The Birch and SwinnertonDyer 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 Lfunction 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 Lfunction 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 TateShafarevich 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 TateSharafevich group $Ш(E/K)$ is finite.
There is a similar conjecture for abelian varieties over number fields.
 Review status: beta
 Last edited by John Cremona on 20201008 05:15:25
 20201008 05:15:25 by John Cremona
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 20201008 04:40:43 by John Cremona