On the local Artin conductor f_{Artin} (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case
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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 fieldK_{k}. 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 wheren_{G} 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) inO_{k} in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]).
After reviewing the Artin charactera_{G} : G → ℂ ofG := Gal(E/K) and Artin representationsA_{g} G → G →GL(V) corresponding toa_{G} : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5)$$\mathfrak{f}_{Artin} (\chi _\rho ) = \mathfrak{p}_K^{dim_\mathbb{C} (V)\left[ {n_{G/\ker (\rho )} + 1} \right]} $$ where Χ_{gr}: G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then$$\mathfrak{f}_{Artin} (\chi _\rho ) = \mathfrak{p}_K^{[E^{\ker (\rho )^ \cdot :K} ](n_{G/\ker (\rho )} + 1)} $$ where Gal(E^{ker(ρ)}/E^{ker(ρ)•}) is any maximal abelian normal subgroup of Gal(E^{ker(ρ)}/K) containing 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} ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then$$A_G \simeq \sum\limits_N {\left[ {(E^N )^ \bullet :K} \right]\left( {n_{G/N} + 1} \right) \times \sum\limits_{\left[ \infty \right] \sim \in V_N } {Ind_{\pi _N^{ - 1} ((G/N)^ \bullet )}^G (\omega \circ \pi _N |_{\pi _N^{ - 1} ((G/N) \bullet )} )} } $$Kâzim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V_{n} denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^{•} → ℂ^{Χ} satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_{δ}} = ω =(G/N) and$$\bigcap\limits_\delta {\ker (\omega _\delta ) = \left\langle {1_{G/N} } \right\rangle } $$ where δ runs over R((G/N)^{•}/(G/N)), a fixed given complete system of representatives of (G/N)^{•}/(G/N), by declaring that ω_{1} ∼ ω_{2} if and only if ω_{1}= ω_{2,δ} for some δ ∈ R((G/N)^{•}/(G/N)).
Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.
Kâzim Ilhan Ikeda^{1} ^{}
Volume 131, 2021
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