Generalized Perrin–Riou conjecture for an elliptic curve over the cyclotomic extension

Let $E/\mathbb{Q}$ be an elliptic curve, let $r={\rm rank}(E(\mathbb{Q}))$, let $I_\infty$ be the augmentation ideal associated with the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_\infty/\mathbb{Q}$, let $\kappa\in E(\mathbb{Q})\otimes_{\mathbb{Z}}I_\infty^{r-1}/I_\infty^r$, and let $R^{\rm Boc}$ be the Bockstein regulator. Let $\mathcal{L}^{\rm an}_\Sigma$ be the $\Sigma$-truncated analytic Birch and Swinnerton-Dyer constant, viewed in $\mathbb{C}_p$. Generalized Perrin–Riou conjecture. One has

$$\kappa=\mathcal{L}^{\rm an}_\Sigma R^{\rm Boc},\qquad R^{\rm Boc}\ne0$$

in

$$E(\mathbb{Q})\otimes_{\mathbb{Z}}\left(I_\infty^{r-1}/I_\infty^r\otimes_{\mathbb{Z}_p}\mathbb{C}_p\right).$$

This is the central conjectural relation between a Kato zeta-element derivative and the leading $L$-value; the paper explains that the nonvanishing prediction is supported by descriptions in terms of classical $p$-adic regulators, but the conjecture is not established in general.