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 realization 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.
Author(s): E. Kowalski
Series: Cambridge Tracts in Mathematics
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 317
Tags: Математика;Теория чисел;
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 13
Acknowledgments......Page 18
Prerequisites and notation......Page 19
1.1 Presentation......Page 25
1.2 Some new applications of the large sieve......Page 28
2.1 Notation and terminology......Page 32
2.2 The large sieve inequality......Page 33
2.3 Duality and ‘exponential sums’......Page 42
2.4 The dual sieve......Page 46
2.5 General comments on the large sieve inequality......Page 49
3.1 Conjugacy sieves......Page 56
3.2 Group sieves......Page 58
3.3 Coset sieves......Page 60
3.4 Exponential sums and equidistribution for group sieves......Page 64
3.5 Self-contained statements......Page 66
4.1 The inclusion-exclusion principle......Page 69
4.2 The classical large sieve......Page 72
4.3 The multiplicative large sieve inequality......Page 81
4.4 The elliptic sieve......Page 83
4.5 Other examples......Page 91
5.1 Introduction......Page 94
5.2 Groups of Lie type with connected centres......Page 96
5.3 Examples......Page 106
5.4 Some groups with disconnected centres......Page 107
6.1 Probabilistic sieves with integers......Page 111
6.2 Some properties of random finitely presented groups......Page 118
7.1 Introduction......Page 125
7.2 Random walks in discrete groups with Property......Page 129
7.3 Applications to arithmetic groups......Page 137
7.4 The cases of (2) and (4)......Page 143
7.5 Arithmetic applications......Page 151
7.6 Geometric applications......Page 156
7.7 Explicit bounds and arithmetic transitions......Page 169
7.8 Other groups......Page 175
8 Sieving for Frobenius over finite fields......Page 178
8.1 A problem about zeta functions of curves over finite fields......Page 179
8.2 The formal setting of the sieve for Frobenius......Page 184
8.3 Bounds for sieve exponential sums......Page 188
8.4 Estimates for sums of Betti numbers......Page 192
8.5 Bounds for the large sieve constants......Page 195
8.6 Application to Chavdarov’s problem......Page 199
8.7 Remarks on monodromy groups......Page 211
8.8 A last application......Page 217
A.1 General results......Page 221
A.2 An application......Page 225
B.1 Density of cycle types for polynomials over finite fields......Page 228
B.2 Some matrix densities over finite fields......Page 234
B.3 Other techniques......Page 242
C.1 Definitions......Page 244
C.2 Harmonic analysis......Page 247
C.3 One-dimensional representations......Page 250
C.4 The character tables of GL(2, Fq) and SL(2, Fq)......Page 251
D.1 Property (T)......Page 256
D.2 Properties and examples......Page 257
D.3 Property (τ)......Page 260
D.4 Shalom’s theorem......Page 262
E.1 Basic terminology......Page 269
E.2 Galois groups of characteristic polynomials......Page 273
F.1 Terminology......Page 278
F.2 The Central Limit Theorem......Page 281
F.3 The Borel–Cantelli lemmas......Page 282
F.4 Random walks......Page 283
G.1 Some basic theorems......Page 286
G.2 An example......Page 288
H.1 The fundamental group......Page 292
H.2 Homology......Page 299
H.3 The mapping class group of surfaces......Page 300
References......Page 307
Index......Page 313
