On Kato’s local 𝜖-isomorphism conjecture for rank-one Iwasawa modules

This paper contains a complete proof of Fukaya and Kato’s $ϵ$ -isomorphism conjecture for invertible $Λ$ -modules (the case of $V = V_{0} (r)$ , 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 Our results rely heavily on Kato’s proof, in an unpublished set of lecture notes, of (commutative) $ϵ$ -isomorphisms for one-dimensional representations of $G_{ℚ_{p}}$ , but apart from fixing some sign ambiguities in Kato’s notes, we use the theory of $(ϕ, Γ)$ -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 $ℚ_{p}$ which trivialises the $p$ -power division points $E (p)$ 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.

Mathematical Subject Classification 2010

Primary: 11R23

Secondary: 11F80, 11R42, 11S40, 11G07, 11G15

Milestones

Received: 17 May 2012

Revised: 21 January 2013

Accepted: 23 February 2013

Published: 18 January 2014

Authors

Otmar Venjakob
Mathematisches Institut Universität Heidelberg Im Neuenheimer Feld 288 D-69120 Heidelberg Germany

Vol. 7, No. 10, 2013

Otmar Venjakob

Abstract

Mathematical Subject Classification 2010

Milestones

Authors