**Graduate Studies in Mathematics**

Volume: 53;
2002;
220 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-3160-1

Product Code: GSM/53

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

**Electronic ISBN: 978-1-4704-1798-7
Product Code: GSM/53.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

# Spectral Methods of Automorphic Forms: Second Edition

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*Henryk Iwaniec*

A co-publication of the AMS and Revista Matemática Iberoamericana (RMI), Madrid, Spain

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.

#### Readership

Graduate students and researchers working in analytic number theory.

#### Reviews & Endorsements

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

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

# Table of Contents

## Spectral Methods of Automorphic Forms: Second Edition

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface ix10 free
- Preface to AMS edition xi12 free
- Introduction 114 free
- Harmonic analysis on the Euclidean plane 316 free
- Harmonic analysis on the hyperbolic plane 720
- Fuchsian groups 3750
- Automorphic forms 5366
- The spectral theorem. Discrete part 6376
- The automorphic Green function 7184
- Analytic continuation of the Eisenstein series 8194
- The spectral theorem. Continuous part 95108
- Estimates for the Fourier coefficients of Maass forms 107120
- Spectral theory of Kloosterman sums 121134
- The trace formula 135148
- The distribution of eigenvalues 157170
- Hyperbolic lattice-point problems 171184
- Spectral bounds for cusp forms 177190
- Classical analysis 185198
- Special functions 197210
- References 209222
- Subject index 215228
- Notation index 219232
- Back Cover Back Cover1234