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Hardcover ISBN:  9780821849477 
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Hardcover ISBN:  9780821849477 
Product Code:  GSM/179 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470440817 
Product Code:  GSM/179.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821849477 
eBook ISBN:  9781470440817 
Product Code:  GSM/179.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 

Book DetailsGraduate Studies in MathematicsVolume: 179; 2017; 700 ppMSC: Primary 11
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 halfintegral 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 littleknown 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 selfcontained, the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on being given in a separate chapter.
Updates and corrections are available for the book; please see errata.
ReadershipGraduate students and researchers interested in modular forms.

Table of Contents

Chapters

Introduction

Elliptic functions, elliptic curves, and theta function

Basic tools

The modular group

General aspects of holomorphic and nonholomorphic modular forms

Sets of $2 \times 2$ integer matrices

Modular forms and functions on subgroups

Eisenstein and Poincaré series

Fourier coefficients of modular forms

Hecke operators and Euler products

Dirichlet series, functional equations, and periods

Unfolding and kernels

Atkin–Lehner–Li theory

Theta functions

More general modular forms: An introduction


Additional Material

Reviews

According to the preface, the authors expect the main use of this book to be for advanced graduate students to learn about the classical theory of modular forms. However, given the tremendous amount of detail provided, the book should also be useful as a reference for established researchers in the area. Further, it can undoubtedly be mined by instructors for a graduate course on modular forms.
Sander Zwegers, Mathematical Reviews 
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


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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 halfintegral 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 littleknown 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 selfcontained, the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on being given in a separate chapter.
Updates and corrections are available for the book; please see errata.
Graduate students and researchers interested in modular forms.

Chapters

Introduction

Elliptic functions, elliptic curves, and theta function

Basic tools

The modular group

General aspects of holomorphic and nonholomorphic modular forms

Sets of $2 \times 2$ integer matrices

Modular forms and functions on subgroups

Eisenstein and Poincaré series

Fourier coefficients of modular forms

Hecke operators and Euler products

Dirichlet series, functional equations, and periods

Unfolding and kernels

Atkin–Lehner–Li theory

Theta functions

More general modular forms: An introduction

According to the preface, the authors expect the main use of this book to be for advanced graduate students to learn about the classical theory of modular forms. However, given the tremendous amount of detail provided, the book should also be useful as a reference for established researchers in the area. Further, it can undoubtedly be mined by instructors for a graduate course on modular forms.
Sander Zwegers, Mathematical Reviews 
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