Connections between Umbral and Classical Moonshine Open Access

Trebat-Leder, Sarah (Spring 2018)

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Both results of this dissertation involve finding unexpected connections between the classical theory of monstrous moonshine and the newer umbral moonshine. In our first result, we use generalized Borcherds products to associate to each pure A-type Niemeier lattice a conjugacy class g of the monster group and give rise to identities relating dimensions of representations from umbral moonshine to values of McKay-Thompson series. Our second result focuses on the Mathieu group M23. While it inherits a moonshine from being a subgroup of M24, we find a new and simpler moonshine for M23 such that the graded traces are, up to constant terms, identical to the monstrous moonshine Hauptmoduln.

Table of Contents

1. Introduction

1.1 Moonshine

1.2 First Result

1.3 Second Result

2. Background

2.1 Vector-Valued Modular Forms

2.1.1 A Lattice Related to Gamma_0(m)

2.1.2 The Weil Representation

2.1.3 Harmonic Weak Maass Forms

2.2 Umbral Moonshine

2.2.1 Mathieu Moonshine

2.2.2 The Objects of Umbral Moonshine

2.2.3 The Conjectures and Proof Strategies of Umbral Moonshine

2.2.4 Defining the Umbral Moonshine Mock Modular Forms

2.3 Rademacher Sums

2.3.1 Introduction to Rademacher Sums

2.3.2 Rademacher Series and Zagier Duality

2.3.3 Monstrous Moonshine Functions as Rademacher Sums

2.3.4 Mathieu Moonshine Functions as Rademacher Sums

2.4 Replicability of Monstrous T_g

3. Proof of First Result

3.1 Relating Umbral and Monstrous Moonshine

3.1.1 Twisted Generalized Borcherds Products

3.1.2 Proofs of Theorem 1.2.1 and Corollary 1.2.2

3.1.3 Examples

3.1.4 Computing the Elements in Q_{Delta, r}/Gamma_0(m)

3.2 p-adic Properties of the Logarithmic Derivative

3.2.1 p-adic Modular Forms

3.2.2 Proof of Theorem 1.2.3

3.2.3 Proofs of Theorem 1.2.4 and Corollary 1.2.5

4. Proof of Second Result

4.1 Proof of Theorem 1.2.1

4.1.1 proof that f_g are Modular

4.1.2 Proof that f_g are Hauptmoduln

4.2 Proof of Theorem 1.2.2

4.2.1 Proof that m_chi are Integral

4.2.2 Estimation Tools

4.2.3 Proof that m_chi are Nonnegative

A. Definitions of Special Functions

B. Character Table of M_23

C. Coefficients of T_chi



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