Analytic order of Sha (reviewed)

For an abelian variety $A/\Q$ the analytic order of Sha is the real number \[ Ш_{an}(A) := \frac{L^{(r)}(A,1)}{r!}\frac{(\#A(\Q)_{\rm tor})^2}{\Omega_AR_AT_A} \] where

• $r$ is the analytic rank of $A$;
• $L(A,s)$ is the L-function of $A$, with leading coefficient $L^{(r)}(A,1)/r!$;
• $A(\Q)_{\rm tor}$ is the torsion subgroup of the Mordell-Weil group $A(\Q)$;
• $\Omega$ is the real period of $A$;
• $R$ is the regulator of $A$;
• $T = \prod_p c_p$ is the Tamagawa product of $A$.

The Birch and Swinnerton-Dyer for abelian varieties implies that the analytic rank of $A$ is equal to its Mordell-Weil rank, and that its analytic order of Sha is equal to the order of its Tate-Shafarevich group.