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Modular Forms: A Classical Approach
 
Henri Cohen Université Bordeaux, Bordeaux, France
Fredrik Strömberg University of Nottingham, Nottingham, United Kingdom
Modular Forms
Hardcover ISBN:  978-0-8218-4947-7
Product Code:  GSM/179
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-1-4704-4081-7
Product Code:  GSM/179.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-4947-7
eBook: ISBN:  978-1-4704-4081-7
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
Modular Forms
Click above image for expanded view
Modular Forms: A Classical Approach
Henri Cohen Université Bordeaux, Bordeaux, France
Fredrik Strömberg University of Nottingham, Nottingham, United Kingdom
Hardcover ISBN:  978-0-8218-4947-7
Product Code:  GSM/179
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-1-4704-4081-7
Product Code:  GSM/179.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-4947-7
eBook ISBN:  978-1-4704-4081-7
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 Details
     
     
    Graduate Studies in Mathematics
    Volume: 1792017; 700 pp
    MSC: 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 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.

    Updates and corrections are available for the book; please see errata.

    Readership

    Graduate 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1792017; 700 pp
MSC: 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 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.

Updates and corrections are available for the book; please see errata.

Readership

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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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