Authors:
- Features profound and recent results of the spectral theory of automorphic surfaces
- Provides a self-contained proof of the so-called Jacquet-Langlands correspondence
- Includes an introduction to Lindenstrauss's ergodic theoretic proof of quantum unique ergodicity for compact arithmetic surfaces, for which he was awarded a Fields medal in 2010
- Includes supplementary material: sn.pub/extras
Part of the book series: Universitext (UTX)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
About this book
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.
Reviews
“This book gives a very nice introduction to the spectral theory of the Laplace-Beltrami operator on hyperbolic surfaces of constant negative curvature. … mainly intended for students with a knowledge of basic differential geometry and functional analysis but also for people doing research in other domains of mathematics or mathematical physics and interested in the present day problems in this very active field of research. … book gives one of the best introductions to this fascinating field of interdisciplinary research.” (Dieter H. Mayer, Mathematical Reviews, August, 2017)
Authors and Affiliations
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IMJ-PRG, Universite Pierre et Marie Curie, Paris, France
Nicolas Bergeron
About the author
Nicolas Bergeron is a Professor at Université Pierre et Marie Curie in Paris. His research interests are in geometry and automorphic forms, in particular the topology and spectral geometry of locally symmetric spaces.
Bibliographic Information
Book Title: The Spectrum of Hyperbolic Surfaces
Authors: Nicolas Bergeron
Series Title: Universitext
DOI: https://doi.org/10.1007/978-3-319-27666-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2016
Softcover ISBN: 978-3-319-27664-9Published: 02 March 2016
eBook ISBN: 978-3-319-27666-3Published: 19 February 2016
Series ISSN: 0172-5939
Series E-ISSN: 2191-6675
Edition Number: 1
Number of Pages: XIII, 370
Number of Illustrations: 8 illustrations in colour
Additional Information: Original French edition published by EDP Sciences, Paris, 2011
Topics: Hyperbolic Geometry, Abstract Harmonic Analysis, Dynamical Systems and Ergodic Theory