An elliptic curve is a cubic curve of the form y^{2} =
x^{3} + Ax + B with A and B
whole numbers. The seminar will explore many incarnations of
elliptic curves in number theory and computer science.
We will spend the first half of the semester building a solid
foundation about the basics of elliptic curves. You will each read
about elliptic curves, then give presentations to each other about
what you read. I will not assume that you come into the seminar
knowing any number theory at all.
Depending on your taste, and how far we get with the basics, we will
examine some of the following topics:
- The ground breaking way Lenstra used elliptic curves to
construct a fast algorithm for finding the "medium-sized" prime
divisors of whole numbers.
- Use of elliptic curves cryptography in one version of Microsoft's
digital rights management scheme.
- The proof of Fermat's Last Theorem by Andrew Wiles.
- The conjecture of Birch and Swinnerton-Dyer about the nature
of the set of rational points on an elliptic curve.
Things To Read
- Silverman and Tate: Rational Points on Elliptic Curves [link to Amazon.com].
- Ribet and Hearst's excellent review of [Silverman-Tate]: PDF, dvi, Postscript
- Barry Mazur's article Number Theory as Gadfly: PDF.
- The book Elementary Number Theory and Elliptic Curves that I'm writing.
- Hellegouarch's book "Invitation to the Mathematics of FERMAT-WILES" (see this review).
Other Resources
- Andrew Wiles's proof of Fermat's Last Theorem: PDF
(860kB pdf file instead of huge JSTOR scan, thanks to Derek Buchanan), TeX
file
- Complete solution to Hilbert's 10 problem in 21 pages: PDF
- Stillwell's article The Evolution of Elliptic Curves: PDF
- Very dated list of elliptic curve resources
Some books you might want to look at
- Mozzochi's Fermat Diary contains lots of photographs of the
people and events surrounding Wiles's proof of Fermat's Last Theorem.
- Devlin's The Millennium Problems is aimed at Joe
Sixpack, and contains a chapter on the Birch and Swinnerton-Dyer
Conjecture, a copy of which I'll hand out to you.
- Silverman-Tate is the course textbook. If your book is at home
and you're in the library, you can look at it.
- Hellegouarch's Invitation to the Mathematics of FERMAT-WILES
contains a discussion, much at the undergraduate level, of some of the
mathematics required to understand the proof of Fermat's Last Theorem.
You might find the background chapters on elliptic curves interesting.
- Cassels's Lectures on Elliptic Curves is a short
alternative undergraduate presentation of elliptic curves, but at a
more sophisticated level than Silvermat-Tate.
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