Generalised Kato classes on CM elliptic curves of rank 2

TL;DR

This paper constructs a generalized Kato class for CM elliptic curves of rank 2 and establishes a criterion linking its nonvanishing to the Selmer group's rank, extending Kolyvagin's results.

Contribution

It introduces a modified construction of generalized Kato classes for CM elliptic curves and proves a rank two analogue of Kolyvagin's theorem connecting these classes to Selmer group dimensions.

Findings

Nonvanishing of the generalized Kato class implies rank 2 Selmer group.

Rank 2 Selmer group dimension implies nonvanishing of the class.

The results extend previous non-CM cases and connect to Iwasawa theory conjectures.

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 $\kappa_p\in{\rm Sel}(\mathbf{Q},V_pE)$, and prove the following rank two analogue of Kolyvagin's result: \[ \kappa_p\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2. \] Conversely, when ${\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2$ we show that $\kappa_p\neq 0$ 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.