#### Title

#### 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

**and finite cyclic groups. The number of finite copies of**

*Z***is called the rank of**

*Z**E*(

**).**

*Q*From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form *E*: *y*^{2} = *x*^{3} + a*x ^{2}* + b

*x*. 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*:

*y*

^{2}=

*x*

^{3}+

*c*. To do this, we review a few well-known homomorphisms on the curve

*E*:

*y*

^{2}=

*x*

^{3}+ a

*x*+ b

^{2}*x*as in Tate and Silverman's

*Elliptic Curves*, and study analogous homomorphisms on

*E*:

*y*

^{2}=

*x*

^{3}+

*c*and relevant facts.

#### Recommended Citation

Mecklenburg, Trinity, "Elliptic Curves" (2015). *Electronic Theses, Projects, and Dissertations*. 186.

http://scholarworks.lib.csusb.edu/etd/186