Mazur–Tate–Teitelbaum ordinary p-adic Birch–Swinnerton-Dyer conjecture

Let $E$ be an elliptic curve with good ordinary reduction at a prime $p$, let $r=\operatorname{rank}E(\mathbb{Q})$, and let $L_p(E,T)$ be the ordinary $p$-adic $L$-series. Write $L_p^*(E,0)$ for its leading term, $N_p$ for the normalization factor used in the source, $\operatorname{Reg}_p(E/\mathbb{Q})$ for the $p$-adic regulator, and $c_v$ for the Tamagawa factors. Mazur–Tate–Teitelbaum's conjecture. The order of vanishing of $L_p(E,T)$ at $T=0$ is $r$, and, up to a $p$-adic unit,

$$L_p^*(E,0)\sim\frac{\prod_v c_v\cdot N_p^2\cdot\#\operatorname{Sha}(E/\mathbb{Q})}{(\#E(\mathbb{Q})[p^\infty])^2}\cdot\frac{\operatorname{Reg}_p(E/\mathbb{Q})}{p}.$$

This is the ordinary $p$-adic Birch–Swinnerton-Dyer formula, connecting the leading term of the $p$-adic $L$-series with arithmetic invariants of $E$. The supplied text gives no resolution status.