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Spectral Methods of Automorphic Forms: Second Edition
Henryk Iwaniec, Rutgers University, Piscataway, NJ
A co-publication of the AMS and Revista Matemática Iberoamericana (RMI), Madrid, Spain.

Graduate Studies in Mathematics
2002; 220 pp; hardcover
Volume: 53
ISBN-10: 0-8218-3160-7
ISBN-13: 978-0-8218-3160-1
List Price: US$60
Member Price: US$48
Order Code: GSM/53
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Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style.

The first edition of this volume was an underground classic, both as a textbook and as a respected source for results, ideas, and references. The book's reputation sparked a growing interest in the mathematical community to bring it back into print. The AMS has answered that call with the publication of this second edition.

In the book, Iwaniec treats the spectral theory of automorphic forms as the study of the space \(L^2 (H\Gamma)\), where \(H\) is the upper half-plane and \(\Gamma\) is a discrete subgroup of volume-preserving transformations of \(H\). He combines various techniques from analytic number theory. Among the topics discussed are Eisenstein series, estimates for Fourier coefficients of automorphic forms, the theory of Kloosterman sums, the Selberg trace formula, and the theory of small eigenvalues.

Henryk Iwaniec was awarded the 2002 AMS Cole Prize for his fundamental contributions to analytic number theory. Also available from the AMS by H. Iwaniec is Topics in Classical Automorphic Forms, Volume 17 in the Graduate Studies in Mathematics series.

The book is designed for graduate students and researchers working in analytic number theory.

This book is copublished by the AMS and Revista Matemática Iberoamericana (RMI), Madrid, Spain.


Graduate students and researchers working in analytic number theory.


"Offers a swift introduction ... The material and the exposition are well suited for use by undergraduate students. Researchers from other fields and graduate students entering the field will also benefit from this well-written book ... comprehensive book ... highly recommend it to students and researchers who are interested in number theory."

-- Bulletin of the LMS

From a review of the first edition:

"The material and exposition are well-suited for second-year or higher graduate students ... This clear and comprehensive book concerning the spectral theory of \(\mathrm{GL} (2)\) automorphic forms belongs on many a bookshelf."

-- Mathematical Reviews

"Gives a superb introduction to real analytic automorphic forms and their applications in number theory starting at a modest level and leading up to topics of current research ... book belongs in any institutional library. If you did not succeed in buying your copy of the first edition hurry up and order your copy of this one."

-- Zentralblatt MATH

Table of Contents

  • Introduction
  • Harmonic analysis on the Euclidean plane
  • Harmonic analysis on the hyperbolic plane
  • Fuchsian groups
  • Automorphic forms
  • The spectral theorem. Discrete part
  • The automorphic Green function
  • Analytic continuation of the Eisenstein series
  • The spectral theorem. Continuous part
  • Estimates for the Fourier coefficients of Maass forms
  • Spectral theory of Kloosterman sums
  • The trace formula
  • The distribution of eigenvalues
  • Hyperbolic lattice-point problems
  • Spectral bounds for cusp forms
  • Classical analysis
  • Special functions
  • References
  • Subject index
  • Notation index
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