Birch Swinnerton-Dyer conjecture (reviewed)

The Birch and Swinnerton-Dyer conjecture (BSD) is one of the Millennium Prize Problems listed by the Clay Mathematics Institute. It relates the order of vanishing (or analytic rank) and the leading coefficient of the L-function associated to an elliptic curve $E$ defined over a number field $K$ at the central point $s=1$ to certain arithmetic data, the BSD invariants of $E$.

• The weak form of the BSD conjecture states just that the analytic rank $r_{an}$ (that is, the order of vanishing of vanishing of $L(E,s)$ at $s=1$), is equal to the rank $r$ of $E/K$.

• The strong form of the conjecture states also that the leading coefficient of the L-function is given by the formula $$ \frac{1}{r!} L^{(r)}(E,1) = |d_K|^{1/2}\cdot \frac{\# Ш(E/K)\cdot \Omega(E/K) \cdot \mathrm{Reg}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}}}{\#E(K)_{\rm tor}^2}. $$

The quantities appearing in this formula are as follows:

• $d_K$ is the discriminant of $K$;
• $r$ is the rank of $E(K)$;
• $Ш(E/K)$ is the Tate-Shafarevich group of $E/K$;
• $\mathrm{Reg}(E/K)$ is the regulator of $E/K$;
• $\Omega(E/K)$ is the global period of $E/K$;
• $c_{\mathfrak{p}}$ is the Tamagawa number of $E$ at each prime $\mathfrak{p}$ of $K$;
• $E(K)_{\rm tor}$ is the torsion order of $E(K)$.

Implicit in the strong form of the conjecture is that the Tate-Sharafevich group $Ш(E/K)$ is finite.

There is a similar conjecture for abelian varieties over number fields.