Integral models of algebraic tori over fields of algebraic numbers
type: Item
instance of: scholarly article
title: Integral models of algebraic tori over fields of algebraic numbers
zbMATH Open document ID: 1386.14169
published in: Journal of Mathematical Sciences (New York)
publication date: 2017-01-23T00:00:00Z
review text: The paper is concerned with integral models of algebraic tori over algebraic number fields together with the comperison of properties of the models. It depends on the scheme which represent the (Néron, Voskresenskií) model under consideration. The author offers a setting for the Néron model of an algebraic torus, namely the standard or the canonical integral models of the torus. The basic result concerns the Néron model of a maximal algebraic torus unaffected in a semisimple algebraic group. Integral models of algebraic tori over Dedekind schemes are presented in the book by \textit{S. Bosch} et al. [Néron models. Berlin etc.: Springer-Verlag (1990; Zbl 0705.14001)], in papers by \textit{V. E. Voskresenskij} [Math. Notes 44, No. 3--4, 651--655 (1988; Zbl 0699.20037); translation from Mat. Zametki 44, No. 3, 309--318 (1988)], by \textit{S. Bosch} and \textit{Q. Liu} [Manuscr. Math. 98, No. 3, 275--293 (1999; Zbl 0934.14029)] and by \textit{S. Yu. Popov} [Vestn. Samar. Gos. Univ., Mat. Mekh. Fiz. Khim. Biol. 2001, No. 4(22), 85--108 (2001; Zbl 1072.14529)]. Let \(T\) be an algebraic torus over a field \(k\), \([L:k]\) its splitting field, \(G = {\mathcal Gal}(L:k)\). Section 1 introduces \(G\)-module \(\hat T\) of rational characters of \(T\), its coordinate ring \(k[T] = L[\hat T]^G\) and affine realization of \(T\). Section 2 concerns definitions of (Voskresenskií, canonical integral, standard integral) models of algebraic tori. Section 3 gives some indications on algorithms of constracting of Néron models of algebraic torus over global field of characteristic 0 and over its localizations. Section 4 presents properties of standard integral model over a number field and of the corresponding models over local fields. The last section contains interesting observations on a family of algebraic tori of a special structure and concludes with an explicit representation of the Néron model of a maximal algebraic torus unaffected in a semisimple algebraic group with the root …
reviewed by: Nikolaj M. Glazunov
Mathematics Subject Classification ID: 14L10
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 6676853
zbMATH Keywords: integral model of algebraic group
zbMATH Keywords: algebraic torus
zbMATH Keywords: algebraic number field
zbMATH Keywords: Néron model
zbMATH Keywords: Voskresenskií model
author: Mikhail Vladimirovich Grekhov
MaRDI profile type: Publication
full work available at URL: s10958-016-3117-2
OpenAlex ID: W2538026592
cites work: Q4210019
cites work: Q4002684
cites work: Néron Models
cites work: Congruences of Néron models for tori and the Artin conductor
cites work: Q5532660
cites work: Q4436146
cites work: Q4831652
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q3720687
cites work: Special fibers of Néron models and wild ramification
DOI: 10.1007/S10958-016-3117-2
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description: scientific article; zbMATH DE number 6676853