Topics in Elliptic Curves 公开

Morrow, Jackson Salvatore (2016)

Permanent URL: https://etd.library.emory.edu/concern/etds/xd07gt36g?locale=zh
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Abstract

In this thesis, the author proves theorems relating to three different areas in the study of elliptic curves: torsion subgroups over number fields, Selmer groups of elliptic curves, and images of Galois. In particular, the thesis contains theorems completing the classification of possible torsion subgroups for elliptic curves defined over cubic number fields; bounding the order of p-Selmer groups for twists of elliptic curves defined over number fields of small degree; and determining the possibilities, indicies, and occurrences of composite level images of Galois for elliptic curves defined over Q.

Table of Contents

1 Introduction 1

1.1 Organization ........................... 2

1.2 Acknowledgements ........................ 3

2 Background 6

2.1 Torsion subgroups......................... 7

2.2 Algebraic study of rank...................... 12

2.3 Class field theory ......................... 14

3 Torsion in cubic number fields 18

3.1 Introduction............................ 18

3.2 Proof technique of Theorem 3.1.3 ................ 20

3.3 Analysis of X_1(45) ........................ 27

4 Selmer groups of twists of elliptic curves over K with K- rational torsion points 30

4.1 Definitions and Notation..................... 30

4.2 Statement of Results ....................... 37

4.3 Proof of Theorem 4.2.1...................... 40

4.4 Proof of Theorem 4.2.2...................... 44

4.5 Elliptic curves satisfying Corollary 4.2.5 . . . . . . . . . . . . 59

5 Composite level images of Galois 66

5.1 Background ............................ 66

5.2 Composite level modular curves ................. 82

5.3 Analysis of Rational Points-Theory. . . . . . . . . . . . . . . 84

5.4 Analysis of Rational Points-Genus2 . . . . . . . . . . . . . . 87

5.5 Analysis of Rational Points-Genus3 . . . . . . . . . . . . . . 90

5.6 Analysis of RationalPoints-HigherGenus. . . . . . . . . . . 92

6 Entanglements 96

6.1 (2,3)-entanglement ........................ 97

6.2 (2,n)-entanglement........................107

7 Future Work 110

Appendix 114

A.1 Tables for Theorem 5.1.12 ....................114

A.2 Tables for Theorem 5.1.15 ....................116

A.3 Applicable subgroup diagrams for Theorem 5.1.12 . . . . . . . 121

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