Moonshine and Elliptic Curves Público
Khaqan, Maryam (Summer 2021)
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
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