Birch and Swinnerton-Dyer conjecture

Let $E$ be an elliptic curve over $\mathbb{Q}$ , and let $L(E,s)$ be the L-series attached to $E$ .

J. Tate said about this conjecture: ``This remarkable conjecture relates the behavior of a function $L$ at a point where it is not at present known to be defined to the order of a group (Sha) which is not known to be finite!'' The precise statement of the conjecture asserts that:

$$\lim_{s\to 1} \frac{L(E,s)}{(s-1)^R}=\frac{|\operatorname{Sha}|\cdot \Omega \cdot \operatorname{Reg}(E/\Q) \cdot \prod_p c_p}{| E_{\operatorname{tors}}(\Q)|^2}$$ where

• $R$ is the rank of $E/\Q$ .
• $\Omega$ is either the real period or twice the real period of a minimal model for $E$ , depending on whether $E(\mathbb{R})$ is connected or not.
• $|\operatorname{Sha}|$ is the order of the Tate-Shafarevich group of $E/\Q$ .
• $\operatorname{Reg}(E/\Q)$ is the elliptic regulator of $E(\Q)$ .
• $|E_{\operatorname{tors}}(\Q)|$ is the number of torsion points on $E/\Q$ (including the point at infinity $O$ ).
• $c_p$ is an elementary local factor, equal to the cardinality of $E(\Q_p)/E_0(\Q_p)$ , where $E_0(\Q_p)$ is the set of points in $E(\Q_p)$ whose reduction modulo $p$ is non-singular in $E(\mathbb{F}_p)$ . Notice that if $p$ is a prime of good reduction for $E/\Q$ then $c_p=1$ , so only $c_p\neq 1$ only for finitely many primes $p$ . The number $c_p$ is usually called the Tamagawa number of $E$ at $p$ .

The following is an easy consequence of the B-SD conjecture:

There has been a great amount of research towards the B-SD conjecture. For example, there are some particular cases which are already known:

Bibliography

1

Claymath Institute, Description, online.

2

J. Coates, A. Wiles, On the Conjecture of Birch and Swinnerton-Dyer, Inv. Math. 39, 223-251 (1977).

3

Keith Devlin, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, 189 - 212, Perseus Books Group, New York (2002).

4

James Milne, Elliptic Curves, online course notes.

5

Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

6

Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.