**Graduate Studies in Mathematics**

Volume: 179;
2017;
700 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-4947-7

Product Code: GSM/179

List Price: $94.00

Individual Member Price: $75.20

**Electronic ISBN: 978-1-4704-4081-7
Product Code: GSM/179.E**

List Price: $94.00

Individual Member Price: $75.20

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#### Supplemental Materials

# Modular Forms: A Classical Approach

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*Henri Cohen; Fredrik Strömberg*

The theory of modular forms is a fundamental tool used in
many areas of mathematics and physics. It is also a very concrete and
“fun” subject in itself and abounds with an amazing number of
surprising identities.

This comprehensive textbook, which includes numerous exercises, aims
to give a complete picture of the classical aspects of the subject,
with an emphasis on explicit formulas. After a number of motivating
examples such as elliptic functions and theta functions, the modular
group, its subgroups, and general aspects of holomorphic and
nonholomorphic modular forms are explained, with an emphasis on
explicit examples. The heart of the book is the classical theory
developed by Hecke and continued up to the Atkin–Lehner–Li
theory of newforms and including the theory of Eisenstein series,
Rankin–Selberg theory, and a more general theory of theta series
including the Weil representation. The final chapter explores in some
detail more general types of modular forms such as half-integral
weight, Hilbert, Jacobi, Maass, and Siegel modular forms.

Some “gems” of the book are an immediately
implementable trace formula for Hecke operators, generalizations of
Haberland's formulas for the computation of Petersson inner products,
W. Li's little-known theorem on the diagonalization of the
full space of modular forms, and explicit algorithms due to
the second author for computing Maass forms.

This book is essentially self-contained, the necessary tools such
as gamma and Bessel functions, Bernoulli numbers, and so on being
given in a separate chapter.

#### Readership

Graduate students and researchers interested in modular forms.

#### Reviews & Endorsements

This book gives a beautiful introduction to the theory of modular forms, with a delicate balance of analytic and arithmetic perspectives. Cohen and Strömberg start with a foundational collection of tools in analysis and number theory, which they use while guiding the reader through a vast landscape of results. They finish by showing us the frontiers of modern research, covering topics generalizing the classical theory in a variety of directions. Throughout, the authors expertly weave fine details with broad perspective. The target readership for this text is graduate students in number theory, though it will also be accessible to advanced undergraduates and will, no doubt, serve as a valuable reference for researchers for years to come.

-- Jennifer Balakrishnan, Boston University

This marvelous book is a gift to the mathematical community and more specifically to anyone wanting to learn modular forms. The authors take a classical view of the material offering extremely helpful explanations in a generous conversational manner and covering such an impressive range of this beautiful, deep, and important subject.

-- Barry Mazur, Harvard University

This book is an almost encyclopedic textbook on modular forms. There are already numerous and some excellent books on the subject. But none of the existing books by themselves contain this much and this detailed information. The authors' knowledge of the subject matter and the experience in writing books are clearly reflected in the end product.

I would not only be very happy to use this book as a textbook next time I teach a course on modular forms, but I am also looking forward to having a hard copy in my library as an extensive reference book.

-- Imamoglu Özlem, ETH Zurich

Modular forms are central to many different fields of mathematics and mathematical physics. Having a detailed and complete treatment of all aspects of the theory by two world experts is a very welcome addition to the literature.

-- Peter Sarnak, Princeton University

#### Table of Contents

# Table of Contents

## Modular Forms: A Classical Approach

- Cover Cover11
- Title page iii4
- Contents v6
- Preface xi12
- Chapter 1. Introduction 114
- Chapter 2. Elliptic Functions, Elliptic Curves, and Theta Functions 1730
- Chapter 3. Basic Tools 7184
- Chapter 4. The Modular Group 115128
- Chapter 5. General Aspects of Holomorphic and Nonholomorphic Modular Forms 129142
- 5.1. Introduction 129142
- 5.2. Examples of Modular Forms: Eisenstein Series 143156
- 5.3. Differential Operators 152165
- 5.4. Taylor Coefficients of Modular Forms 169182
- 5.5. Modular Forms on the Modular Group and Its Subgroups 174187
- 5.6. Zeros, Poles, and Dimension Formulas 177190
- 5.7. The Modular Invariant 𝑗 188201
- 5.8. The Dedekind 𝜂-Function and the Product Formula for Δ 190203
- 5.9. Eta Quotients 192205
- 5.10. A Brief Introduction to Complex Multiplication 199212
- Exercises 203216

- Chapter 6. Sets of 2×2 Integer Matrices 215228
- Chapter 7. Modular Forms and Functions on Subgroups 253266
- Chapter 8. Eisenstein and Poincaré Series 269282
- 8.1. Definitions 269282
- 8.2. Basic Results on Poincaré and Eisenstein Series 273286
- 8.3. Poincaré and Eisenstein Series for Congruence Subgroups 279292
- 8.4. Fourier Expansions 281294
- 8.5. Eisenstein and Poincaré Series in 𝑀_{𝑘}(\G₀(𝑁),𝜒) 288301
- 8.6. Generalization of the Petersson Scalar Product 303316
- Exercises 307320

- Chapter 9. Fourier Coefficients of Modular Forms 311324
- Chapter 10. Hecke Operators and Euler Products 341354
- 10.1. Introduction 341354
- 10.2. Introduction to Hecke Operators 343356
- 10.3. The Hecke Operators Are Hermitian 349362
- 10.4. Eigenvalues and Eigenfunctions of Hecke Operators on \G 360373
- 10.5. Double Coset Operators 362375
- 10.6. Bases of Modular Forms for the Full Modular Group 363376
- 10.7. Euler Products 369382
- 10.8. Convolutions 374387
- Exercises 382395

- Chapter 11. Dirichlet Series, Functional Equations, and Periods 383396
- 11.1. Introduction 384397
- 11.2. The Main Theorem 386399
- 11.3. Weil’s Theorem 390403
- 11.4. Application to the Riemann Zeta Function 397410
- 11.5. Periods and Antiderivatives of Modular Forms 398411
- 11.6. The Case of Eisenstein Series 402415
- 11.7. Transformation under an Arbitrary \ga∈\G 404417
- 11.8. Eichler Cohomology 406419
- 11.9. Interpretation in Terms of Periods 414427
- 11.10. Action of Hecke Operators on Periods 419432
- 11.11. Rationality and Parity Theorems 425438
- 11.12. Rankin–Selberg Theory 431444
- Exercises 438451

- Chapter 12. Unfolding and Kernels 441454
- Chapter 13. Atkin–Lehner–Li Theory 515528
- Chapter 14. Theta Functions 557570
- Chapter 15. More General Modular Forms: An Introduction 593606
- Bibliography 679692
- Index of Notation 693706
- General Index 697710
- Back Cover Back Cover1714