Applications of Harmonic Maass Forms Open Access

Griffin, Michael John (2015)

Permanent URL: https://etd.library.emory.edu/concern/etds/ng451j49n?locale=en
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Abstract

In this thesis, we prove various results in the theory of modular forms and harmonic Maass forms, representation theory, elliptic curves and differential geometry. In particular, we give a broad framework of Rogers-Ramanujan identities and algebraic values; we prove that Ramanujan's mock theta functions satisfy his original conjectured definition; and we show that certain harmonic Maass forms which arise naturally from the arithmetic of elliptic curves encode central L-values and L-derivatives involved in the Birch and Swinnerton-Dyer conjecture. We also prove a conjecture of Moore and Witten connecting the regularized u-plane integral on the complex projective plane with Donaldson invariants for the SU(2)-gauge theory. In our final two applications, we turn to moonshine phenomena. Monstrous Moonshine relates the Fourier coefficients of certain modular functions to values of the irreducible characters of the Monster group--the largest of the sporadic simple groups. We give the asymptotic distribution of these character values, answering a question of Witten with applications to mathematical physics. The Umbral Moonshine conjectures relate the the values of irreducible characters of prescribed finite groups with the Fourier coefficients of distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. We complete the proof in the remaining cases.

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . 1

2 Background . . . . . . . . . . . . . . . . . . .51

2.1 Harmonic Maass forms . . . . . . . . . . . . . . . . . . . . . . 51

2.2 Maass-Poincar'e series . . . . . . . . . . . . . . . . . . . . . . . 55

2.3 Holomorphic projection . . . . . . . . . . . . . . . . . . . . . . 58

3 RogersRamanujan identities . . . . . . . . . . . . . . . . . . .61

3.1 The HallLittlewood polynomials . . . . . . . . . . . . . . . . 61

3.2 Proof of Theorems 1.21.5 . . . . . . . . . . . . . . . . . . . . 66

3.2.1 The WatsonAndrews approach . . . . . . . . . . . . . 66

3.2.2 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . 70

3.2.3 Proof of Theorem 1.2 (1.8a) . . . . . . . . . . . . . . . 74

3.2.4 Proof of Theorem 1.2 (1.8b) . . . . . . . . . . . . . . . 79

3.2.5 Proof of Theorem 1.5 . . . . . . . . . . . . . . . . . . . 81

3.3 Proof of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . . . 86

3.4 Siegel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.4.1 Basic Facts about Siegel functions . . . . . . . . . . . . 89

3.4.2 Galois theory of singular values of Siegel functions . . . 92

3.5 Proofs of Theorems 1.8 and 1.15 . . . . . . . . . . . . . . . . . 94

3.5.1 Reformulation of the (m; n; ) series . . . . . . . . . 94

3.5.2 Proofs of Theorems 1.8 and 1.15 . . . . . . . . . . . . 98

3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4 Ramanujan's mock theta functions . . . . . . . . . . . . . . . . . . .104

4.1 Proof of Theorem 1.11 . . . . . . . . . . . . . . . . . . . . . . 104

4.2 Proof of Corollary 4.1 . . . . . . . . . . . . . . . . . . . . . . . 106

4.3 The f(q) example . . . . . . . . . . . . . . . . . . . . . . . . . 107

5 Weierstrass mock modular forms . . . . . . . . . . . . . . . . . . .108

5.1 Weierstrass Theory and the proof of Theorems 1.14, 1.15 and 1.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.1 Basic facts about Weierstrass theory . . . . . . . . . . 108

5.1.2 Proofs of Theorems 1.14 and 1.15 . . . . . . . . . . . . 109

5.1.3 Proof of Theorem 1.16 . . . . . . . . . . . . . . . . . . 111

5.2 Vector valued harmonic Maass forms . . . . . . . . . . . . . . 113

5.2.1 A lattice related to \Gamma_0(N) . . . . . . . . . . . . . . . . 115

5.2.2 The Weil representation and vector-valued automorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.3 Poincar'e series and Whittaker functions . . . . . . . . 118

5.2.4 Twisted theta series . . . . . . . . . . . . . . . . . . . . 119

5.3 Theta lifts of harmonic Maass forms . . . . . . . . . . . . . . . 122

5.3.1 Fourier expansion of the holomorphic part . . . . . . . 127

5.3.2 Lift of Poincar'e series and constants . . . . . . . . . . 131

5.4 General version of Theorem 1.17 and its proof . . . . . . . . . 135

5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6 SU(2)-Donaldson Invariants . . . . . . . . . . . . . . . . . . .142

6.1 Some relevant classical functions . . . . . . . . . . . . . . . . . 142

6.2 SU(2)-Donaldson invariants on CP2 . . . . . . . . . . . . . . . 143

6.2.1 The work of Gottsche and his collaborators . . . . . . . 145

6.2.2 The u-plane integral and the work of Moore and Witten147

6.2.3 Criterion for proving Theorem 1.20 . . . . . . . . . . . 153

6.3 The proof of Theorem 1.20 . . . . . . . . . . . . . . . . . . . . 154

6.3.1 q-series identities . . . . . . . . . . . . . . . . . . . . . 154

6.3.2 Work of Zwegers . . . . . . . . . . . . . . . . . . . . . 155

6.3.3 Modularity Properties of K0(\tau) . . . . . . . . . . . . . 158

6.3.4 The proof of Theorem 1.20 . . . . . . . . . . . . . . . . 159

6.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 163

7 Moonshine . . . . . . . . . . . . . . . . . . .164

7.1 Vertex operators and the proof of classical moonshine . . . . . 164

7.2 Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.3 Rademacher Sums . . . . . . . . . . . . . . . . . . . . . . . . 173

7.4 Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.5 Moonshine Tower . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.6 Monstrous Moonshine's Distributions . . . . . . . . . . . . . . 188

7.6.1 The modular groups in monstrous moonshine . . . . . 189

7.6.2 Exact formulas for Tg(-m) . . . . . . . . . . . . . . . . . 191

7.6.3 Exact formulas for Ug up to a theta function . . . . . . 194

7.6.4 Proof of Theorem 7.10 . . . . . . . . . . . . . . . . . . 195

7.6.5 Examples of the exact formulas . . . . . . . . . . . . . 197

8 Umbral Moonshine . . . . . . . . . . . . . . . . . . .199

8.1 Proof of Theorem 1.32 . . . . . . . . . . . . . . . . . . . . . . 199

8.2 Proof of Theorem 1.30 . . . . . . . . . . . . . . . . . . . . . . 202

Appendix . . . . . . . . . . . . . . . . . . .208

A.1 Monstrous Groups . . . . . . . . . . . . . . . . . . . . . . . . 208

A.2 The Umbral Groups . . . . . . . . . . . . . . . . . . . . . . . 209

A.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 210

A.2.2 Character Tables . . . . . . . . . . . . . . . . . . . . . 213

A.2.3 Twisted Euler Characters . . . . . . . . . . . . . . . . 221

A.3 The Umbral McKay-Thompson Series . . . . . . . . . . . . . . 229

A.3.1 Special Functions . . . . . . . . . . . . . . . . . . . . . 229

A.3.2 Shadows . . . . . . . . . . . . . . . . . . . . . . . . . . 232

A.3.3 Rademacher Sums . . . . . . . . . . . . . . . . . . . . 233

A.3.4 Explicit Prescriptions . . . . . . . . . . . . . . . . . . . 235

Bibliography . . . . . . . . . . . . . . . . . . . 264

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