You are here
HomeBirch and SwinnertonDyer conjecture
Primary tabs
Birch and SwinnertonDyer conjecture
Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $L(E,s)$ be the Lseries attached to $E$.
Conjecture 1 (Birch and SwinnertonDyer).
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. $\lim_{{s\to 1}}(s1)^{{R}}L(E,s)$ has a concrete expression involving the following invariants of $E$: the real period, the TateShafarevich 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:
$\lim_{{s\to 1}}\frac{L(E,s)}{(s1)^{R}}=\frac{\operatorname{Sha}\cdot\Omega% \cdot\operatorname{Reg}(E/\mathbb{Q})\cdot\prod_{p}c_{p}}{E_{{\operatorname{% tors}}}(\mathbb{Q})^{2}}$ 
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 TateShafarevich group of $E/\mathbb{Q}$.

$\operatorname{Reg}(E/\mathbb{Q})$ is the elliptic 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 nonsingular 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 BSD 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 BSD 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 Lseries of $E$. If $L(E,1)\neq 0$ then $E(K)$ is finite.
References
 1 Claymath Institute, Description, online.
 2 J. Coates, A. Wiles, On the Conjecture of Birch and SwinnertonDyer, Inv. Math. 39, 223251 (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. SpringerVerlag, New York, 1986.
 6 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. SpringerVerlag, New York, 1994.
Mathematics Subject Classification
14H52 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections