Document Type

Honors Project - Open Access

Comments

Thank you to my advisor, Dr. Bressoud, and to my readers, Dr. Doyle and Dr. Flath.

Abstract

The study of elliptic curves grows out of the study of elliptic functions which dates back to work done by mathematicians such as Weierstrass, Abel, and Jacobi. Elliptic curves continue to play a prominent role in mathematics today. An elliptic curve E is defined by the equation, y2 = x3 + ax + b, where a and b are coefficients that satisfy the property 4a3 + 27b2 = 0. The rational solutions of this curve form a group. This group, denoted E(Q), is known as the Mordell-Weil group and was proved by Mordell to be isomorphic to Zr ⊕ E(Q)tors where the group of rational torsion points consists of all points of finite order. The rank r is difficult to compute and the main goal of this research is to explore the relationship between ranks of elliptic curves and values of a and b. Specifically, we have put a lower bound on the ranks of equations of the form Cm : y2 = x3 − m2 x + 1 and Km : y2 = x3 + m3 x − m3.

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