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 ${\rm Sel}(\mathbf{Q},V_pE)$ of elliptic curves $E/\mathbf{Q}$ of positive even analytic rank, where $p>3$ is any prime of good ordinary reduction for $E$. In some cases, their conjectured that $\kappa_p(E)\neq 0$ precisely when ${\rm Sel}(\mathbf{Q},V_pE)$ 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 $$\kappa_p(E)\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2$$ established in op. cit., and show that the converse implication holds if \emph{and only if} the restriction map ${\rm loc}_p:{\rm Sel}(\mathbf{Q},V_pE)\rightarrow E(\mathbf{Q}_p)\hat\otimes\mathbf{Q}_p$ 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.