Algebraic Groups and Number Theory: Volume 1

The first volume of a two-volume book offering a comprehensive account of the arithmetic theory of algebraic groups.

Vladimir Platonov (Author), Andrei Rapinchuk (Author), Igor Rapinchuk (Author)

9780521113618, Cambridge University Press

Hardback, published 31 July 2023

381 pages
28 x 19 x 2.6 cm, 0.768 kg

'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

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.

1. Algebraic number theory
2. Algebraic groups
3. Algebraic groups over locally compact fields
4. Arithmetic groups and reduction theory
5. Adeles
References
Index.

Subject Areas: Number theory [PBH]