Winter/Spring, 2002, MWF 10:30-11:20
Elliptic curves have been a central object of study in
number theory for many years. This course will be an introduction to several aspects
of this topic. One of the basic questions which has been of continual interest
is:
If E is an elliptic curve defined over a field K, what can
one say about the set of points on the elliptic curve which have their
coordinates in K ?
This set of points is denoted by E(K) and turns out to be
an abelian group (under a certain,
naturally defined group operation).
This question has been
extensively studied when K is a finite field, a local field, and is especially
interesting when K is an algebraic number field. In the last case, one of the
fundamental theorems is the Mordell-Weil theorem which asserts that E(K) is
finitely generated whenever K is a finite extension of the rational numbers.
This theorem (which we will prove early in the course)
then leads to questions such as:
What can one say about the rank of the abelian group E(K)?
What can one say about its torsion subgroup?
The proof of the Mordell-Weil theorem involves studying
another group associated to the elliptic curve E and K - the so-called Selmer
group Sel(E,K). It is a torsion group. One can extract information about the
rank of E(K) by determining the number of elements of order n in Sel(E,K) for
any n > 1. In principle, this is always possible. In practice, it's rather
difficult. The Selmer group then becomes an interesting object of study itself.
The definition of the Selmer group is based on Kummer
theory. The familiar form of Kummer theory concerns the multiplicative group of
a field K. For elliptic curves, one considers the group E(K) instead. One
interesting development in recent years is that the notion of a Selmer group
can be vastly generalized - a topic which will also be discussed in the course.
The course will also survey what is known about the above
questions concerning the rank and the torsion. Quite a variety of approaches
have been brought to bear on those questions.
Prerequisites for the course: A good background in algebra
- especially Galois theory. A knowledge of algebraic number theory would be
helpful, although the course will review the main theorems of that subject. A
familiarity with completions of number fields would be helpful too. No prior
knowledge of elliptic curves will be assumed.
There will be no textbook for the course.