Nonvanishing of generalised Kato classes and Iwasawa main conjectures

A construction due to Darmon–Rotger gives rise to generalised Kato classes ${\kappa }_p(E)$ in the $p$-adic Selmer group $\mathrm{S}\mathrm{e}\mathrm{l}(\mathbf{Q},V_{p}E)$ of elliptic curves $E\mathrm{/}\mathbf{Q}$ of positive even analytic rank, where $p>3$ is any prime of good ordinary reduction for $E$. In some cases, their conjectured that ${\kappa }_p(E)\mathrm{\ne }0$ precisely when $\mathrm{S}\mathrm{e}\mathrm{l}(\mathbf{Q},V_{p}E)$ is two-dimensional. The first cases of this conjecture were obtained in a joint work of the author with M.-L. Hsieh. In this note we give a new proof of the implication $${\kappa }_p(E)\mathrm{\ne }0⟹{\mathrm{d}\mathrm{i}\mathrm{m}}_{{\mathbf{Q}}_p}\mathrm{S}\mathrm{e}\mathrm{l}(\mathbf{Q},V_{p}E)=2$$ established in op. cit., and show that the converse implication holds if \emph{and only if} the restriction map ${\mathrm{l}\mathrm{o}\mathrm{c}}_p:\mathrm{S}\mathrm{e}\mathrm{l}(\mathbf{Q},V_{p}E)\to E({\mathbf{Q}}_p)\widehat{\otimes }{\mathbf{Q}}_p$ is nonzero. The present approach is an adaptation to the non-CM case of the method introduced by the author in the case of CM elliptic curves.