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

Cambridge University Press, 2008. — 316 p. — (Cambridge Tracts in Mathematics 175). — ISBN: 978-0521888516, 978-0511398872.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 (τ ) for discrete groups.Presentation, Some new applications of the large sieve.
The principle of the large sieve.
Notation and terminology, The large sieve inequality, Duality and 'exponential sums',
The dual sieve, General comments on the large sieve inequality.
Group and conjugacy sieves.
Conjugacy sieves, Group sieves, Coset sieves,
Exponential sums and equidistribution for group sieves, Self-contained statements.
Elementary and classical examples.
The inclusion-exclusion principle, The classical large sieve, The multiplicative large sieve inequality,
The elliptic sieve, Other examples.
Degrees of representations of finite groups.
Introduction, Groups of Lie type with connected centres, Examples, Some groups with disconnected centres.
Probabilistic sieves.
Probabilistic sieves with integers, Some properties of random finitely presented groups.
Sieving in discrete groups.
Introduction, Random walks in discrete groups with Property (τ ), Applications to arithmetic groups,
The cases of SL(2) and Sp(4), Arithmetic applications, Geometric applications,
Explicit bounds and arithmetic transitions, Other groups.
Sieving for Frobenius over finite fields.
A problem about zeta functions of curves over finite, Fields, The formal setting of the sieve for Frobenius,
Bounds for sieve exponential sums, Estimates for sums of Betti numbers, Bounds for the large sieve constants,
Application to Chavdarov’s problem, Remarks on monodromy groups, A last application.
Appendix A. Small sieves:
General results, An application.
Appendix B. Local density computations over finite fields:
Density of cycle types for polynomials over finite fields, Some matrix densities over finite fields, Other techniques.
Appendix C. Representation theory:
Definitions, Harmonic analysis, One-dimensional representations,
The character tables of GL(2, F_q) and SL(2, F_q).
Appendix D. Property (T ) and Property (τ ):
Property (T ), Properties and examples, Property (τ ), Shalom’s theorem.
Appendix E. Linear algebraic groups:
Basic terminology, Galois groups of characteristic polynomials.
Appendix F. Probability theory and random walks:
Terminology, The Central Limit Theorem, The Borel–Cantelli lemmas, Random walks.
Appendix G. Sums of multiplicative functions:
Some basic theorems, An example.
Appendix H. Topology:
The fundamental group, Homology, The mapping class group of surfaces.