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 $E$ .

Conjecture 1 (Birch and Swinnerton-Dyer).

1.

$L(E,s)$ has a zero at $s=1$ of order equal to the rank of $E(\mathbb{Q})$ .
2.

Let $R=\operatorname{rank}(E(\mathbb{Q}))$ . Then the residue of $L(E,s)$ at $s=1$ , i.e. $\lim_{s\to 1}(s-1)^{-R}L(E,s)$ has a concrete expression involving the following invariants of $E$ : the real period, the Tate-Shafarevich group, the elliptic regulator and the Neron model of $E$ .

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/\mathbb{Q})\cdot\prod_{p}c_{p}}{|E_{\operatorname{% tors}}(\mathbb{Q})|^{2}}

where

•

$R$ is the rank of $E/\mathbb{Q}$ .
•

$\Omega$ 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 $E/\mathbb{Q}$ .
•

$\operatorname{Reg}(E/\mathbb{Q})$ is the http://planetmath.org/node/RegulatorOfAnEllipticCurveelliptic regulator of $E(\mathbb{Q})$ .
•

$|E_{\operatorname{tors}}(\mathbb{Q})|$ is the number of torsion points on $E/\mathbb{Q}$ (including the point at infinity $O$ ).
•

$c_{p}$ is an elementary local factor, equal to the cardinality of $E(\mathbb{Q}_{p})/E_{0}(\mathbb{Q}_{p})$ , where $E_{0}(\mathbb{Q}_{p})$ is the set of points in $E(\mathbb{Q}_{p})$ whose reduction modulo $p$ is non-singular in $E(\mathbb{F}_{p})$ . Notice that if $p$ is a prime of good reduction for $E/\mathbb{Q}$ 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:

Conjecture 2 (Parity Conjecture).

The root number of $E$ , denoted by $w$ , indicates the parity of the rank of the elliptic curve, this is, $w=1$ if and only if the rank is even.

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

Theorem 1 (Coates, Wiles).

Suppose $E$ is an elliptic curve defined over an imaginary quadratic field $K$ , with complex multiplication by $K$ , and $L(E,s)$ is the L-series of $E$ . If $L(E,1)\neq 0$ then $E(K)$ is finite.

References

1 Claymath Institute, Description, http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/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, http://www.jmilne.org/math/CourseNotes/math679.htmlonline 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.

Title	Birch and Swinnerton-Dyer conjecture
Canonical name	BirchAndSwinnertonDyerConjecture
Date of creation	2013-03-22 13:49:46
Last modified on	2013-03-22 13:49:46
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	16
Author	alozano (2414)
Entry type	Conjecture
Classification	msc 14H52
Synonym	BS-D conjecture
Related topic	EllipticCurve
Related topic	RegulatorOfAnEllipticCurve
Related topic	MordellCurve
Related topic	ArithmeticOfEllipticCurves
Defines	Birch and Swinnerton-Dyer conjecture
Defines	parity conjecture