Maass forms and quantum modular forms Open Access
Rolen, Larry (2013)
Published
Abstract
This thesis describes several new results in the theory of harmonic Maass forms and related objects. Maass forms have recently led to a flood of applications throughout number theory and combinatorics in recent years, especially following their development by the work of Bruinier and Funke [10] the modern understanding Ramanujan's mock theta functions due to Zwegers [36,37]. The first of three main theorems discussed in this thesis concerns the integrality properties of singular moduli. These are well-known to be algebraic integers, and they play a beautiful role in complex multiplication and explicit class field theory for imaginary quadratic fields. One can also study "singular moduli'' for special non-holomorphic functions, which are algebraic but are not necessarily algebraic integers. Here we will explain the phenomenon of integrality properties and provide a sharp bound on denominators of symmetric functions in singular moduli. The second main theme of the thesis concerns Zagier's recent definition of a quantum modular form. Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous different topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions. Motivated by Zagier's example of the quantum modularity of Kontsevich's "strange'' function F(q), we revisit work of Andrews, Jiménez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components. The final chapter of this thesis is devoted to a study of asymptotics of mock theta functions near roots of unity. In his famous deathbed letter, Ramanujan introduced the notion of a mock theta function, and he offered some alleged examples. The theory of mock theta functions has been brought to fruition using the framework of harmonic Maass forms, thanks to Zwegers [36,37]. Despite this understanding, little attention has been given to Ramanujan's original definition. Here we prove that Ramanujan's examples do indeed satisfy his original definition.
Table of Contents
1 Introduction ... 1
1.1 Classical Modular Forms ... 1
1.2 Harmonic Maass Forms ... 11
1.3 Mock Theta Functions ... 13
1.4 Quantum Modular Forms ... 15
1.5 Main Results ... 17
2 Basic Facts from the Theory of Harmonic Maass Forms ... 30
2.1 Raising and Lowering Operators ... 30
2.2 Fourier Expansions ... 31
2.3 The ξ-operator and "Shadows" ... 34
2.4 The Bruinier-Funke Pairing ... 36
3 Integrality Properties of Singular Moduli ... 38
3.1 The Spectral Decomposition ... 39
3.2 Integrality Results of Duke and Jenkins ... 41
3.3 A Useful Vanishing Condition ... 42
3.3.1 Rankin-Cohen Brackets ... 43
3.4 The Vanishing Lemma ... 44
3.5 Hecke Structure of the Zagier Lift, and a Special Family of Operators ... 45
3.6 Poincaré Series and the Hecke Structure of Zagier Lifts ... 46
3.7 A Sprecial Family of Hecke Operators ... 50
3.8 Integrality of the Coefficients ... 53
3.8.1 Integrality of ΖD(F) ... 56
4 A New Quantum Modular Form ... 58
4.1 Preliminaries ... 58
4.1.1 Sums of Tails Identities ... 58
4.2 Properties of Eichler Integrals ... 59
4.3 Proof of Theorem 1.14 ... 62
4.3.1 Proof of Theorem 1.14 (1) ... 62
4.3.2 Proof of Theorem 1.14 (2) ... 63
4.4 Proof of Corollary 1.15 ... 64
5 Ramanujan's Mock Theta Functions ... 66
5.1 Poincaré series ... 67
5.2 The Proof of Theorem 1.16 and Corollary 1.17 ... 70
5.2.1 Proof of Thorem 1.16 ... 70
5.2.2 Proof of Corollary 1.17 ... 71
5.3 Proof of Theorem 1.18 ... 71
Bibliography ... 73
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