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2013 On Kato's local $\epsilon$-isomorphism conjecture for rank-one Iwasawa modules
Otmar Venjakob
Algebra Number Theory 7(10): 2369-2416 (2013). DOI: 10.2140/ant.2013.7.2369


This paper contains a complete proof of Fukaya and Kato’s ϵ-isomorphism conjecture for invertible Λ-modules (the case of V=V0(r), where V0 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 Our results rely heavily on Kato’s proof, in an unpublished set of lecture notes, of (commutative) ϵ-isomorphisms for one-dimensional representations of Gp, but apart from fixing some sign ambiguities in Kato’s notes,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 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.


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Otmar Venjakob. "On Kato's local $\epsilon$-isomorphism conjecture for rank-one Iwasawa modules." Algebra Number Theory 7 (10) 2369 - 2416, 2013.


Received: 17 May 2012; Revised: 21 January 2013; Accepted: 23 February 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1305.11095
MathSciNet: MR3194646
Digital Object Identifier: 10.2140/ant.2013.7.2369

Primary: 11R23
Secondary: 11F80 , 11G07 , 11G15 , 11R42 , 11S40

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.7 • No. 10 • 2013
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