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The Spectrum of Hyperbolic Surfaces

  • Nicolas Bergeron

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Nicolas Bergeron
    Pages 1-29
  3. Nicolas Bergeron
    Pages 31-52
  4. Nicolas Bergeron
    Pages 53-98
  5. Nicolas Bergeron
    Pages 99-151
  6. Nicolas Bergeron
    Pages 153-192
  7. Nicolas Bergeron
    Pages 193-211
  8. Nicolas Bergeron
    Pages 213-265
  9. Nicolas Bergeron
    Pages 267-293
  10. Nicolas Bergeron
    Pages 295-342
  11. Back Matter
    Pages 343-370

About this book

Introduction

This text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces. For some of these surfaces, called “arithmetic hyperbolic surfaces”, the eigenfunctions are of arithmetic nature, and one may use analytic tools as well as powerful methods in number theory to study them.

After an introduction to the hyperbolic geometry of surfaces, with a special emphasis on those of arithmetic type, and then an introduction to spectral analytic methods on the Laplace operator on these surfaces, the author develops the analogy between geometry (closed geodesics) and arithmetic (prime numbers) in proving the Selberg trace formula. Along with important number theoretic applications, the author exhibits applications of these tools to the spectral statistics of the Laplacian and the quantum unique ergodicity property. The latter refers to the arithmetic quantum unique ergodicity theorem, recently proved by Elon Lindenstrauss.

The fruit of several graduate level courses at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces allows the reader to review an array of classical results and then to be led towards very active areas in modern mathematics.

Keywords

Hyperbolic surfaces Arithmetic manifolds Laplacian Maass forms Quantum chaos Selberg trace formula

Authors and affiliations

  • Nicolas Bergeron
    • 1
  1. 1.IMJ-PRGUniversite Pierre et Marie CurieParisFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-27666-3
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-27664-9
  • Online ISBN 978-3-319-27666-3
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site