Strong Birch–Swinnerton-Dyer conjecture for abelian varieties

Let $A$ be an abelian variety of dimension $g$ over a number field $F$, let $A^t$ be its dual abelian variety, and let $L^*(A,1)$ denote the leading coefficient of $L(A,s)$ at $s=1$. Let $P_A$ be the global volume, $\Theta_{NT}(A)$ the determinant of the Néron–Tate height pairing, and $\Sha(A/F)$ the Tate–Shafarevich group. Strong Birch–Swinnerton-Dyer conjecture. The group $\Sha(A/F)$ is finite and

$$L^*(A,1)=\frac{P_A\,\Theta_{NT}(A)\,[\Sha(A/F)]}{[A(F)_{\mathrm{tor}}]\,[A^t(F)_{\mathrm{tor}}]}.$$

The conjecture refines the rank prediction by giving the leading coefficient in terms of periods, regulators, torsion, and the Tate–Shafarevich group. The source attributes this formulation to Tate and Gross and gives no resolution.