Math 662 (Section 602) -- Spring 2014

Seminar in Algebra
Elliptic Curves

MW 4:10-5:25

Course Description:

Topics of Study: Elliptic curves make up an important class of geometric objects, which have connections in many areas of mathematics, especially number theory and algebraic geometry, as well as several applications, including to cryptography and integer factorization. At the outset elliptic curves are merely solutions of certain cubic polynomial equations, but the points on these curves possess a natural abelian group structure, which leads to many mesmerizing problems.

This course will focus on the study of elliptic curves over various fields, including the rational numbers, finite fields, the p-adic numbers, and the complex numbers. The two main goals of the course will be to prove the Mordell-Weil theorem, which states that the group of points over the rational numbers is finitely generated, and to discuss the Birch and Swinnerton-Dyer conjecture, which relates the rank of the Mordell-Weil group to the vanishing of the L-function of the elliptic curve.

The course will cover the following topics as time permits:

  • Elliptic curves as plane curves
  • The group law on an elliptic curve
  • Torsion points
  • Isogenies and the Tate module
  • Elliptic curves over finite fields
  • Hasse's bound on points over finite fields
  • Congruence zeta functions and the Riemann hypothesis over finite fields
  • Elliptic functions and elliptic curves over the complex numbers
  • Elliptic curve cryptography
  • Reduction modulo primes (good, bad, and not so bad)
  • Height functions
  • Mordell's theorem
  • The Hasse-Weil L-function and the Birch and Swinnerton-Dyer conjecture

Prerequisites: The course prerequisites are Math 653/654 (Graduate Algebra I & II), or consent of the instructor. Otherwise the course will be fairly self-contained, and necessary elements of algebra, number theory, and algebraic geometry will be covered during the semester.

Students interested in doing research in Number Theory are especially encouraged to attend this course.

Course Information:

Instructor: Matthew Papanikolas

Office Hours: Mon. 2-3, Wed. 11-12; also by appointment

Office: 321 Milner


Textbook: The Arithmetic of Elliptic Curves, 2nd Ed., by Joseph H. Silverman, Springer, 2009, ISBN 978-0-387-09493-9.

Prerequisites: Math 653 & 654 (Graduate Algebra I & II); or equivalents

Course Webpage:

Course Work and Grades: There will be regular homework assignments. These will serve as the basis for grades in the course.

Page maintained by Matt Papanikolas, Dept. of Mathematics, Texas A&M University.