Non-abelian local reciprocity law
type: Item
instance of: scholarly article
title: Non-abelian local reciprocity law
zbMATH Open document ID: 1205.11128
author: Erol Serbest
author: Kazım İlhan İkeda
published in: Manuscripta Mathematica
publication date: 2010-04-30T00:00:00Z
review text: This paper is the continuation of previous work of the authors [Algebra Anal. 20, No. 3, 112--162 (2008); translation in St. Petersburg Math. J. 20, No. 3, 407--445 (2009; Zbl 1206.11138); Algebra Anal. 20, No. 4, 118--159 (2008); translation in St. Petersburg Math. J. 20, No. 4, 593--624 (2009; Zbl 1206.11140)]. Let \(K\) be a field complete under a discrete valuation with finite residue field \(k\) of \(q=p^r\) elements. Fix an extension \(\varphi\) of the Frobenius automorphism of the maximal unramified extension \(K^{\text{un}}\) of \(K\) to the separable closure \(K^{\text{sep}}\) of \(K\). The authors construct the non-abelian reciprocity map \(\Phi_ K^{(\varphi)}\) for \(K\), which is an isomorphism of the absolute Galois group \(G_ K\) of \(K\) onto a topological group \(\Delta_{K,Y} ^ {(\varphi)}\) described in Section 6 of the paper. This is achieved under the assumption that the multiplicative group \(\mu_ p(K^{\text{sep}})\) of the \(p\)--th roots of unity of \(K^ {\text{sep}}\) are already contained in \(K\). It is also sketched the construction of the non-abelian local reciprocity map \(\Phi_ K^{(\varphi)}\) of \(K\) when \(K\) does not necessarily satisfy that \(\mu_ p(K^ {\text{sep}}) \subseteq K\) via non-abelian Schreier theory. After reviewing the main results on Fesenko reciprocity map and studying APF Galois extensions \(\Gamma _ d^ {(n)}/K\), the authors give the construction of \(\Phi_ K ^ {(\varphi)}\) in terms of the generalized reciprocity maps \(\Phi_ {\Gamma_ d^ {(n)}/K}^ {(\varphi)}\) and study the basic properties of this map. In the final section it is sketched the construction of the non-abelian local reciprocity map \(\Phi_ K^ {(\varphi_ 0)}\) for any local field by generating the non-abelian reciprocity map \(\Phi _ {K_ 0}^{(\varphi_ 0)}\) of \(K_ 0\) and the local Artin map \(\roman{Art}_ {K_ 0/K}\) of \(K_ 0/K\), where \(\zeta _ p\in K^ {\text{sep}}\) is a fixed primitive root of unity and \(K_ 0=K(\zeta _ p)\).
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 11S31
zbMATH DE Number: 5702983
zbMATH Keywords: Reciprocity law
zbMATH Keywords: Langlands program
zbMATH Keywords: non-abelian class field theory
zbMATH Keywords: Fesenko reciprocity map
zbMATH Keywords: totally ramified arithmetically profinite (APF) Galois extension
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: s00229-010-0336-6
OpenAlex ID: W2055511070
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DOI: 10.1007/S00229-010-0336-6
description: scientific article; zbMATH DE number 5702983