The **Birch and Swinnerton-Dyer** conjecture is one of the Millennium Prize Problems listed by the Clay Mathematics Institute. It relates the order of vanishing and the first non-zero Taylor series coefficient of the L-function associated to an elliptic curve $E$ defined over $\Q$ at the central point $s=1$ to certain arithmetic data, the BSD invariants of $E$.

Specifically, the BSD conjecture states that the order $r$ of vanishing of $L(E,s)$ at $s=1$ is equal to the rank of the Mordell-Weil group $E(\Q)$, and that

$\displaystyle \frac{1}{r!} L^{(r)}(E,1)= \displaystyle \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2}. $

The quantities appearing in this formula are the BSD invariants of $E$:

- $r$ is the rank of $E(\Q)$ (a non-negative integer);
- Ш is the Tate-Shafarevich group of $E$ (which is conjectured to always be finite, a positive integer);
- $\mathrm{Reg}(E/\Q)$ is the regulator of $E/\Q$;
- $\Omega_E$ is the real period of $E/\Q$ (a positive real number);
- $c_p$ is the Tamagawa number of $E$ at each prime $p$ (a positive integer which is $1$ for all but at most finitely many primes);
- $E(\Q)_{\rm tor}$ is the torsion order of $E(\Q)$ (a positive integer).

There is a similar conjecture for abelian varieties, in which the real period is replaced by the regulator of the period lattice.

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- Review status: reviewed
- Last edited by Vishal Arul on 2019-09-20 16:29:45

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- 2019-09-20 16:29:45 by Vishal Arul (Reviewed)
- 2019-09-05 20:19:33 by Kiran S. Kedlaya
- 2019-02-08 11:36:42 by John Cremona (Reviewed)

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