Modular Forms

Henri Cohen, Universite Bordeaux, France.Fredrik Stromberg, University of Nottingham, United Kingdom.

“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

• IntroductionElliptic functions, elliptic curves, and theta functionBasic toolsThe modular groupGeneral aspects of holomorphic and nonholomorphic modular formsSets of $2 \times 2$ integer matricesModular forms and functions on subgroupsEisenstein and Poincare seriesFourier coefficients of modular formsHecke operators and Euler productsDirichlet series, functional equations, and periodsUnfolding and kernelsAtkin-Lehner-Li theoryTheta functionsMore general modular forms: An introductionBibliographyIndex of notationGeneral index.