Topics in Elliptic Curves Público

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

Bibliography

About this Master's Thesis

Rights statement

• Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.

School	Laney Graduate School
Department	Math and Computer Science
Degree	MS
Submission	Master's Thesis
Language	English
Research Field	Mathematics
Palavra-chave	image of Galois elliptic curves torsion subgroups Selmer groups
Committee Chair / Thesis Advisor	Zureick-Brown, David M, Emory University
Committee Members	Ono, Ken, Emory University Raman, Parimala, Emory University

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