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.
| Identifer | oai:union.ndltd.org:csusb.edu/oai:scholarworks.lib.csusb.edu:etd-1226 |
| Date | 01 June 2015 |
| Creators | Mecklenburg, Trinity |
| Publisher | CSUSB ScholarWorks |
| Source Sets | California State University San Bernardino |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Electronic Theses, Projects, and Dissertations |
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