Modular Forms: A Classical Approach (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 179)

The theory of modular forms is a fundamental tool used inmany areas of mathematics and physics. It is also a very concrete and“fun” subject in itself and abounds with an amazing number ofsurprising identities.This comprehensive textbook, which includes numerous exercises, aimsto give a complete picture of the classical aspects of the subject,with an emphasis on explicit formulas. After a number of motivatingexamples such as elliptic functions and theta functions, the modulargroup, its subgroups, and general aspects of holomorphic andnonholomorphic modular forms are explained, with an emphasis onexplicit examples. The heart of the book is the classical theorydeveloped by Hecke and continued up to the Atkin–Lehner–Litheory of newforms and including the theory of Eisenstein series,Rankin–Selberg theory, and a more general theory of theta seriesincluding the Weil representation. The final chapter explores in somedetail more general types of modular forms such as half-integralweight, Hilbert, Jacobi, Maass, and Siegel modular forms.Some “gems” of the book are an immediatelyimplementable trace formula for Hecke operators, generalizations ofHaberland's formulas for the computation of Petersson inner products,W. Li's little-known theorem on the diagonalization of thefull space of modular forms, and explicit algorithms due tothe second author for computing Maass forms.This book is essentially self-contained, the necessary tools suchas gamma and Bessel functions, Bernoulli numbers, and so on beinggiven in a separate chapter.Graduate students and researchers interested in modularforms. 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. This book is essentially self-contained, the necessary tools being given in a separate chapter. This book gives a beautiful introduction to the theory of modular forms, with a delicate balance of analytic and arithmetic perspectives. The target readership 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 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 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.--Publisher website 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. He theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and 2fun3 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 2gems3 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 Henri Cohen, Fredrik Strömberg. Includes Bibliographical References And Indexes.