Algebra & Number Theory

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

Otmar Venjakob

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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 Our results rely heavily on Kato’s proof, in an unpublished set of lecture notes, of (commutative) ϵ-isomorphisms for one-dimensional representations of Gp, but apart from fixing some sign ambiguities in Kato’s notes,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.

Article information

Algebra Number Theory, Volume 7, Number 10 (2013), 2369-2416.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory
Secondary: 11F80: Galois representations 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27] 11G07: Elliptic curves over local fields [See also 14G20, 14H52] 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]


Venjakob, Otmar. On Kato's local $\epsilon$-isomorphism conjecture for rank-one Iwasawa modules. Algebra Number Theory 7 (2013), no. 10, 2369--2416. doi:10.2140/ant.2013.7.2369.

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