Nonvanishing of generalised Kato classes and Iwasawa main conjectures

Abstract

A construction due to Darmon--Rotger gives rise to generalised Kato classes ${\kappa }_p(E)$ in the $p$-adic Selmer group $\mathrm{Sel}(\mathbf{Q},V_{p}E)$ 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 ${\kappa }_p(E)\ne 0$ precisely when $\mathrm{Sel}(\mathbf{Q},V_{p}E)$ 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 \[ κ_p(E)\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2 \] established in op. cit., and show that the converse implication holds if \emph{and only if} the restriction map ${\mathrm{loc}}_p:\mathrm{Sel}(\mathbf{Q},V_{p}E)\to E({\mathbf{Q}}_p)\hat{\otimes }{\mathbf{Q}}_p$ 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.

arXiv comment20 pages, submitted to the proceedings in celebration of Massimo Bertolini's 60th birthday