EAN: 9783319276649

UNIVERSITEXT THE SPECTRUM OF HYPERBOLIC SURFA...

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.

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