The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and improvements, as well as new material covering more recent developments. Volume I begins with chapters covering background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic subgroups and reduction theory in both the real and adelic settings. Volume I includes new material on groups with bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for researchers in number theory, algebraic geometry, and related areas.

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Vladimir Platonov is Principal Research Fellow at the Steklov Mathematical Institute and the Scientific Research Institute for System Analysis of the Russian Academy of Sciences. He has made fundamental contributions to the theory of algebraic groups, including the resolution of the Kneser–Tits problem, a criterion for strong approximation in algebraic groups, and the analysis of the rationality of group varieties. His recent work brought about major progress on the problem of periodicity of continued fractions in hyperelliptic fields and investigation of torsion in the Jacobian varieties of hyperelliptic curves. A recipient of the Lenin Prize (1978) and the Chebyshev Gold Medal for outstanding results in mathematics (2022), he is currently an academician of the Russian Academy of Sciences and of the National Academy of Sciences of Belarus, and a member of the Indian National Academy of Sciences.

Andrei Rapinchuk is McConnell-Bernard Professor of Mathematics at the University of Virginia. His contributions to the arithmetic theory of algebraic groups include a variety of results concerning the normal subgroup structure of the groups of rational points of algebraic groups, the congruence subgroup and metaplectic problems, and different aspects of the local-global principle. He has also applied the theory of arithmetic groups to investigate isospectral locally symmetric spaces.

Igor Rapinchuk is Associate Professor of Mathematics at Michigan State University. His current research deals mainly with the emerging arithmetic theory of algebraic groups over higher-dimensional fields, focusing on finiteness properties of groups with good reduction, local-global principles, and abstract homomorphisms.

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