Graduation Year
2015
Date of Submission
4-2015
Document Type
Open Access Senior Thesis
Degree Name
Bachelor of Arts
Department
Mathematics
Reader 1
David Krumm
Terms of Use & License Information
Rights Information
© 2015 Jonathan S. Star
Abstract
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.
Recommended Citation
Star, Jonathan, "Elliptic Curves and The Congruent Number Problem" (2015). CMC Senior Theses. 1120.
https://scholarship.claremont.edu/cmc_theses/1120
