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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.

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The Upper Half Plane and Fuchsian Groups.- Automorphic Forms.- L-Functions.- Modular Groups and Modular Forms.- Unit Groups of Quaternion Algebras.- Traces of Hecke Operators.- Eisenstein Series.

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From the reviews of the first edition:

"Toshitune Miyake´s book is a useful introductory book on the subject of modular forms. ... Some key features of Miyake´s book are that it is accessible, well written, and well organized. ... As a result, Miyake´s book is both an excellent first text from which to learn the subject and a very good reference. I have found myself returning to the book repeatedly for tidbits that are hard to find in the literature. ... Miyake´s book is an excellent introduction to ... modular forms." (Eknath Ghate, SIAM Review, Vol. 48 (3), 2006)

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