Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $L(E,s)$ be the L-series attached to $E$.

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:

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 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 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:

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