This paper contains a complete proof of Fukaya and Kato’s
-isomorphism conjecture
for invertible
-modules
(the case of
, where
is unramified
of dimension ).
Our results rely heavily on Kato’s proof, in an unpublished set of lecture notes, of (commutative)
-isomorphisms for one-dimensional
representations of
,
but apart from fixing some sign ambiguities in Kato’s notes, we use the theory of
-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
over the extension
of
which trivialises
the
-power
division points
of
. 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.
