Atnaujinkite slapukų nuostatas

BruhatTits Theory: A New Approach [Kietas viršelis]

(University of Michigan, Ann Arbor), (University of Michigan, Ann Arbor)
  • Formatas: Hardback, 700 pages, aukštis x plotis x storis: 235x158x50 mm, weight: 1260 g, Worked examples or Exercises
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 26-Jan-2023
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108831966
  • ISBN-13: 9781108831963
Kitos knygos pagal šią temą:
BruhatTits Theory: A New ApproachDidesnis paveikslėlis
  • Formatas: Hardback, 700 pages, aukštis x plotis x storis: 235x158x50 mm, weight: 1260 g, Worked examples or Exercises
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 26-Jan-2023
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108831966
  • ISBN-13: 9781108831963
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781108831963.html
Raktažodžiai:
"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 areference 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. For more experienced readers looking to learn the essentials for use in their own work, there is also 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 the integral models of tori"--

Daugiau informacijos

Comprehensive treatment of BruhatTits theory for graduate students and researchers in number theory, representation theory, and algebraic geometry.
List of illustrations
xii
List of tables
xiii
Introduction xv
PART ONE BACKGROUND AND REVIEW
1(138)
1 Affine Root Systems and Abstract Buildings
3(65)
1.1 Metric Spaces
3(5)
1.2 Affine Spaces
8(6)
1.3 Affine Root Systems
14(33)
1.4 Tits Systems
47(7)
1.5 Abstract Buildings
54(12)
1.6 The Monoid R
66(2)
2 Algebraic Groups
68(71)
2.1 Henselian Fields
68(2)
2.2 Bounded Subgroups of Reductive Groups
70(8)
2.3 Fields of Dimension
78(3)
2.4 Affine Group Schemes over Perfect Fields
81(5)
2.5 Tori
86(6)
2.6 Reductive Groups
92(16)
2.7 The Group SU3
108(2)
2.8 Separable Quadratic Extensions of Local Fields
110(2)
2.9 Chevalley Systems
112(9)
2.10 Integral Models
121(9)
2.11 Group Scheme Actions and the Dynamic Method
130(9)
PART TWO BRUHAT-TITS THEORY
139(248)
3 Examples: Quasi-split Simple Groups of Rank 1
141(16)
3.1 The Example of SL2
141(6)
3.2 The Example of SU3
147(10)
4 Overview and Summary of Bruhat-Tits Theory
157(29)
4.1 Axiomatization of Bruhat-Tits Theory
157(11)
4.2 Metric
168(10)
4.3 The Enlarged Building
178(2)
4.4 Uniqueness of the Apartment and the Building
180(6)
5 Bruhat. Cartan, and Iwasawa Decompositions
186(9)
5.1 The (affine) Bruhat Decomposition
186(1)
5.2 The Cartan Decomposition
187(1)
5.3 The Iwasawa Decomposition
188(3)
5.4 The Intersection of Cartan and Iwasawa Double Cosets
191(4)
6 The Apartment
195(40)
6.1 The Apartment of a Quasi-split Reductive Group
196(16)
6.2 Affine Reflections and Uniqueness of Valuations
212(5)
6.3 Affine Roots and Affine Root Groups
217(6)
6.4 The Affine Root System of a Quasi-split Group
223(6)
6.5 Change of Valuation
229(1)
6.6 The Affine Weyl Group
230(3)
6.7 Projection to a Levi Subgroup
233(2)
7 The Bruhat-Tits Building for a Valuation of the Root Datum
235(48)
7.1 Commutator Computations
236(9)
7.2 A Filtration of Z(k)
245(1)
7.3 Concave Functions
246(9)
7.4 Parahoric Subgroups
255(4)
7.5 The Iwahori-Tits System
259(4)
7.6 The (Reduced) Building
263(4)
7.7 Disconnected Parahoric Subgroups
267(5)
7.8 The Iwahori-Weyl Group
272(1)
7.9 Change of Base Field and Automorphisms
273(3)
7.10 Passage to Completion
276(2)
7.11 Absolutely Special Points
278(5)
8 Integral Models
283(40)
8.1 Preliminaries
283(3)
8.2 General Properties of Smooth Models of G
286(9)
8.3 Parahoric Integral Models
295(7)
8.4 The Structure of the Special Fiber of gω
302(8)
8.5 Integral Models Associated to Concave Functions
310(11)
8.6 Passage to Completion
321(2)
9 Unramified Descent
323(64)
9.1 Preliminaries
324(1)
9.2 Statement of the Main Result
325(5)
9.3 The Building and its Apartments
330(15)
9.4 The Affine Root System
345(18)
9.5 Completion of the Proof of the Main Result
363(1)
9.6 Valuation of Root Datum
364(4)
9.7 Levi Subgroups
368(6)
9.8 Concave Function Groups
374(5)
9.9 Special, Superspecial, and Hyperspecial Points
379(2)
9.10 Residually Split and Residually Quasi-split Groups
381(4)
9.11 Restriction of Scalars
385(2)
PART THREE ADDITIONAL DEVELOPMENTS
387(158)
