On the local Artin conductor fArtin (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

Abstract

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldKk. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wherenG is the break in the upper ramification filtration ofG = Gal(E/K) defined by\(G^{n_G } \ne G^{n_{G + \delta } } = 1,\forall \delta \in \mathbb{R}_{_ \ne ^ > 0} \). Next, we study the basic properties of the idealf(E/K) inOk in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]).