Random Matrices, Frobenius Eigenvalues, and Monodromy

Random Matrices, Frobenius Eigenvalues, and Monodromy

by Nicholas M. Katz, Peter Sarnak
     
 

ISBN-10: 0821810170

ISBN-13: 9780821810170

Pub. Date: 09/01/1998

Publisher: American Mathematical Society

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

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Overview

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 infinity and related techniques from orthogonal polynomials and Fredholm determinants.

Product Details

ISBN-13:
9780821810170
Publisher:
American Mathematical Society
Publication date:
09/01/1998
Series:
Colloquium Publications Series, #45
Pages:
419
Product dimensions:
7.09(w) x 10.63(h) x (d)

Table of Contents

Introduction1
Ch. 1Statements of the Main Results17
Ch. 2Reformulation of the Main Results35
Ch. 3Reduction Steps in Proving the Main Theorems73
Ch. 4Test Functions101
Ch. 5Haar Measure107
Ch. 6Tail Estimates141
Ch. 7Large N Limits and Fredholm Determinants197
Ch. 8Several Variables245
Ch. 9Equidistribution267
Ch. 10Monodromy of Families of Curves293
Ch. 11Monodromy of Some Other Families323
Ch. 12GUE Discrepancies in Various Families351
Ch. 13Distribution of Low-lying Frobenius Eigenvalues in Various Families361
AppDensities373
App: Graphs411
References417

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