10 Residue Field f of Dimension 1
389(31)
10.1 Conjugacy of Special Tori
389(2)
10.2 Superspecial Points
391(1)
10.3 Anisotropic Groups
391(2)
10.4 Fixed Points of Large Subgroups of Tori
393(2)
10.5 Existence of Anisotropic Tori
395(2)
10.6 Cohomological Results
397(4)
10.7 Classification of Connected Reductive /c-Groups
401(19)
11 Component Groups of Integral Models
420(16)
11.1 The Kottwitz Homomorphism for Tori
421(5)
11.2 The Component Groups of Jft and Jlft
426(1)
11.3 The Algebraic Fundamental Group
427(1)
11.4 z-Extensions
428(1)
11.5 The Kottwitz Homomorphism for Reductive Groups
429(3)
11.6 The Component Groups of Parahoric Integral Models
432(1)
11.7 The Case of dim(f) ≤ 1
433(3)
12 Finite Group Actions and Tamely Ramified Descent
436(30)
12.1 Preliminaries
437(2)
12.2 Certain Group Schemes Associated to H and G
439(4)
12.3 A Reduction
443(2)
12.4 Apartments of B
445(4)
12.5 The Polyhedral Structure on B
449(7)
12.6 Identification of Parahoric Subgroups
456(1)
12.7 The Main Theorem
457(2)
12.8 The Case of a Finite Cyclic Group
459(4)
12.9 Tamely Ramified Descent
463(3)
13 Moy-Prasad Filtrations
466(24)
13.1 Filtrations of Tori
466(2)
13.2 Filtrations of Parahoric Subgroups
468(2)
13.3 Filtrations of the Lie Algebra and its Dual
470(1)
13.4 Optimal Points
471(3)
13.5 The Moy-Prasad Isomorphism
474(4)
13.6 Semi-stability
478(3)
13.7 G-Domains in the Lie Algebra g
481(4)
13.8 Vanishing of Cohomology
485(5)
14 Functorial Properties
490(23)
14.1 Quotient Maps
493(1)
14.2 Embeddings: Isometric Properties
494(2)
14.3 Embeddings: Factorization through a Levi Subgroup
496(1)
14.4 Embeddings of Apartments
497(3)
14.5 Adapted Points: Definition and Properties
500(2)
14.6 Embeddings of Buildings via Adapted Points
502(2)
14.7 The Space of Embeddings and Galois Descent
504(1)
14.8 Existence of Adapted Points
505(4)
14.9 Uniqueness of Admissible Embeddings
509(4)
15 The Buildings of Classical Groups via Lattice Chains
513(32)
15.1 The Special and General Linear Groups
513(11)
15.2 Symplectic, Orthogonal, and Unitary groups
524(21)
PART FOUR APPLICATIONS
545(46)
16 Classification of Maximal Unramified Tori (d'apres DeBacker)
547(6)
17 Classification of Tamely Ramified Maximal Tori
553(5)
18 The Volume Formula
558(33)
18.1 Remarks on Arithmetic Subgroups
559(5)
18.2 Notations, Conventions and Preliminaries
564(2)
18.3 Tamagawa Forms on Quasi-split Groups
566(7)
18.4 Volumes of Parahoric Subgroups
573(5)
18.5 Covolumes of Principal S-Arithmetic Subgroups
578(6)
18.6 Euler-Poincare characteristic of 5-arithmetic subgroups
584(1)
18.7 Bounds for the Class Number of Simply Connected Groups
585(2)
18.8 The Discriminant Quotient Formula for Global Fields
587(4)
PART FIVE APPENDICES
591(2)
A Operations on Integral Models
593(60)
A.1 Base Change
593(1)
A.2 Schematic Closure
593(2)
A.3 Weil Restriction of Scalars
595(8)
A.4 The Greenberg Functor
603(29)
A.5 Dilatation
632(13)
A.6 Smoothening
645(2)
A.7 Schematic Subgroups
647(2)
A.8 Reductive Models
649(4)
B Integral Models of Tori
653(43)
B.1 Preliminaries
653(1)
B.2 Split Tori
654(2)
B.3 Induced Tori
656(3)
B.4 The Standard Model
659(8)
B.5 The Standard Filtration
667(1)
B.6 Weakly induced tori
668(7)
B.7 The ft-Neron Model
675(5)
B.8 The N6ron Mapping Properties and the lft-Neron Model
680(4)
B.9 The pro-unipotent radical
684(2)
B.10 The Minimal Congruent Filtration
686(10)
C Integral Models of Root Groups
696(12)
C.1 Introduction
696(1)
C.2 Integral Models for Filtration Subgroups of Ga
697(1)
C.3 Integral Models for Filtration Subgroups of R0L/KGa
698(1)
C.4 Integral Models for Filtration Subgroups of UL/k
699(4)
C.5 Summary
703(5)
References 708(7)
Index of Symbols 715(2)
General Index 717
Tasho Kaletha is Professor of Mathematics at the University of Michigan. He is an expert on the Langlands program, and has studied arithmetic and representation-theoretic aspects of the local Langlands correspondence for p-adic groups. Gopal Prasad is Raoul Bott Professor Emeritus of Mathematics at the University of Michigan. He is a leading expert on real and p-adic Lie groups and algebraic groups. Together with Ofer Gabber and Brian Conrad, he published the complete classification and structure theory of pseudo-reductive groups in the books Pseudo-reductive Groups (2010, 2015) and Classification of Pseudo-reductive Groups (2015).