Elliptic Curves

Abstract

The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q). From John Tate and Joseph Silverman [ST92], we have a formula to compute the rank of curves of the form E : y2 = x3 + ax2 + bx. In this thesis, we generalize this formula, using a purely group theoretic approach, and utilize this generalization to find the rank of curves of the form E : y2 = x3 + c. To do this, we review a few well-known homomorphisms on the curve E : y2 = x3 + ax2 + bx as in Tate and Silverman’s Elliptic Curves [ST92], and study analogous homomorphisms on E : y2 = x3 + c and relevant facts.