Mathieu Moonshine: Mock Modular Lifts Público

Abstract

Classical Moonshine describes the remarkable phenomenon that the coefficients of Hauptmoduln are graded traces for the action of the Monster group, the largest of the 26 sporadic groups, on a graded infinite-dimensional module. A similar phenomena has been shown to hold for other sporadic groups, particularly the Mathieu group M24 where instead of Hauptmoduln, the graded traces are shown to be coefficients of mock modular forms. In a recent paper, Ono, Rolen, and Trebat-Leder relate a monstrous moonshine function to one of the Mathieu moonshine functions by constructing products of rational functions of the monstrous Hauptmodul via generalized Borcherds lifts on mock modular forms. The lifting procedure is that introduced in a paper by Bruinier-Ono. We conjecture a generalization of this lift for all the mock modular forms of Mathieu moonshine. In particular, our generalization relates Mathieu moonshine mock modular forms to the modular forms of Conway moonshine by evaluating Heegner points corresponding to various congruence subgroups of the modular group. We present data that supports our conjecture.

Table of Contents

1 Introduction 1

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

1.2 Mathieu Moonshine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Conway Moonshine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Our Conjecture 6

2.1 Calculating Heegner Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Generalized Borcherds Products 15

3.1 A Lattice Related to 􀀀0(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Weil Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Vector Valued Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Lifting He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Generalized Borcherds Products . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Definitions Modular Forms, Jacobi Forms, Theta Functions, etc. 23

4.1 Mathieu (Mock) Modular Functions . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

List of Tables

1 Cycle Shapes and Eta-functions attached to M24 . . . . . . . . . . . . . . . . 3

2 Modular Subgroups attached to elements of M24 . . . . . . . . . . . . . . . . 12

3 Quadratic representatives given � = 􀀀7 . . . . . . . . . . . . . . . . . . . . 14

4 Quadratic representatives given � = 􀀀15 . . . . . . . . . . . . . . . . . . . . 14

5 Mathieu Moonshine Mock Modulars . . . . . . . . . . . . . . . . . . . . . . . 24

About this Master's Thesis

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School	Laney Graduate School
Department	Mathematics
Degree	M.S.
Submission	Master's Thesis
Language	English
Research Field	Theoretical Mathematics
Palabra Clave	moonshine modular forms
Committee Chair / Thesis Advisor	Duncan, Advisor, Emory University
Committee Members	Zureick-Brown, David, Emory University Martin, Gregory, Emory University

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