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 $R=\mathrm{rank}(E(\mathbb{Q}))$. 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:
$$\underset{s\to 1}{lim}\frac{L(E,s)}{{(s1)}^{R}}=\frac{\mathrm{Sha}\cdot \mathrm{\Omega}\cdot \mathrm{Reg}(E/\mathbb{Q})\cdot {\prod}_{p}{c}_{p}}{{{E}_{\mathrm{tors}}(\mathbb{Q})}^{2}}$$ 
where

•
$R$ is the rank of $E/\mathbb{Q}$.

•
$\mathrm{\Omega}$ 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.

•
$\mathrm{Sha}$ is the order of the TateShafarevich group of $E/\mathbb{Q}$.

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

•
${E}_{\mathrm{tors}}(\mathbb{Q})$ 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 ${c}_{p}\ne 1$ 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, $w\mathrm{=}\mathrm{1}$ 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 $L\mathit{}\mathrm{(}E\mathrm{,}s\mathrm{)}$ is the Lseries of $E$. If $L\mathit{}\mathrm{(}E\mathrm{,}\mathrm{1}\mathrm{)}\mathrm{\ne}\mathrm{0}$ then $E\mathit{}\mathrm{(}K\mathrm{)}$ is finite.
References
 1 Claymath Institute, Description, http://www.claymath.org/millennium/Birch_and_SwinnertonDyer_Conjecture/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, http://www.jmilne.org/math/CourseNotes/math679.htmlonline 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.
Title  Birch and SwinnertonDyer conjecture 
Canonical name  BirchAndSwinnertonDyerConjecture 
Date of creation  20130322 13:49:46 
Last modified on  20130322 13:49:46 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  16 
Author  alozano (2414) 
Entry type  Conjecture 
Classification  msc 14H52 
Synonym  BSD conjecture 
Related topic  EllipticCurve 
Related topic  RegulatorOfAnEllipticCurve 
Related topic  MordellCurve 
Related topic  ArithmeticOfEllipticCurves 
Defines  Birch and SwinnertonDyer conjecture 
Defines  parity conjecture 