Adele groups and Siegel-Tamagawa formulas
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.
scientific article; zbMATH DE number 859201
scientific article