 1 · 23 2 β(23y)q 24 − 2 · 47 2 β(47y)q 24 +··· . (1.3) We will construct the functions hm(z) in Section 3 using Maass-Poincaré series. Unfor- tunately, the standard construction does not produce any nonholomorphic forms in ! (ε) H 5 , and we must therefore consider derivatives of these series with respect to an aux- 2 iliary parameter. This method was recently used by Duke, Imamoglu¯ and Tóth [16] in the case of weight 2 (see also [5, 20]). The construction provides an exact formula for ±( ) = the coefficients pm n .Inparticular,whenm 1 we obtain the famous exact formula of Rademacher [23] for p(n) as a corollary (see Section 4 for details). −( ) Work of Bruinier and Ono [11] provides an algebraic formula for the coefficients pm 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 [3] investigates the analogous arithmetic +( ) and geometric nature of the coefficients pm n . In analogy with [14] and [9], the coef- ficients are interpreted as the real quadratic traces (i.e. sums of cycle integrals) of weak Maass forms. Ahlgren and Andersen Research in Number Theory (2015) 1:10 Page 4 of 16 Remark. When m < 0, we can construct the forms hm directly as in (1.2). We list a few =− d examples here. Let j : q dq j.Then h− = η j 23 − 23 1 25 49 73 97 = q 24 − q 24 − 196885q 24 − 42790636q 24 − 2549715506q 24 + O q 24 , h− = h− ( j − 743) 47 23 − 47 1 25 49 73 = q 24 − 2q 24 − 21690645q 24 − 40513206272q 24 + O q 24 , h− = h− ( j2 − 1487j + 355910) 71 23 − 71 1 25 49 73 = q 24 − 3q 24 − 886187500q 24 − 8543738297129q 24 + O q 24 . Together with the family (1.2), these form a “grid” in the sense of Guerzhoy [17] or Zagier [25] (note that the integers appearing as coefficients are the same up to sign as those in (1.2)). > +( ) For m 0, the formula for pm n which results from the construction is an infinite series whose terms are Kloosterman sums multiplied by a derivative of the J-Bessel func- tion in its index. Here we give an alternate interpretation of these coefficients involving the regularized Petersson inner product, in analogy with [16] and [15]. For modular forms f and g of weight k,define f , g = lim f (z)g(z)yk dx dy , (1.4) reg →∞ y2 Y F(Y) provided that this limit exists. Here F(Y) denotes the usual fundamental domain for SL2(Z) truncated at height Y. Theorem 3. For positive m, n ≡ 1 (mod 24) with m = n we have + 3 ( ) = 6 2 · pm n : πm gm, gn reg.
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