In this paper we explain the congruent number problem and its connection to elliptic curves. We begin with a brief history of the problem and some early attempts to understand congruent numbers. We then introduce elliptic curves and many of their basic properties, as well as explain a few key theorems in the study of elliptic curves. Following this, we prove that determining whether or not a number n is congruent is equivalent to determining whether or not the algebraic rank of a corresponding elliptic curve En is 0. We then introduce L-functions and explain the Birch and Swinnerton- Dyer (BSD) Conjecture. We then explain the machinery needed to understand an algorithm by Tim Dokchitser for evaluating L-functions at 1. We end by computing whether or not a given number n is congruent by implementing Dokchitser’s algorithm with Sage and by using Tunnel’s Theorem.
| Identifer | oai:union.ndltd.org:CLAREMONT/oai:scholarship.claremont.edu:cmc_theses-2207 |
| Date | 01 January 2015 |
| Creators | Star, Jonathan |
| Publisher | Scholarship @ Claremont |
| Source Sets | Claremont Colleges |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | CMC Senior Theses |
| Rights | © 2015 Jonathan S. Star, default |
Page generated in 0.0018 seconds