The Large Sieve and its Applications: Arithmetic Geometry, Random Walks and Discrete Groups

by E. Kowalski

• ISBN13: 9780521888516
• ISBN10: 0521888514
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-07-14
• Publisher: Cambridge University Press

Summary

Among the modern methods used to study prime numbers, the sieve has been one of the most efficient. Originally conceived by Linnik in 1941, the large sieve has developed extensively since the 1960s, with a recent realisation that the underlying principles were capable of applications going well beyond prime number theory. This book develops a general form of sieve inequality, and describes its varied applications, including the study of families of zeta functions of algebraic curves over finite fields; arithmetic properties of characteristic polynomials of random unimodular matrices; homological properties of random 3-manifolds; and the average number of primes dividing the denominators of rational points on elliptic curves. Also covered in detail are the tools of harmonic analysis used to implement the forms of the large sieve inequality, including the Riemann Hypothesis over finite fields, and Property (T) or Property (tau) for discrete groups.

Table of Contents

Preface

Prerequisites and notation

Introduction

The principle of the large sieve

Group and conjugacy sieves

Elementary and classical examples

Degrees of representations of finite groups

Probabilistic sieves

Sieving in discrete groups

Sieving for Frobenius over finite fields

Small sieves

Local density computations over finite fields

Representation theory

Property (T) and Property (t)

Linear algebraic groups

Probability theory and random walks

Sums of multiplicative functions

Topology

Bibliography

Index

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