Introduction
Elliptic curves are some of the most deeply-studied objects in number
theory (and, for that matter, complex analysis and algebraic
geometry). They have a very down-to-earth description as the
solution sets of certain cubics in two variables. In 1621, Bachet was
able to write down a formula by which, given a (rational) solution to such an
equation, he was able to produce infinitely many. The reason for his
success is that the set of points on an elliptic curve is actually a
group; exploiting the interaction between these algebraic and
geometric structures gives rise to gorgeous number theory.
This course focuses on the structure of points on elliptic curves over
the rational numbers and finite fields; relevant ideas from algebraic
geometry and complex analysis will be introduced as needed. The
course moves from antiquity to the present day and even beyond
(!) -- the last of the Clay Institute's Millennium Problems concerns
the number of points on an elliptic curve over various fields.
Requirements
Each is worth half your grade:
-
For much of the semester, homework will be posted weekly. It is
expected that for each problem set you will at least look at ever problem, and hand in
(or be prepared to present) two problems.
- There will be a final project due at the end of the semester,
with written and oral components. More details will be available
later.
Resources
While there is no official textbook for the course, I will often
consult the following, and encourage you to, also: