Modular Forms
A Classical Approach
Published on 30. August 2017
Book
Hardback
700 pages
978-0-8218-4947-7 (ISBN)
Description
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 are given in a separate chapter.
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 are given in a separate chapter.
Reviews / Votes
"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 Stroemberg 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 OEzlem, 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
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Dimensions
Height: 254 mm
Width: 178 mm
Weight
1400 gr
ISBN-13
978-0-8218-4947-7 (9780821849477)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Persons
Henri Cohen, Universite Bordeaux, France.
Fredrik Stromberg, University of Nottingham, United Kingdom.
Fredrik Stromberg, University of Nottingham, United Kingdom.
Content
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 Poincare 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
Bibliography
Index of notation
General index.
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 Poincare 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
Bibliography
Index of notation
General index.