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 $R=\mathrm{rank}(E(\mathbb{Q}))$. Then the residue of $L(E,s)$ at$s=1$, i.e. ${lim}_{s\to 1}{(s-1)}^{-R}L(E,s)$ has a concreteexpression involving the following invariants of $E$: the realperiod, the Tate-Shafarevich group, the elliptic regulator and theNeron 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:

$$\underset{s\to 1}{lim}\frac{L(E,s)}{{(s-1)}^R}=\frac{|\mathrm{Sha}|\cdot \mathrm{\Omega }\cdot \mathrm{Reg}(E/\mathbb{Q})\cdot {\prod }_p{c}_p}{{|E_{\mathrm{tors}}(\mathbb{Q})|}^2}$$

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 $E(ℝ)$ is connected or not.

• $|\mathrm{Sha}|$ is the order of the Tate-Shafarevich 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 non-singular in $E({𝔽}_p)$. 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 B-SD conjecture:

Conjecture 2 (Parity Conjecture).

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

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

Theorem 1 (Coates, Wiles).

Suppose $E$ is an elliptic curve defined over an imaginary quadraticfield $K$, with complex multiplication by $K$, and $L\mathit{}\mathrm{(}E\mathrm{,}s\mathrm{)}$ is theL-series of $E$. If $L\mathit{}\mathrm{(}E\mathrm{,}\mathrm{1}\mathrm{)}\mathrm{\ne }\mathrm{0}$ then $E\mathit{}\mathrm{(}K\mathrm{)}$ is finite.