Low-degree points on some rank 0 modular curves 公开

Abstract

Let E be an elliptic curve defined over a number field K. We present some new progress on the classification of the finite groups which appear as the torsion subgroup of E(K) as K ranges over quartic, quintic and sextic number fields. In particular, we concentrate on determining the quartic, quintic and sextic points on certain modular curves X_1(N) for which the rank of the Jacobian is zero. 

Table of Contents

1 Introduction 1

2 Background 6

2.1 Elliptic Curves .............................. 6 2.2 Modular Curves.............................. 11 2.2.1 Intermediate Modular Curves .................. 12 2.2.2 The Jacobian ........................... 12 2.3 The Cuspidal Subscheme Lemma .................... 13

3 Methods 17

3.1 Introduction................................ 17 3.2 Direct Analysis .............................. 19 3.2.1 Intermediate Modular Curves .................. 21

4 Results 22

4.1 The Case of N=25 ........................... 22

4.2 The Case of N=26 ........................... 23

4.3 The Case of N=27 ........................... 24

4.4 The Case of N=28 ........................... 24

4.5 The Case of N=34 ........................... 25

4.6 The Case of N=35 ........................... 26

4.7 The Case of N=40 ........................... 26

5 Future Work 28

Appendix A Code 30

Bibliography 38

About this Dissertation

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

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