Birch and Swinnerton-Dyer conjecture (reviewed)

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);
• $\#Ш(E/\Q)$ is the order of 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 covolume of the period lattice.