Number Theory Commons^™

Elliptic Curves And The Congruent Number Problem, Jonathan Star

CMC Senior Theses

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 E_n is 0. We then introduce L-functions and explain the Birch and …