On Kato's local epsilon-isomorphism Conjecture for rank one Iwasawa modules

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This paper contains a complete proof of Fukaya's and Kato's epsilon-isomorphism conjecture in [23] for invertible Λ-modules (the case of V = V₀(r) where V₀ is unramified of dimension 1). Our results rely heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for one dimensional representations of GQₚ in [27], but apart from fixing some sign-ambiguities in (loc.cit.) 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 [23] and extend it to the (slightly non-commutative) semi-global 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 Qₚ which trivialises the p-power division points E(p) of E. In this sense the paper is complimentary to the joint work [7] on noncommutative Main Conjectures for CM elliptic curves.