Math 662 (Section 601) -- Spring 2010

Elliptic Curves

TuTh 12:45-2:00
BLOC 624


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.

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

The course will cover the following topics as time permits:

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

Prerequisites: The course prerequisites are Math 653 (1st semester graduate algebra), 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.

Course Information:

Instructor: Dr. Matthew Papanikolas

Office Hours: Tues. 3:00-4:00, Wed. 11:00-12:00; also by appointment

Office: 321 Milner

Office Phone: 845-1615


Textbook: Elliptic Curves, 2nd Ed., by Dale Husemöller, Springer-Verlag, 2004, ISBN 0-387-95490-2.

Prerequisites: Math 653 or equivalent

Course Webpage:

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

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