Moonshine and Elliptic Curves Público

Abstract

In this dissertation, we characterize all infinite-dimensional graded virtual

modules for Thompson’s sporadic simple group, whose graded traces are

weight 3/2 weakly holomorphic modular forms satisfying certain special properties.

We then use these modules to detect the non-triviality of Mordell-

Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic

curves. Thus proving the existence of a new kind of moonshine as well as

establishing applications of moonshine to number theory.

Table of Contents

Abstract 5

Acknowledgements 3

Contents 9

List of Tables 11

1 Introduction 1

1.1 Moonshine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Weight 3

2 moonshine for Th . . . . . . . . . . . . . . . . . . . 3

2 Background and Notation 7

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Rational Characters. . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Mock Modular Forms . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Rademacher sums . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Kohnen Plus Space Condition . . . . . . . . . . . . . . . . . . 13

2.6 Eta-Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Moonshine for Thompson’s sporadic simple group 17

3.1 McKay–Thompson Series . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Using Rademacher Sums . . . . . . . . . . . . . . . . . 20

3.1.2 Using Eta-Quotients. . . . . . . . . . . . . . . . . . . . 22

3.2 Proof of Proposition 3.1.1 . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Cusp forms . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.2 Integer Coefficients. . . . . . . . . . . . . . . . . . . . 25

3.3 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Generalized Characters . . . . . . . . . . . . . . . . . 27

4 Elliptic Curves 33

4.1 Statements of Theorems . . . . . . . . . . . . . . . . . . . . . 33

4.2 Traces of Singular Moduli . . . . . . . . . . . . . . . . . . . . 36

4.3 Background on Elliptic Curves . . . . . . . . . . . . . . . . . . 38

4.4 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . 39

4.5 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . 40

A Tables 45

A.1 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

A.2 p-regular sections . . . . . . . . . . . . . . . . . . . . . . . . . 46

A.3 Cusp forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Palabra Clave	Moonshine Modular Forms Elliptic curves
Committee Chair / Thesis Advisor	John Duncan, Emory University
Committee Members	David Borthwick, Emory University David Zureick-Brown, Emory University

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