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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:

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.

Knowl status:
  • Review status: reviewed
  • Last edited by John Cremona on 2020-10-08 05:15:25
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