Lecture 34: The Birch and Swinnerton-Dyer Conjecture, Part 1

William Stein

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 E be the elliptic curve defined by

y^{2} + xy = x^{3} − x^{2} − 79x + 289.

Denote by L(E, s) = ∑_{n=1}^{∞} a_{n}n^{−s} the corresponding L-series, which extends to a function everywhere. The graph of L(E, s) for s ∈ (0, 5) is given on the next page. It can be proved that E(ℚ) ≈ ℤ^{4} by showing that

(8, 7), (\frac{120}{27}, \frac{29}{27}) , (\frac{70}{8}, \frac{81}{8}) , and (\frac{564}{8}, \frac{665}{64})

generate a “subgroup of finite index” in E(ℚ). The Birch and Swinnerton-Dyer Conjecture then predicts that

ord_{s=1} L(E, s) = 4,

which looks plausible from the shape of the graph on the next page. It is relatively easy to prove that the following is equivalent to showing that ord_{s=1} L(E, s) = 4:

Open Problem: Prove that L′′(E, 1) = 0.

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 L-series with ord_{s=1} L(E, s) = 3, 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

y  x

The L-series of the “simplest” known elliptic curve of rank 4.