Preface

Acknowledgments

Prerequisites and notation

1 Introduction

1.1 Presentation

1.2 Some new applications of the large sieve

2 The principle of the large sieve

2.1 Notation and terminology

2.2 The large sieve inequality

2.3 Duality and 'exponential sums'

2.4 The dual sieve

2.5 General comments on the large sieve inequality

3 Group and conjugacy sieves

3.1 Conjugacy sieves

3.2 Group sieves

3.3 Coset sieves

3.4 Exponential sums and equidistribution for group sieves

3.5 Self-contained statements

4 Elementary and classical examples

4.1 The inclusion-exclusion principle

4.2 The classical large sieve

4.3 The multiplicative large sieve inequality

4.4 The elliptic sieve

4.5 Other examples

5 Degrees of representations of finite groups

5.l Introduction

5.2 Groups of Lie type with connected centres

5.3 Examples

5.4 Some groups with disconnected centres

6 Probabilistic sieves

6.1 Probabilistic sieves with integers

6.2 Some properties of random finitely presented groups

7 Sieving in discrete groups

7.1 Introduction

7.2 Random walks in discrete groups with Property (τ)

7.3 Applications to arithmetic groups

7.4 The cases of SL(2) and Sp(4)

7.5 Arithmetic applications

7.6 Geometric applications

7.7 Explicit bounds and arithmetic transitions

7.8 Other groups

8 Sieving for Frobenius over finite fields

8.1 A problem about zeta functions of curves over finite fields

8.2 The formal setting of the sieve for Frobenius

8.3 Bounds for sieve exponential sums

8.4 Estimates for sums or Betti numbers

8.5 Bounds for the large sieve constants

8.6 Application to Chavdarov's problem

8.7 Remarks on monodromy groups

8.8 A last application

Appendix A Small sieves

A.I General results

A.2 An application

Appendix B Local density computations over finite fields

B.I Density of cycle types for polynomials over finite fields

B.2 Some matrix densities over finite fields

B.3 Other techniques

Appendix C Representation theory

C.1 Definitions

C.2 Harmonic analysis

C.3 One-dimensional representations

C.4 The character tables of GL(2, **F**_{q}) and SL(2, **F**_{q})

Appendix D Property (T) and Property (τ)

D.1 Property (T)

D.2 Properties and examples

D.3 Property (τ)

D.4 Shalom's theorem

Appendix E Linear algebraic groups

E.1 Basic terminology

E.2 Galois groups of characteristic polynomials

Appendix F Probability theory and random walks

F.1 Terminology

F.2 The Central Limi! Theorem

F.3 The Borel-Cantelli lemmas

F.4 Random walks

Appendix G Sums of multiplicative functions

G.1 Some basic theorems

G.2 An example

Appendix H Topology

H.1 The fundamental group

H.2 Homology

H.3 The mapping class group of surfaces

References

Index