Date of Award
Honors Thesis (Open Access)
Colby College. Mathematics and Statistics Dept.
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 Fq-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.
Elliptic curves, Finite fields
Recommended CitationCalger, Christopher S., "Elliptic Curves Over Finite Fields" (2023). Honors Theses. Paper 1413.
Algebra Commons, Algebraic Geometry Commons, Number Theory Commons