Secord-order cusp forms and mixed mock modular forms
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Secord-order cusp forms and mixed mock modular forms Kathrin Bringmann · Ben Kane Dedicated to Mourad Ismail and Dennis Stanton Received: 1 January 2012 / Accepted: 24 May 2012 / Published online: 6 September 2012 © Springer Science+Business Media, LLC 2012
Abstract In this paper, we consider the space of second order cusp forms. We determine that this space is precisely the same as a certain subspace of mixed mock modular forms. Based upon Poincaré series of Diamantis and O’Sullivan (Trans. Am. Math. Soc. 360:5629–5666, 2008) which span the space of second order cusp forms, we construct Poincaré series which span a natural (more general) subspace of mixed mock modular forms. Keywords Modular forms · Second-order modular forms · Mixed mock modular forms · Poincaré series · Harmonic Maass forms Mathematics Subject Classification 11F11 · 11F12 · 11F37 1 Introduction In his last letter to Hardy (see pp. 57–61 of [34]), Ramanujan described 17 q-hypergeometric series which he called mock theta functions. For example, denoting q := e2πiτ for τ ∈ H, one such function is f (τ ) := 1 +
∞ n=1
2
qn . (1 + q)2 (1 + q 2 )2 · · · (1 + q n )2
Ramanujan mysteriously said that these q-series “enter mathematics as beautifully as ordinary theta-functions” (Ramanujan referred to all modular forms as “theta functions”), but only gave a vague definition. While many of his claims were proven K. Bringmann () · B. Kane Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany e-mail: [email protected] B. Kane e-mail: [email protected]
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K. Bringmann, B. Kane
throughout the years, the exact role of his functions within the theory of automorphic forms remained a mystery until Zwegers [39] showed that each of the mock theta functions is the “holomorphic part” of a weight 12 harmonic weak Maass form. A harmonic weak Maass form of weight κ is a certain non-holomorphic automorphic form which is annihilated by the weight κ hyperbolic Laplacian (see Sect. 2 for a definition). Following Zagier [35], we call the holomorphic part of a harmonic weak Maass form a mock modular form. The non-holomorphic part of a harmonic weak Maass form is related to a non-holomorphic Eichler integral g ∗ (see Sect. 2) arising from a weakly holomorphic modular form (i.e., a meromorphic modular form whose only possible poles occur at cusps) g of weight 2 − κ. One refers to g as the shadow of the mock modular form. As evidence of their influence, mock modular forms and harmonic weak Maass forms naturally appear in partition theory (for example, [2, 4, 7, 11, 13]), Zagier’s duality [36] (for example, [12]), and derivatives of L-functions (for example, [16, 17]). Extending work of Conway and Norton [19] and Borcherds [3] on classical Monstrous Moonshine, Eguchi, Ooguri, and Tachikawa [25] have recently observed a connection between mock modular forms and Monstrous Moonshine of the largest Mathieu group M24 . To expound upon another application, an exact formula for the Fourier coefficients α(n) of f (τ ) (as a sum invol
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