The analytic-rank-zero Birch and Swinnerton-Dyer conjecture for abelian varieties

Let $A/\mathbb{Q}$ be an abelian variety with $L$-function $L(A,s)$, and let $\Omega(A)$ denote its real period. For each prime $p$, let $c_p(A)$ be the Tamagawa number of $A/\mathbb{Q}$ at $p$, let $\Sha(A/\mathbb{Q})$ be its Shafarevich–Tate group, and let $A^{\vee}/\mathbb{Q}$ be the abelian variety dual to $A/\mathbb{Q}$. The analytic-rank-zero Birch and Swinnerton-Dyer conjecture. If $L(A,1)\neq 0$, then

$$\frac{L(A,1)}{\Omega(A)}=\frac{|\Sha(A/\mathbb{Q})|\cdot\prod_p c_p(A)}{|A(\mathbb{Q})_{\textrm{tors}}|\cdot|A^{\vee}(\mathbb{Q})_{\textrm{tors}}|}.$$

This is the second part of the Birch and Swinnerton-Dyer conjecture in analytic rank zero; the paper uses it to derive a divisibility statement that is then studied unconditionally for optimal modular quotients.