On the nonvanishing of generalised Kato classes for elliptic curves of rank 2

Abstract

AbstractLet$E/\mathbf {Q}$be an elliptic curve and$p>3$be a good ordinary prime forEand assume that$L(E,1)=0$with root number$+1$(so$\text {ord}_{s=1}L(E,s)\geqslant 2$). A construction of Darmon–Rotger attaches toEand an auxiliary weight 1 cuspidal eigenformgsuch that$L(E,\text {ad}^{0}(g),1)\neq 0$, a Selmer class$\kappa _{p}\in \text {Sel}(\mathbf {Q},V_{p}E)$, and they conjectured the equivalence$$ \begin{align*} \kappa_{p}\neq 0\quad\Longleftrightarrow\quad{\textrm{dim}}_{{\mathbf{Q}}_{p}}\textrm{Sel}(\mathbf{Q},V_{p}E)=2. \end{align*} $$In this article, we prove the first cases on Darmon–Rotger’s conjecture when the auxiliary eigenformghas complex multiplication. In particular, this provides a new construction of nontrivial Selmer classes for elliptic curves of rank 2.