Adele groups and Siegel-Tamagawa formulas
type: Item
instance of: scholarly article
title: Adele groups and Siegel-Tamagawa formulas
zbMATH Open document ID: 0892.20027
DOI: 10.1007/BF02366354
author: V. E. Voskresenskiĭ
published in: Journal of Mathematical Sciences (New York)
publication date: 1996-04-23T00:00:00Z
review text: The paper is a survey in arithmetic of linear algebraic groups focussed on the notion of Tamagawa number. The author discusses such important subjects as adele groups and Haar measures, integer structures in linear algebraic groups, Artin \(L\)-functions. He gives examples of computation of Tamagawa numbers and derives formulae for computation of local volumes. Langlands' method for computing the Tamagawa number of a split semisimple group \(G\) is presented in detail, with heavy use of harmonic analysis on semisimple groups (Eisenstein series, Gindikin-Karpelevich integrals). The author gives an alternative (more elementary) computation in the case \(G=\text{SL}(n)\). He discusses the Minkowski-Siegel-Tamagawa formula and its relation to classical class number problems and computes the weight of a genus (for odd and even lattices) which leads to classical exact formulae for the representation of a number as a sum of \(n\) squares. Yet another subject is the computation of the class group of an algebraic torus and its relationship with classical Hasse's index-formulae.
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 859201
zbMATH Keywords: linear algebraic groups
zbMATH Keywords: Tamagawa numbers
zbMATH Keywords: class numbers
zbMATH Keywords: split semisimple groups
zbMATH Keywords: Hasse index formula
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02366354
OpenAlex ID: W1993501913
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description: scientific article; zbMATH DE number 859201