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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.
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