William Stein
Date: Math 124 HARVARD UNIVERSITY Fall 2001
The next three lectures will be about the Birch and Swinnerton-Dyer conjecture, which is considered by many people to be the most important accessible open problem in number theory. Today I will guide you through Wiles's Clay Math Institute paper on the Birch and Swinnerton-Dyer conjecture.
On Friday, I will talk about the following open problem, which is a frustrating specific case of the Birch and Swinnerton-Dyer conjecture. Let be the elliptic curve defined by
Open Problem:
Prove that
.
If you could solve this open problem, people like Gross, Tate, Mazur, Zagier, Wiles, me, etc., would be very excited. The related problem of giving an example of an -series with , was solved as a consequence of a very deep theorem of Gross and Zagier, and resulting in an effective solution to Gauss's class number problem.
John Tate gave a talk about the BSD conjecture for the Clay Math
Institute. I strongly encourage you to watch it online at
http://www.msri.org/publications/ln/hosted/cmi/2000/cmiparis/index-tate.html |