Hardcover ISBN:  9780821810170 
Product Code:  COLL/45 
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Electronic ISBN:  9781470431914 
Product Code:  COLL/45.E 
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Book DetailsColloquium PublicationsVolume: 45; 1999; 419 ppMSC: Primary 11; 14; 60; Secondary 82;
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 MontgomeryOdlyzko 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.
ReadershipResearch mathematicians and graduate students interested in varieties over finite and local fields, zetafunctions, limit theorems and structure of families.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Statements of the main results

Chapter 3. Reformulation of the main results

Chapter 4. Reduction steps in proving the main theorems

Chapter 5. Test functions

Chapter 6. Haar measure

Chapter 7. Tail estimates

Chapter 8. Large N limits and Fredholm determinants

Chapter 9. Several variables

Chapter 10. Equidistribution

Chapter 11. Monodromy of families of curves

Chapter 12. Monodromy of some other families

Chapter 13. GUE discrepancies in various families

Chapter 14. Distribution of lowlying Frobenius eigenvalues in various families

Appendix: Densities

Appendix: Graphs


Additional Material

Reviews

[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 zetafunctions.
Bulletin of the London Mathematical Society


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 Book Details
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 Get Permissions
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 MontgomeryOdlyzko 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, zetafunctions, limit theorems and structure of families.

Chapters

Chapter 1. Introduction

Chapter 2. Statements of the main results

Chapter 3. Reformulation of the main results

Chapter 4. Reduction steps in proving the main theorems

Chapter 5. Test functions

Chapter 6. Haar measure

Chapter 7. Tail estimates

Chapter 8. Large N limits and Fredholm determinants

Chapter 9. Several variables

Chapter 10. Equidistribution

Chapter 11. Monodromy of families of curves

Chapter 12. Monodromy of some other families

Chapter 13. GUE discrepancies in various families

Chapter 14. Distribution of lowlying Frobenius eigenvalues in various families

Appendix: Densities

Appendix: Graphs

[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 zetafunctions.
Bulletin of the London Mathematical Society