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Algebraic groups and number theory. Transl. from the Russian by Rachel Rowen. (English) Zbl 0841.20046
Pure and Applied Mathematics, 139. Boston, MA: Academic Press. xi, 614 p. (1994).
The arithmetic theory of algebraic groups combines elements of group theory, algebraic number theory and of algebraic geometry. This part of mathematics arises from the arithmetic theory of quadratic forms beginning with Gauss and Hermite, Minkowski, Hasse and Siegel and also from the theory of discrete subgroups of Lie groups in connection with the theory of automorphic functions (Riemann, Klein, PoincarĂ©). The publication of the book sums up and also intends to unify different journal articles of the last decade as well as some new material due to the authors (cf. chap. 9) and it is tried to give a nearly self contained exposition. Some well known assertions are presented with new proofs and a number of open problems is mentioned. At least for the reviewer the most important parts of the book concerns the Galois cohomology of algebraic groups over local and over number fields (including the proof of the Hasse principle for simply connected semisimple groups of type \(E_8\), firstly published by Chernousov) and the weak and strong approximation properties in algebraic varieties (Chap. 7) and the normal subgroup structure of groups of rational points of algebraic groups (Chap. 9). It is nearly impossible to embrace all main results in the arithmetic theory of algebraic groups in one textbook, so some results (for instance strong approximation) in the case of global fields of positive characteristic are only mentioned, since the proofs in that case require considerable modifications. Other topics like Margulis’ and Mostow-Margulis rigidity theorems are not the subject of this monograph (for this the nice book of R. J. Zimmer on “Ergodic Theory and Semisimple Groups” [Basel 1984; Zbl 0571.58015] is recommended). As the authors also mentioned in the introduction the structure and the exposition of the book was strongly influenced by the first author’s survey article “Arithmetic theory of algebraic groups” which appeared in 1982 [in Usp. Mat. Nauk 37, No. 3(225), 3-54 (1982; Zbl 0502.20025)].

MSC:
20G30 Linear algebraic groups over global fields and their integers
20-02 Research exposition (monographs, survey articles) pertaining to group theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11E72 Galois cohomology of linear algebraic groups
11Rxx Algebraic number theory: global fields
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