For the most part, this book is the translation from Japanese of the earlier book written jointly by Koji Doi and the author who revised it substantially for the English edition. It sets out to provide the reader with the basic knowledge of elliptic modular forms necessary to understand the recent developments in number theory. The first part gives the general theory of modular groups, modular forms and Hecke operators, with emphasis on the Hecke-Weil theory of the relation between modular forms and Dirichlet series. The second part is on the unit groups of quaternion algebras, which are seldom dealt with in books. The so-called Eichler-Selberg trace formula of Hecke operators follows next and the explicit computable formula is given. In the last chapter, written for the English edition, Eisenstein series with parameter are discussed following the recent work of Shimura: Eisenstein series are likely to play a very important role in the future progress of number theory, and this chapter provides a good introduction to the topic. For the most part, this book is the translation from Japanese of the earlier book written jointly by Koji Doi and the author who has revised it substantially for the English edition. It sets out to provide the reader with the basic knowledge of elliptic modular forms necessary to understand the recent developments in number theory. The first part gives the general theory of modular groups, modular forms and Hecke operators, with emphasis on the Hecke-Weil theory of the relation between modular forms and Dirichlet series. The second part is on the unit groups of quaternion algebras, which are seldom dealt with in books. The so-called Eichler-Selberg trace formula of Hecke operators follows next and the explicit computable formula is given. In the last chapter, Eisenstein series with parameter are discussed following the recent work of Shimura: Eisenstein series are likely to play a very important role in the future progress of number theory, and this chapter provides a good introduction to the topic This Book Is A Translation Of The Earlier Book Written By Koji Doi And The Author, Who Revised It Substantially For This English Edition. It Offers The Basic Knowledge Of Elliptic Modular Forms Necessary To Understand Recent Developments In Number Theory. It Also Treats The Unit Groups Of Quaternion Algebras, Rarely Dealt With In Books; And In The Last Chapter, Eisenstein Series With Parameter Are Discussed Following The Recent Work Of Shimura. Front Matter....Pages I-IX The Upper Half Plane and Fuchsian Groups....Pages 1-36 Automorphic Forms....Pages 37-78 L -Functions....Pages 79-95 Modular Groups and Modular Forms....Pages 96-194 Unit Groups of Quaternion Algebras....Pages 195-218 Traces of Hecke Operators....Pages 219-267 Eisenstein Series....Pages 268-293 Back Matter....Pages 295-338 Provides the basic knowledge of elliptic modular forms necessary to understand the developments in number theory. This book includes general theory of modular groups, Hecke operators, the unit groups of quaternion algebras, and more. Eisenstein series with parameter are discussed following the work of Shimura: Eisenstein series
