Generalised Kato classes on CM elliptic curves of rank 2

Abstract

Let $E/\mathbf{Q}$ be a CM elliptic curve and let $p\geq 5$ be a prime of good ordinary reduction for $E$. Suppose that $L(E,s)$ vanishes at $s=1$ and has sign $+1$ in its functional equation, so in particular ${\rm ord}_{s=1}L(E,s)\geq 2$. In this paper we slightly modify a construction of Darmon--Rotger to define a generalised Kato class $κ_p\in{\rm Sel}(\mathbf{Q},V_pE)$, and prove the following rank two analogue of Kolyvagin's result: \[ κ_p\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2. \] Conversely, when ${\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2$ we show that $κ_p\neq 0$ if and only if the restriction map \[ {\rm Sel}(\mathbf{Q},V_pE)\rightarrow E(\mathbf{Q}_p)\hat{\otimes}\mathbf{Q}_p \] is nonzero. The proof of these results, which extend and strenghten similar results of the author with Hsieh in the non-CM case, exploit a new link between the nonvanishing of generalised Kato classes and a main conjecture in anticyclotomic Iwasawa theory.