Topics in Abelian Varieties Open Access

Abstract

We prove theorems and present progress on problems in the broad category of abelian varieties. The flavor of these problems and the techniques used to solve them vary, but a common theme is the use of geometric techniques (and in particular moduli theory) to solve concrete questions from arithmetic. The first two problems involve section rings of algebraic varieties. These rings are classical objects of study and play a central role in the minimal model program. In the first of these problems, we describe the section ring of elliptic curves for arbitrary divisors, and give a complete description when the underlying divisor is supported by up to two points. In the second, we investigate canonical rings of moduli stacks of principally polarized abelian varieties, with particular focus on the g = 2 case. These have additional arithmetic significance: the canonical ring of modular curves, when equipped with the structure of an algebraic stack, gives rise to rings of modular forms. By considering this higher dimensional analogue, we can determine explicit presentations for rings of Siegel modular forms. After these problems, we present progress on classifying torsion subgroups for elliptic curves over quartic fields, extending work of Mazur and Merel. Finally, we investigate under what conditions a Weil polynomial of degree 2g occurs as the characteristic polynomial of Frobenius for a simple abelian variety of dimension g over a given finite field, and give an answer for the g = 7 case. 

Table of Contents

1 Section rings of Q-Divisors on elliptic curves 1

1.1 Introduction and Background ...................... 1

1.2 One-point support ............................ 4

1.3 The one-point case ............................ 4

1.3.1 Generators............................. 5

1.3.2 Relations ............................. 11

1.4 The effective two-point case ....................... 20

1.4.1 Generators............................. 20

1.4.2 Relations ............................. 25

1.5 The subtle behavior of the ineffective two-point case . . . . . . . . . . 33

1.5.1 A conjecture on generators.................... 35

1.6 Arbitrary effective Q-divisors ...................... 38

2 The Canonical Ring of Ag 45

2.1 Introduction, Background, and Setup.................. 45

2.1.1 Siegel Modular Forms ...................... 46

2.1.2 The g=2 case .......................... 47

3 Quartic Torsion 53

3.1 Introduction................................ 53

3.2 Strategy.................................. 56

3.2.1 Direct Analysis .......................... 56

4 Weil polynomials of abelian varieties over finite fields 59

4.1 Introduction................................ 59

4.2 Answer for g=7 ............................. 61

Bibliography 77 

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	arithmetic geometry number theory algebraic geometry
Committee Chair / Thesis Advisor	David Zureick-Brown, Emory University
Committee Members	Suresh Venapally, Emory University Parimala Raman, Emory University

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