 Spring 2003 Freshman Seminar 21n: Mathematical and Computational Aspects of Elliptic Curves William Stein The Student's FINAL PROJECTS Class Photo Lecture Notes / Handouts / Homework

An elliptic curve is a cubic curve of the form y2 = x3 + 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.

1. Silverman and Tate: Rational Points on Elliptic Curves [link to Amazon.com].
2. Ribet and Hearst's excellent review of [Silverman-Tate]: PDF, dvi, Postscript
3. Barry Mazur's article Number Theory as Gadfly: PDF.
4. The book Elementary Number Theory and Elliptic Curves that I'm writing.
5. Hellegouarch's book "Invitation to the Mathematics of FERMAT-WILES" (see this review).

Other Resources

1. Andrew Wiles's proof of Fermat's Last Theorem: PDF (860kB pdf file instead of huge JSTOR scan, thanks to Derek Buchanan), TeX file
2. Complete solution to Hilbert's 10 problem in 21 pages: PDF
3. Stillwell's article The Evolution of Elliptic Curves: PDF
4. 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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