On the Local Artin Conductor $mathfrak{f}$ Artin (𝜒) of a Character 𝜒 of Gal(𝐸/𝐾) - II: Main Results for the Metabelian Case

[摘要] This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. Let 𝐾 be a local field with finite residue class field $k_K$. We first define (cf. Definition 2.4) the conductor $mathfrak{f}(E/K)$ of an arbitrary finite Galois extension 𝐸/𝐾 in the sense of non-abelian local class field theory as$$mathfrak{f}(E/K)=mathfrak{p}_K^{[[n_G]]+1},$$where $n_G$ is the break in the upper ramification filtration of $G = mathrm{Gal}(E/K)$ defined by $G^{n_G} ≠ G^{n_G+𝛿} = 1, forall𝛿 in mathbb{R}_{gneq 0}$. Next, we study the basic properties of the ideal $mathfrak{f}(E/K)$ 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 $a_G:G → mathbb{C}$ of $G:=mathrm{Gal}(E/K)$ and Artin representations $A_G:G → GL(V)$ corresponding to $a_G:G → mathbb{C}$, we prove that (Proposition 3.2 and Corollary 3.5)$$mathfrak{f}mathrm{Artin}(𝜒_ρ)=mathfrak{p}_K^{dim_mathbb{C}(V)[n_{G/ker(ρ)}+1]},$$where 𝜒ρ : $G → mathbb{C}$ is the character associated to an irreducible representation $ρ : G → GL(V)$ of $G (ext{over} mathbb{C})$. The first main result (Theorem 1.2) of the paper states that, if in particular, $ρ : G → GL(V)$ is an irreducible representation of $G (ext{over} mathbb{C})$ with metabelian image, then$$mathfrak{f}mathrm{Artin}(𝜒_ρ)=mathfrak{p}_K^{[E^{ker(ρ)^ullet}:K](n_{G/ker(ρ)}+1)},$$where $mathrm{Gal}(E^{ker(ρ)}/E^{ker(ρ)^ullet})$ is any maximal abelian normal subgroup of $mathrm{Gal}(E^{ker(ρ)}/K)$ containing $mathrm{Gal}(E^{ker(ρ)}/K)'$, and the break $n_{G/ker(ρ)}$ in the upper ramification filtration of $G/ker(ρ)$ 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 $mathbb{C}$ (Problem 1.3). More precisely, we prove in Theorem 1.4 that if 𝐸/𝐾 is a metabelian extension with Galois group 𝐺, thenegin{align*}A_Gsimeq & sumlimits_N[(E^N)^ullet :K](n_{G/N}+1)\ & ×sumlimits_{[𝜔]simin𝜈_N}mathrm{Ind}_{𝜋_N^{-1}((G/N)^ullet)}^Gleft(𝜔circ𝜋_N|_{𝜋_N^{-1}((G/N)^ullet)}ight),end{align*}where 𝑁 runs over all normal subgroups of 𝐺, and for such an $N, 𝜈_N$ denotes the collection of all ∼-equivalence classes $[𝜔]_sim$, where `∼' denotes the equivalence relation on the set of all representations $𝜔 :(G/N)^ullet→mathbb{C}^×$ satisfying the conditions$$mathrm{Inert}(𝜔)={𝛿in G/N :𝜔_𝛿=𝜔}=(G/N)^ullet$$and$$igcaplimits_𝛿 ker(𝜔_𝛿)=langle 1_{G/N}angle,$$where 𝛿 runs over $mathcal{R}((G/N)^ulletackslash(G/N))$, a fixed given complete system of representatives of $(G/N)^ulletackslash(G/N)$, by declaring that $𝜔_1 sim 𝜔_2$ if and only if 𝜔1 = 𝜔2,𝛿 for some $𝛿 in mathcal{R}((G/N)^ulletackslash(G/N))$.Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.

[学科分类] 数学（综合）

[关键词] Local fields;higher-ramification groups;local Artin conductor;metabelian local class field theory;non-abelian local class field theory;local Langlands correspondence for $GL(n)$.