Perrin-Riou–Bernardi supersingular p-adic Birch–Swinnerton-Dyer conjecture

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at $p$, and let $r=\operatorname{rank}E(\mathbb{Q})$. Let $L_p(E,T)$ be the supersingular $p$-adic $L$-series, let $L_p^*(E,0)$ denote its leading term at $T=0$, let $\phi$ be Frobenius on the Dieudonné module $D_pE$, and let $\operatorname{Reg}_p(E/\mathbb{Q})$ be the $p$-adic regulator. Perrin-Riou–Bernardi's conjecture. The order of vanishing of $L_p(E,T)$ at $T=0$ is $r$, and

$$(1-\phi)^{-2}L_p^*(E,0)=\frac{\prod_v c_v\cdot\operatorname{Sha}(E/\mathbb{Q})}{|E(\mathbb{Q})_{\mathrm{tors}}|^2}\cdot\frac{\operatorname{Reg}_p(E/\mathbb{Q})}{p^r}.$$

This is the supersingular $p$-adic analogue of the Birch–Swinnerton-Dyer conjecture, relating the analytic leading term to Tamagawa factors, the Tate–Shafarevich group, torsion, and the $p$-adic regulator. The supplied text gives no resolution status.