Modular Forms

by Toshitsune Miyake, Yoshitaka Maeda (Translator)

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.

335 pages, Hardcover

First published October 4, 1989

Reviews

[Wissam Raji · Rating 4 out of 5]

Another classic in the theory of modular forms with a unique approach to the theory from a geometric point of view. The geometric treatment of moduli spaces is definitely one of the best in any book on the theory of modular forms. The book focuses on Eisenstein series on subgroups and mainly those of low weights mainly weights 1 and 2. It is considered a hard book for beginners in the theory of modular forms but the treatment is definitely elegant.