Bruhat–Tits Theory
A New Approach

Comprehensive treatment of Bruhat–Tits theory for graduate students and researchers in number theory, representation theory, and algebraic geometry.

Tasho Kaletha (Author), Gopal Prasad (Author)

9781108831963, Cambridge University Press

Hardback, published 26 January 2023

700 pages
23.5 x 15.8 x 5 cm, 1.26 kg

Bruhat–Tits theory is an important topic in number theory, representation theory, harmonic analysis, and algebraic geometry. This book gives the first comprehensive treatment of this theory over discretely valued Henselian fields. It can serve both as a reference for researchers in the field and as a thorough introduction for graduate students and early career mathematicians. Part I of the book gives a review of the relevant background material, touching upon Lie theory, metric geometry, algebraic groups, and integral models. Part II gives a complete, detailed, and motivated treatment of the core theory as well as an axiomatic summary of Bruhat–Tits theory that suffices for the main applications. Part III treats modern topics that have become important in current research. Part IV provides a few sample applications of the theory. The appendices contain further details on the topic of integral models, including a detailed study of integral models.

Introduction
Part I. Background and Review: 1. Affine root systems and abstract buildings
2. Algebraic groups
Part II. Bruhat–Tits theory: 3. Examples: Quasi-split groups of rank 1
4. Overview and summary of Bruhat–Tits theory
5. Bruhat, Cartan, and Iwasawa decompositions
6. The apartment
7. The Bruhat–Tits building for a valuation of the root datum
8. Integral models
9. Unramified descent
Part III. Additional Developments: 10. Residue field f of dimension ? 1
11. The buildings of classical groups via lattice chains
12. Component groups of integral models
13. Finite group actions and tamely ramified descent
14. Moy–Prasad filtrations
15. Functorial properties
Part IV. Applications: 16. Classification of maximal unramified tori (d'après DeBacker)
17. Classification of tamely ramified maximal tori
18. The volume formula
Part V. Appendices: A. Operations on integral models
B. Integral models of tori
C. Integral models of root subgroups
References
Index.

Subject Areas: Algebraic topology [PBPD], Number theory [PBH], Algebra [PBF]