Monstrous moonshine, elliptic curves and vertex algebras Öffentlichkeit

Abstract

The topic of this dissertation is monstrous moonshine, which refers to the unexpected relationship between the monster sporadic group and the modular j-invariant. Despite being the first and best understood example of moonshine, monstrous moonshine has aspects that lack complete understanding. We investigate here three such aspects.

First we revisit a theorem of Ogg on supersingular j-invariants, and generalize it to supersingular elliptic curves with level structure. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We now know that this is partly explained by monstrous moonshine. Here we show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g. baby monster, Fischer's largest group). More generally we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine.

Second we propose a definition of moonshine with a higher genus property that naturally extends the genus zero property of monstrous moonshine. We outline a method, inspired by monstrous moonshine, for searching for examples of moonshine with the higher genus property. We demonstrate the success of this strategy by employing it to obtain several examples of moonshine with the genus one property.

Third we generalize to certain vertex operator algebras the results of Duncan, Griffin and Ono regarding the asymptotic structure of the homogeneous subspaces of the moonshine module. We prove that an analogous result holds for any vertex operator algebra satisfying certain hypotheses.

Table of Contents

1. Introduction and summary of results . . . . . . . . . . . . . . . . . . 1

2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3. Supersingular elliptic curves and moonshine . . . . . . . . . . . 41

4. Towards higher genus moonshine . . . . . . . . . . . . . . . . . . . . 53

5. Homogeneous subspaces of vertex algebras . . . . . . . . . . . . 62

A. Appendices for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

B. Appendices for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Stichwort	elliptic curves moonshine modular forms monster group modular curves vertex algebras
Committee Chair / Thesis Advisor	John Duncan, Emory University
Committee Members	David Borthwick, Emory University David Zureick-Brown, Emory University

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