Algebraic groups and their birational invariants / V.E. Voskresenskiĭ.

Format:

Book

Author/Creator:

Voskresenskiĭ, Valentin Evgenʹevich.

Contributor:

Voskresenskiĭ, Valentin Evgenʹevich.

Series:

Translations of mathematical monographs 0065-9282 ; v. 179.

Translations of mathematical monographs, 0065-9282 ; v. 179

Standardized Title:

Algebraicheskie gruppy i ikh birat͡sionalʹnye invarianty. English

Language:

English

Russian

Subjects (All):

Linear algebraic groups.

Geometry, Algebraic.

Physical Description:

xiii, 218 pages : illustrations ; 27 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [1998]

Contents:

Chapter 1. Forms and Galois Cohomology 1

1 Group schemes and their cohomology

1.1 Group objects in a category 1

1.2 Group schemes 3

1.3 Affine groups, Hopf algebras 4

1.4 Group schemes over a field, algebraic groups 7

1.5 Frobenius morphisms 7

1.6 Diagonal groups 10

1.7 Characters of group schemes 11

1.8 Bicharacters 13

1.9 Exactness of the functor D 14

1.10 Galois cohomology 16

1.11 Sheaves and cohomology in the etale topology 17

1.12 Cartier divisors and Weil divisors 19

2 The Brauer group of a projective variety 20

2.1 The unramified Brauer group of a function field 20

2.2 The Kummer exact sequence 20

2.3 The Tate group, the Picard number, the Lefschetz number 21

3 The theory of k-forms 23

3.1 Forms and one-dimensional cohomology 23

3.2 Splitting fields of a k-form 24

3.3 Forms of group schemes 25

3.4 Groups of multiplicative type 25

3.5 Principal homogeneous spaces 27

3.6 Projective groups and associated k-forms 29

3.7 The Brauer group of a field 30

3.8 Chevalley groups 32

3.9 Semisimple groups 35

3.10 Inner and outer forms 36

3.11 Almost simple semisimple groups 37

3.12 The Weil restriction 37

Chapter 2. Birational Geometry of Algebraic Tori 41

4 Birational invariants of linear algebraic groups 41

4.1 The variety of maximal tori of a reductive group 41

4.2 Structure of the generic torus of a semisimple group 42

4.3 The Picard group and the Brauer group of a linear algebraic group 44

4.4 Criteria for birational equivalence of algebraic varieties 46

4.5 Projective models of linear algebraic groups 47

4.6 Flasque resolutions of a module 49

4.7 Stable equivalence 51

4.8 Chevalley modules 52

4.9 Tori of small dimension 57

4.10 Tori with a biquadratic splitting field 58

4.11 The semigroup of stable equivalence 59

5 Tori with a cyclic splitting field 60

5.1 "Devissage" of a quasi-split torus 60

5.2 Invertibility of the Picard class 62

5.3 The Chistov multiplication 62

6 Stable rationality of varieties 65

6.1 Stably rational tori as orbit varieties 65

6.2 Covariants of linear represntations 67

6.3 Rationality of tori of type pq 69

6.4 Universal torsors 71

6.5 Counterexamples to Zariski's conjecture 73

Chapter 3. Invariants of Finite Transformation Groups 75

7 Fields of invariants of finite transformation groups 75

7.1 Fields of invariants and their models 75

7.2 Invariants of finite abelian groups 76

7.3 The fields (k,p[superscript alpha]), p > 2 78

7.4 The fields (k, 2[superscript alpha]) 79

7.5 General case 79

7.6 Invariants of finite groups over an algebraically closed field 81

7.7 Invariants of finite linear groups 82

7.8 Invariants of finite groups acting on tori 85

7.9 Invariants of connected algebraic groups 87

8 Invariant projective Demazure models 90

8.1 Cones and fans 90

8.2 Projective invariant fans 93

8.3 Birational invariants of tori without affect 97

