**Graduate Studies in Mathematics**

Volume: 17;
1997;
259 pp;
Hardcover

MSC: Primary 32;

Print ISBN: 978-0-8218-0777-4

Product Code: GSM/17

List Price: $50.00

AMS Member Price: $40.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-2074-1
Product Code: GSM/17.E**

List Price: $50.00

AMS Member Price: $40.00

MAA Member Price: $45.00

# Topics in Classical Automorphic Forms

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*Henryk Iwaniec*

The book is based on the notes from the graduate course given by the author at Rutgers University in the fall of 1994 and the spring of 1995. The main goal of the book is to acquaint the reader with various perspectives of the theory of automorphic forms. In addition to detailed and often nonstandard exposition of familiar topics of the theory, particular attention is paid to such subjects as theta-functions and representations by quadratic forms.

#### Readership

Graduate students and research mathematicians working in number theory and related topics of algebraic geometry.

#### Reviews & Endorsements

The author discusses many important topics in the theory of automorphic forms which are rarely seen in the textbooks available on the subject … the presentation of the proofs … is … unusual, and this may give the reader a different flavor of the subject … graduate students will certainly benefit from this book.

-- Mathematical Reviews

An excellent place to begin the study of the analytic approach to modular forms … a welcome addition to this growing expository of literature.

-- Bulletin of the AMS

[T]his is an excellent book, requiring hard work from the reader and giving rich reward for her or his effort.

-- Zentralblatt MATH

An excellent graduate text. The book by Iwaniec provides the graduate student and the researcher wishing to acquire the basics on automorphic forms with a beautifully written and self-contained treatment of the classical modular and automorphic forms, Kloosterman sums, Hecke operators, automorphic L-functions, cusp forms and Eisenstein series, spherical functions, theta functions and convolution L-functions.

-- Bulletin of the London Mathematical Society

#### Table of Contents

# Table of Contents

## Topics in Classical Automorphic Forms

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xi12 free
- Chapter 0. Introduction 114 free
- Chapter 1. The Classical Modular Forms 316 free
- Chapter 2. Automorphic Forms in General 2336
- Chapter 3. The Eisenstein and the Poincaré Series 4760
- Chapter 4. Kloosterman Sums 5770
- Chapter 5. Bounds for the Fourier Coefficients of Cusp Forms 6982
- Chapter 6. Hecke Operators 91104
- 6.1. Introduction 91104
- 6.2. Hecke operators T[sub(n)] 92105
- 6.3. The Hecke operators on periodic functions 94107
- 6.4. The Hecke operators for the modular group 98111
- 6.5. The Hecke operators with a character 101114
- 6.6. An overview of newforms 107120
- 6.7. Hecke eigencuspforms for a primitive character 108121
- 6.8. Final remarks 118131

- Chapter 7. Automorphic L-functions 119132
- Chapter 8. Cusp Forms Associated with Elliptic Curves 133146
- Chapter 9. Spherical Functions 147160
- Chapter 10. Theta Functions 165178
- Chapter 11. Representations by Quadratic Forms 179192
- Chapter 12. Automorphic Forms Associated with Number Fields 203216
- 12.1. Automorphic forms attached to Dirichlet L-functions 203216
- 12.2. Hecke L-functions with Grossencharacters 206219
- 12.3. Automorphic forms associated with quadratic fields 211224
- 12.4. Class group L-functions reconsidered 215228
- 12.5. L-functions for genus characters 219232
- 12.6. Automorphic forms of weight one 224237

- Chapter 13. Convolution L-functions 231244
- 13.1. Introduction 231244
- 13.2. Rankin-Selberg integrals 232245
- 13.3. Selberg's theory of Eisenstein series 235248
- 13.4. Statement of general results 240253
- 13.5. The scattering matrix for Γ[sub(0)](N) 240253
- 13.6. Functional equations for the convolution L-functions 243256
- 13.7. Metaplectic Eisenstein series 246259
- 13.8. Symmetric power L-functions 248261

- Bibliography 255268
- Index 257270