BSD invariants (reviewed)

The BSD invariants of an abelian variety $A/\Q$ include:

• the analytic rank of $A$;
• the Mordell-Weil rank of $A$;
• the leading coefficient $L^{(r)}(A,1)/r!$ of the L-function of $A$;
• the order of the torsion subgroup $A(\Q)_{\rm tor}$ of the Mordell-Weil group $A$;
• the real period $\Omega_A$ of $A$;
• the regulator $R_A$ of $A$;
• the Tamagawa product $T_A:=\prod_p c_p$ of $A$;
• the order of the Tate-Shafarevich group $Ш(A)$ of $A$.

The weak form of the Birch and Swinnerton-Dyer conjecture predicts that the first two invariants are equal, while the strong form of their conjecture relates the last six invariants via the BSD formula \[ \frac{L^{(r)}(A,1)}{r!} = \frac{\#Ш(A)\Omega_AR_AT_A}{(\#A(\Q)_{\rm tor})^2}. \] No effective method to compute $Ш(A)$ is currently known (indeed, it is not even known that $Ш(A)$ is finitely generated, although the BSD conjecture requires this). However, one can compute (approximations of) the the five remaining quantities and use this to compute the analytic order of Sha, and under the assumption that $Ш(A)$ is finite, one can determine whether its order is a square or not. One can also compute the rank of the 2-Selmer group of $A$, which constrains the 2-part of $Ш(A)$.