Elliptic Curves Over Finite Fields

Author (Your Name)

Christopher S. Calger, Colby CollegeFollow

Date of Award

2023

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Changningphaabi Namoijam

Second Advisor

Nora Youngs

Abstract

The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of F_q-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use of the Frobenius endomorphism. While the earlier sections are written so that the material is applicable to arbitrary fields, much of the presented information was chosen because of its utility to the theory of elliptic curves over finite fields.

Keywords

Elliptic curves, Finite fields

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