Zeta functions of groups and rings / Marcus du Sautoy, Luke Woodward.

Format:

Book

Author/Creator:

Du Sautoy, Marcus.

Contributor:

Woodward, Luke, D. Phil.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1925.

Lecture notes in mathematics ; 1925

Language:

English

Subjects (All):

Group theory.

Functions, Zeta.

Rings (Algebra).

Noncommutative algebras.

Physical Description:

xii, 208 pages ; 24 cm.

Place of Publication:

Berlin : Springer, [2008]

Summary:

Zeta functions have been a powerful tool in mathematics over the last two centuries. This book considers a new class of non-commutative zeta functions which encode the structure of the subgroup lattice in infinite groups. The book explores the analytic behaviour of these functions together with an investigation of functional equations. Many important examples of zeta functions are calculated and recorded providing an important data base of explicit examples and methods for calculation.

Contents:

1.1 A Brief History of Zeta Functions 1

1.1.1 Euler, Riemann 1

1.1.2 Dirichlet 3

1.1.3 Dedekind 4

1.1.4 Artin, Weil 5

1.1.5 Birch, Swinnerton-Dyer 6

1.2 Zeta Functions of Groups 6

1.2.1 Zeta Functions of Algebraic Groups 7

1.2.2 Zeta Functions of Rings 9

1.2.3 Local Functional Equations 10

1.2.4 Uniformity 11

1.2.5 Analytic Properties 12

1.3 p-Adic Integrals 14

1.4 Natural Boundaries of Euler Products 16

2 Nilpotent Groups: Explicit Examples 21

2.1 Calculating Zeta Functions of Groups 21

2.2 Calculating Zeta Functions of Lie Rings 23

2.2.1 Constructing the Cone Integral 23

2.2.2 Resolution 25

2.2.3 Evaluating Monomial Integrals 31

2.2.4 Summing the Rational Functions 32

2.3 Explicit Examples 32

2.4 Free Abelian Lie Rings 33

2.5 Heisenberg Lie Ring and Variants 34

2.6 Grenham's Lie Rings 38

2.7 Free Class-2 Nilpotent Lie Rings 40

2.7.1 Three Generators 40

2.7.2 n Generators 41

2.8 The 'Elliptic Curve Example' 42

2.9 Other Class Two Examples 43

2.10 The Maximal Class Lie Ring M[subscript 3] and Variants 45

2.11 Lie Rings with Large Abelian Ideals 48

2.12 F[subscript 3,2] 51

2.13 The Maximal Class Lie Rings M[subscript 4] and Fil[subscript 4] 52

2.14 Nilpotent Lie Algebras of Dimension [less than or equal] 6 55

2.15 Nilpotent Lie Algebras of Dimension 7 62

3 Soluble Lie Rings 69

3.2 Proof of Theorem 3.1 71

3.2.1 Choosing a Basis for tr[subscript n](Z) 71

3.2.2 Determining the Conditions 72

3.2.3 Constructing the Zeta Function 74

3.2.4 Transforming the Conditions 74

3.2.5 Deducing the Functional Equation 75

3.3 Explicit Examples 77

3.4 Variations 78

3.4.1 Quotients of tr[subscript n](Z) 78

3.4.2 Counting All Subrings 82

4 Local Functional Equations 83

4.2 Algebraic Groups 83

4.3 Nilpotent Groups and Lie Rings 83

4.4 The Conjecture 84

4.5 Special Cases Known to Hold 86

4.6 A Special Case of the Conjecture 87

4.6.1 Projectivisation 88

4.6.2 Resolution 89

4.6.3 Manipulating the Cone Sums 91

4.6.4 Cones and Schemes 93

4.6.5 Quasi-Good Sets 95

4.6.6 Quasi-Good Sets: The Monomial Case 97

4.7 Applications of Conjecture 4.5 98

4.8 Counting Subrings and p-Subrings 102

4.9 Counting Ideals and p-Ideals 103

4.9.1 Heights, Cocentral Bases and the [pi]-Map 104

4.9.2 Property ([dagger]) 107

4.9.3 Lie Rings Without ([dagger]) 119

5 Natural Boundaries I: Theory 121

5.1 A Natural Boundary for [zeta]GSp[subscript 6] (s) 121

5.2 Natural Boundaries for Euler Products 123

5.2.2 Distinguishing Types I, II and III 136

5.3 Avoiding the Riemann Hypothesis 139

5.4 All Local Zeros on or to the Left of R(s) = [beta] 142

5.4.1 Using Riemann Zeros 143

5.4.2 Avoiding Rational Independence of Riemann Zeros 145

5.4.3 Continuation with Finitely Many Riemann Zeta Functions 149

5.4.4 Infinite Products of Riemann Zeta Functions 150

6 Natural Boundaries II: Algebraic Groups 155

6.2 G = GO[subscript 2l+1] of Type B[subscript l] 159

6.3 G = GSp[subscript 2l] of Type C[subscript l] or G = GO[superscript +][subscript 2l] of Type D[subscript l] 161

6.3.1 G = GSp[subscript 2l] of Type C[subscript l] 162

6.3.2 G = GO[superscript + subscript 2l] of Type D[subscript l] 165

7 Natural Boundaries III: Nilpotent Groups 169

7.2 Zeta Functions with Meromorphic Continuation 169

7.3 Zeta Functions with Natural Boundaries 170

7.3.1 Type I 171

7.3.2 Type II 171

7.3.3 Type III 173

7.4 Other Types 177

7.4.1 Types IIIa and IIIb 177

7.4.2 Types IV, V and VI 177

A Large Polynomials 179

A.1 H[superscript 4], Counting Ideals 179

A.2 g[subscript 6,4], Counting All Subrings 180

A.3 T[subscript 4], Counting All Subrings 180

A.4 L[subscript (3,2,2)], Counting Ideals 181

A.5 G[subscript 3] x g[subscript 5,3], Counting Ideals 182

A.6 g[subscript 6,12], Counting All Subrings 183

A.7 g[subscript 1357G], Counting Ideals 184

A.8 g[subscript 1457A], Counting Ideals 186

A.9 g[subscript 1457B], Counting Ideals 187

A.10 tr[subscript 6](Z), Counting Ideals 188

A.11 tr[subscript 7](Z), Counting Ideals 188

B Factorisation of Polynomials Associated to Classical Groups 191.

Notes:

Includes bibliographical references and index.

ISBN:

9783540747017

354074701X

3540747761

9783540747765

OCLC:

183145208