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Random Matrices, Frobenius Eigenvalues, and Monodromy
Nicholas M. Katz and Peter Sarnak, Princeton University, NJ

Colloquium Publications
1999; 419 pp; hardcover
Volume: 45
ISBN-10: 0-8218-1017-0
ISBN-13: 978-0-8218-1017-0
List Price: US$89
Member Price: US$71.20
Order Code: COLL/45
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See also:

Eigenvalue Distribution of Large Random Matrices - Leonid Pastur and Mariya Shcherbina

Random Matrix Theory: Invariant Ensembles and Universality - Percy Deift and Dimitri Gioev

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.

To view the Index, click on the PDF or PostScript file above.


Research mathematicians and graduate students interested in varieties over finite and local fields, zeta-functions, limit theorems and structure of families.


"[F]or research workers interested in the Riemann Hypothesis, or in the arithmetic of varieties over finite fields, this work has important messages which may help to shape our thinking on fundamental issues on the nature of zeta-functions."

-- Bulletin of the London Mathematical Society

Table of Contents

  • 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 AD: Densities
  • Appendix AG: Graphs
  • References
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