Birational geometry and arithmetic of linear algebraic groups. III (Q2761462)

[For part I see ibid. 1997, No. 2(4), 18--98 (1997; Zbl 1029.14015).]NEWLINENEWLINEThis is the third part of a survey concerning the geometry and the arithmetic of linear algebraic groups. This part has just one chapter divided in two sections: Haar measures on the group of adèles, and Minkowski-Siegel-Tamagawa formula. The first section has a general character introducing the basic concepts such as: products of local measures, Tamagawa measures, Tamagawa numbers and their computation for algebraic tori or for semisimple algebraic groups (following Langlands' method). In the second section the author proves the Minkowski-Siegel-Tamagawa formula and applies it in a number of important examples. For instance he gives a detailed calculation of the local \(p\)-adic volumes which are necessary for obtaining very explicit formulas.