On the image of noncommutative local reciprocity map

Abstract

First steps in the direction of an arithmetic noncommutative local class field theory were described in [2] as an attempt to find an arithmetic generalization of the classical abelian class field theory; see [3] for an exposition of its main features. In particular, [2] clarified and simplified the metabelian local class field theory of H. Koch and E. de Shalit [7], [8]. In the noncommutative local class field theory [2] a direct arithmetic description of Galois extensions of a fixed local field F is given by means of noncommutative reciprocity maps between the Galois group Gal(L/F ) of a totally ramified arithmetically profinite Galois extension L/F and a certain subquotient of formal power series in one variable over the algebraic closure of the residue field of F (which, more precisely, is the completion of the maximal unramified extension of the field of norms of L/F ). One of the reciprocity maps (see below for definitions) is