On Kato's local $\epsilon$-isomorphism conjecture for rank-one Iwasawa modules

Abstract

This paper contains a complete proof of Fukaya and Kato’s $ϵ$-isomorphism conjecture for invertible $\Lambda$-modules (the case of $V=V_0\left(r\right)$, where $V_0$ is unramified of dimension $1$). Our results rely heavily on Kato’s proof, in an unpublished set of lecture notes, of (commutative) $ϵ$-isomorphisms for one-dimensional representations of $G_{{\mathbb{Q}}_p}$, but apart from fixing some sign ambiguities in Kato’s notes, we use the theory of $\left(\varphi ,\Gamma \right)$-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of Fukuya and Kato and extend it to the (slightly noncommutative) semiglobal setting. Finally we discuss some direct applications concerning the Iwasawa theory of CM elliptic curves, in particular the local Iwasawa Main Conjecture for CM elliptic curves $E$ over the extension of ${\mathbb{Q}}_p$ which trivialises the $p$-power division points $E\left(p\right)$ of $E$. In this sense the paper is complimentary to our work with Bouganis (Asian J. Math. 14:3 (2010), 385–416) on noncommutative Main Conjectures for CM elliptic curves.