Non-abelian class field theory for arithmetic surfaces
type: Item
instance of: scholarly article
title: Non-abelian class field theory for arithmetic surfaces
zbMATH Open document ID: 1080.14029
author: Walter Hofmann
author: Götz Wiesend
published in: Mathematische Zeitschrift
publication date: 2005-05-04T00:00:00Z
review text: Let \(K\) be a number field with ring of integers \({\mathcal O}\), let \(X\) be a smooth projective curve with function field \(F\) and field of constants \(K\) and let \(\pi:{\mathcal X}\to \text{Spec}({\mathcal O})\) be an integral, normal proper and flat model; the scheme \({\mathcal X}\) is called an arithmetic surface. Let \(U\subset X\) be an open subscheme which is assumed to be regular. The main result of this paper gives a description of the étale fundamental group \(\pi_1({\mathcal U})\) of \({\mathcal U}\), and the projective limit of \(\pi_1({\mathcal U})\) over all such \({\mathcal U}\) is used to give a description of the absolute Galois group of \(F\). These results are applied to develop abelian class field theory for arithmetic surfaces using \(K_0\)- and \(K_1\)- groups of local and global fields. In the last section the authors give a comparison of their approach with the class field theory of \textit{K. Kato} and \textit{S. Saito} [in: Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 103--152 (1983; Zbl 0544.12011)].
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11G45
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 2165577
reviewed by: Hans Opolka
MaRDI profile type: Publication
full work available at URL: s00209-004-0751-z
OpenAlex ID: W2004556131
cites work: Q4692374
cites work: Algebraic K-theory and classfield theory for arithmetic surfaces
cites work: Unramified class field theory of arithmetical surfaces
cites work: Q4722183
cites work: Q3871006
cites work: Class field theory for curves over local fields
DOI: 10.1007/S00209-004-0751-Z
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description: scientific article; zbMATH DE number 2165577