8.4 The graded ring of a toric variety 99

Chapter 4. Arithmetic of Linear Algebraic Groups 103

9 Tori over a finite field 103

9.1 Number of rational points 103

9.2 Zeta function 104

10 Tori over local fields 106

10.1 Tori over reals 106

10.2 Tori over a nonarchimedean field 107

10.3 Integer structures in linear algebraic groups 107

10.4 Canonical integer form of a quasisplit torus 109

10.5 Canonical form of a norm torus 111

11 Tori over global fields 111

11.1 Adele groups 111

11.2 Canonical integer model of a torus over a number field 113

11.3 Cohomology of adele groups 114

11.4 Descent of the ground field 118

11.5 Approximation problems 119

11.6 Arithmetical meaning of the birational invariant H[superscript 1] (k,p(T)) 120

12 Arithmetic of semisimple groups 122

12.1 Cohomology of semisimple groups 122

12.2 Weak approximation 124

12.3 The group H[superscript 1] (k, Pic X) 125

13 Artin L-functions 127

13.1 Partial Artin L-functions 127

13.2 Theorems of Artin and Brauer 129

13.3 Global zeta function of a torus 131

Chapter 5. Tamagawa Numbers 133

14 Haar measure on adele groups 133

14.1 Product of local measures 133

14.2 Computation of local volumes 134

14.3 Canonical convergence factors 136

14.4 The Tamagawa measure 137

14.5 Properties of Tamagawa numbers 142

14.6 Tamagawa numbers of algebraic tori 142

14.7 The group [phi] 147

14.8 Further development of the method 148

14.9 Chevalley group Z-schemes 148

14.10 Gindikin-Karpelevich integrals 149

14.11 Langlands' method of computing Tamagawa numbers 153

14.12 Elementary computations of volumes of some classical quotients 160

15 The Minkowski-Siegel-Tamagawa formula 163

15.1 Infinite products 163

15.2 The weight of a genus of an odd positive lattice 165

15.3 The weight of a genus of an even positive unimodular lattice 169

15.4 Sums of squares 169

15.5 Sum of two squares 172

15.6 Sum of four squares 173

15.7 Sum of six squares 173

15.8 Sum of eight squares 174

15.9 Sum of three squares 174

15.10 Sum of five squares 175

15.11 Sum of seven squares 176

Chapter 6. R-equivalence in Algebraic Groups 177

16 The group of R-equivalence classes 177

16.1 First properties of R-equivalence on varieties 177

16.2 Birational invariance of R-equivalence in groups 179

17 R-equivalence on algebraic tori 180

17.1 Flasque resolution of a torus and R-equivalence 180

17.2 Some special tori 182

17.3 The group T(k(t)) 184

18 The unimodular group of a simple algebra 185

18.1 Reduction to the anisotropic kernel 185

18.2 The Whitehead group of a simple algebra 185

18.3 Platonov's examples 187

18.4 The Whitehead group of an isotropic group 188

18.5 R-equivalence over special fields 188

19 Algebras with involutions and groups of adjoint type 190

19.1 Algebras with involutions 190

19.2 Indecomposable algebras with involutions 190

19.3 Automorphisms of indecomposable algebras with involutions 192

19.4 Forms of algebras with involutions 192

19.5 The covering of G[subscript 0] 193

19.6 Merkurjev's theorems 194

Chapter 7. Index Formulas in Arithmetic of Algebraic Tori 197

20 Arithmetic of the projective group of a field 197

20.1 Ratio of class numbers 197

20.2 Index formulas for quadratic extensions 200

20.3 The Hasse relations for an imaginary extension 201

21 Arithmetic of a norm hypersurface 202.

Notes:

Rev. ed. of: Algebraicheskie tory. 1977.

Includes bibliographical references (pages 211-218).

Related Work:

Voskresenskiĭ, Valentin Evgenʹevich. Algebraicheskie tory.

ISBN:

0821809059

OCLC:

39189925