Mazur--Tate--Teitelbaum's $p$-adic Birch--Swinnerton-Dyer conjecture

Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$, let $L_p(E)$ be its $p$-adic $L$-function, and choose a topological generator $\gamma$ of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, identifying $L_p(E)$ with an element of $\mathbb{Z}_p\llbracket X\rrbracket$. Let $\alpha$ be the unit root of $x^2-a_p(E)x+p$, and let $\operatorname{Reg}_\gamma(E/\mathbb{Q})$ be the normalized $p$-adic regulator. Mazur--Tate--Teitelbaum conjecture.

$$\operatorname{rk}_{\mathbb{Z}}E(\mathbb{Q})=\operatorname{ord}_{X=0}L_p(E)=r.$$

If this holds and $\Sha(E/\mathbb{Q})[p^\infty]$ is finite, then

$$L_p^{(r)}(E)=\left(\frac{1}{1-\alpha}\right)^2\frac{\#\Sha(E/\mathbb{Q})[p^\infty]}{(\#E(\mathbb{Q})_{\mathrm{tor}})^2}\prod_{\ell\mid N}c_\ell\,\operatorname{Reg}_\gamma(E/\mathbb{Q}).$$

This is the ordinary $p$-adic analogue of the classical Birch--Swinnerton-Dyer conjecture. The supplied text does not give a resolution of the full statement, although it records substantial progress in related cases.