Bertolini–Darmon–Prasanna p-adic BSD conjecture over an imaginary quadratic field

Let $E/\mathbb{Q}$ have good reduction at an odd prime $p$, and let $K$ be an imaginary quadratic field satisfying the stated splitting and coprimality conditions. Let $L_{\mathfrak{p}}^{\mathrm{BDP}}(t)$ be the square of the anticyclotomic Bertolini–Darmon–Prasanna $p$-adic $L$-function. Assume $r_{\mathrm{an}}(E/K)=1$ and $E(K)[p]=0$, and let $P$ generate $E(K)_{/\mathrm{tors}}$.

Bertolini–Darmon–Prasanna p-adic BSD conjecture. One has

$$L_{\mathfrak{p}}^{\mathrm{BDP}}(0)=\left(1-a_p(E)p^{-1}+p^{-1}\right)^2\cdot\#\Sha(E/K)\cdot\log_{\omega}(P)^2\cdot\operatorname{Tam}(E/\mathbb{Q})^2.$$

This is the anticyclotomic $p$-adic analogue of BSD; the paper later invokes its $p$-part under an Iwasawa main-conjecture hypothesis.