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.