On the local Artin conductor fArtin (?) of a character ? of Gal(E/K) -: II

Abstract

This paper which is a continuation of [21, is essentially expository in nature, although some new results are presented. Let K be a local field with finite residue class field kappa(K) - We first define (cf. Definition 2.4) the conductor f (E/K) of an arbitrary finite Galois extension E/K in the sense of non-abelian local class field theory as [GRAPHICS] where n(G) is the break in the upper ramification filtration of G = Gal(E/K) defined by G(nG) not equal G(nG+delta) = 1, For Alldelta epsilon R (greater than or equal to 0). Next, we study the basic properties of the ideal f (E/K) in O-K in case E/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin character a(G) : G --> C of G := Gal(E/K) and Artin representations A(G) : G --> GL(V) corresponding to a(G) : G --> C, we prove that (Proposition 3.2 and. Corollary 3.5) [GRAPHICS] where chi(rho) : G ---> C is the character associated to an irreducible representation rho G --> GL(V) of G (over C). The first main result (Theorem, 1.2) of the paper states that, if in particular, rho : G --> GL(V) is an irreducible representation of G (over C) with metabelian image, then [GRAPHICS] where Gal (E-ker(rho)/ E-ker(rho).) is any maximal abelian normal subgroup of Gal (E-ker(rho) / K) containing Gal (E-ker(rho) / K)', and the break n(G/ker(rho)) in the upper ramification filtration of G/ker(rho) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji's theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a 'natural' A(G) of G over C (Problem 1.3). More precisely, we prove in Theorem 1.4 that if E/K is a metabelian extension with Galois group G, then [GRAPHICS] where N runs over all normal subgroups of G, and for such an N, nu(N) denotes the collection of all similar to-equivalence classes [omega] similar to, where 'similar to' denotes the equivalence relation on the set of all representations omega : (G/N)(.) --> C-x satisfying the conditions inert (omega) = [delta epsilon G / N : omegadelta = omega] = (G/N)(.) and [GRAPHICS] where delta runs over R((G/N)(.)\(G/N)), a fixed given complete system of representatives of (G/N)(.)\(G/N), by declaring that omega(1) similar to omega(2) if and only if omega(1) = omega(2,delta) for some delta epsilon R((G/N)(.)\(G/N)). Finally, we conclude our paper with certain remarks on Problem 1. 1 and Problem 1.3.