Random Matrices, Frobenius Eigenvalues, and Monodromy

✍ Scribed by Nicholas M. Katz, Peter Sarnak

• Publisher: AMS
• Year: 1999
• Tongue: English
• Leaves: 441
• Series: Colloquium Publications 45
• Category: Library

✦ Synopsis

The main topic of this book is the deep relation between the spacings between zeros of zeta and L-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and L-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinit.

✦ Table of Contents

Cover
Title page
Contents
Introduction
Statements of the main results
Reformulation of the main results
Reduction steps in proving the main theorems
Test functions
Haar measure
Tail estimates
Large N limits and Fredholm determinants
Several variables
Equidistribution
Monodromy of families of curves
Monodromy of some other families
GUE discrepancies in various families
Distribution of low-lying Frobenius eigenvalues in various families
Appendix: Densities
Appendix: Graphs
References
Back Cover