Rational Points on Curves Open Access

Etropolski, Anastassia (2016)

Permanent URL: https://etd.library.emory.edu/concern/etds/fx719m861?locale=en


This thesis investigates three areas of arithmetic geometry, all of which fit under the umbrella of ``rational points on curves," yet are distinct and require completely different methods of proof. The first is a generalization of a theorem of Drew Sutherland (which generalizes a theorem of Nicholas Katz) on a local-global question that arises when studying Galois representations associated to elliptic curves. The second is a recent joint result with David Zureick-Brown and Jackson Morrow on cubic torsion on elliptic curves. In particular, we resolve an open problem in the field by classifying the subgroups which can occur as the torsion subgroup for an elliptic curve over a cubic number field. The final project is also the resolution of an open problem; in particular, the full classification of algebraic function fields with class number 3.

Table of Contents

1. A Local-Global principle for Galois representations

1.1 Introduction

1.2 Preliminaries

1.3 Local-Global principle for subgroups of GL_2(F_ell)

1.4 Modular curves and specific counterexamples

2. Torsion on Elliptic Curves

2.1 Introduction and Background

2.2 Classification of cubic torsion

3. Class Numbers of Function Fields

3.1 Introduction

3.2 Background

3.3 The class number 3 problem

A. L-polynomial Equations

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