Elliptic Curves

Author

Trinity MecklenburgFollow

Date of Award

6-2015

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Han, Ilseop

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 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, and study analogous homomorphisms on E: y² = x³ + c and relevant facts.

Recommended Citation

Mecklenburg, Trinity, "Elliptic Curves" (2015). Electronic Theses, Projects, and Dissertations. 186.
https://scholarworks.lib.csusb.edu/etd/186