Algebraic Groups and Number Theory: Volume 1: (Series Number 205 Cambridge Studies in Advanced Mathematics)

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.

Table of Contents:
1. Algebraic number theory; 2. Algebraic groups; 3. Algebraic groups over locally compact fields; 4. Arithmetic groups and reduction theory; 5. Adeles; Bibliography; Index.

About the Author :
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.

Review :
'The original English version of the book 'Algebraic Groups and Number Theory' by Platonov and Rapinchuk was a go to reference for graduate students and senior researchers alike working in areas of arithmetic and algebraic groups, discrete subgroups of Lie groups, and connections with number theory. The second edition, which will be split into two volumes, and also co-authored with I. Rapinchuk, is a welcome and timely update to the original. The first volume of the second edition, consists of an update to chapters 1-5 of the original with an additional section 4.9 to include new material on the structure of extensions of arithmetic groups. There is no doubt in my mind that this first volume of the second edition will again take on the role of a go to text for those working in an area of huge ongoing interest and importance, and be at the forefront training new generations of mathematicians working in the areas of arithmetic and algebraic groups, discrete subgroups of Lie groups, and connections with number theory.' Alan Reid, Rice University 'The arithmetic theory of algebraic groups is a beautiful area of mathematics: a crossroad of number theory, groups, geometry, representation theory, and more. Not surprisingly it attracted some of the greatest mathematicians of the last few generations. The first edition of the book 'Algebraic Groups and Number Theory' by Vladimir Platonov and Andrei Rapinchuk which came out in the early 90s has quickly become the standard reference of the field. It presents in a clear way several deep topics. The book was one of the reasons the area attracted more researchers and expanded to new directions. This made an updated version much needed. The original authors and Igor Rapinchuk should be thanked by the mathematical community for carrying out this monumental job.' Alex Lubotzky, Hebrew University of Jerusalem