Elliptic Curves

<p>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 <em>E</em>(<strong><em>Q</em></strong>) of an elliptic curve <em>E</em> 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 <strong><em>Z </em></strong><em></em>and finite cyclic groups. The number of finite copies of <strong><em>Z</em></strong> is called the rank of <em>E</em>(<strong><em>Q</em></strong>).</p> <p>From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form <em>E</em>: <em>y</em>² = <em>x</em>³ + a<em>x²</em> + b<em>x</em>. 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 <em>E</em>: <em>y</em>² = <em>x</em>³ + <em>c</em>. To do this, we review a few well-known homomorphisms on the curve <em>E</em>: <em>y</em>² = <em>x</em>³ + a<em>x²</em> + b<em>x</em> as in Tate and Silverman's <em>Elliptic Curves</em>, and study analogous homomorphisms on <em>E</em>: <em>y</em>² = <em>x</em>³ + <em>c </em>and relevant facts.</p>

Name thesis:degree_name

Master of Arts in Mathematics

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2015

Author dc:creator

• Mecklenburg, Trinity

Contributors dc:contributor

• Han, Ilseop

• rank
• elliptic curves
• y^2=x^3+c
• Algebra

Repository record dc:identifier

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

OAI identifier oai:identifier

oai:scholarworks.lib.csusb.edu:etd-1226