**Graduate Studies in Mathematics** 2007; 268 pp; hardcover Volume: 79 ISBN-10: 0-8218-3960-8 ISBN-13: 978-0-8218-3960-7 List Price: US$57 Member Price: US$45.60 Order Code: GSM/79
| *This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments. Its novelty lies in its computational emphasis throughout: Stein not only defines what modular forms are, but shows in illuminating detail how one can compute everything about them in practice. This is illustrated throughout the book with examples from his own (entirely free) software package SAGE, which really bring the subject to life while not detracting in any way from its theoretical beauty. The author is the leading expert in computations with modular forms, and what he says on this subject is all tried and tested and based on his extensive experience. As well as being an invaluable companion to those learning the theory in a more traditional way, this book will be a great help to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations. There is also a useful Appendix by Gunnells on extensions to more general modular forms, which has enough in it to inspire many PhD theses for years to come. While the book's main readership will be graduate students in number theory, it will also be accessible to advanced undergraduates and useful to both specialists and non-specialists in number theory.* --*John E. Cremona, University of Nottingham* William Stein is an associate professor of mathematics at the University of Washington at Seattle. He earned a PhD in mathematics from UC Berkeley and has held positions at Harvard University and UC San Diego. His current research interests lie in modular forms, elliptic curves, and computational mathematics. Readership Graduate students and research mathematicians interested in modular forms. Reviews "Stein fills the gap in the literature on classical modular forms with this unique approach that centers on computation throughout, defining what modular forms are and showing in detail how one can compute everything about them in practice. He uses examples from his own free software package, making this a suitable text for beyond the introductory level." *-- Book News Inc.* "The author, a leading expert in the field of computational arithmetic, presents here the first comprehensive textbook about algorithms for computing with modular forms, together with an accompanying introduction to the underlying theory of modular forms. ... Due to the author's vivid and lucid style of exposition, his mastery in the field, and the minimum of assumed prerequisites, this textbook for graduate students should be even easily accessible to advanced undergraduates, or to interested mathematicians in general." *-- Zentralblatt MATH* "The exposition is very clear and vivid showing the author's mastery in the subject. It contains illustrating examples, many methodological comments and references to the relevant research literature for further reading." *-- European Mathematical Society Newsletter* "...it must be noted that the author gives a very lucid and user-friendly introduction to the required background on modular forms and modular symbols, with reference to standard textbooks for detailed proofs. Throughout the book, the comprehensive and up-to-date citations to open questions, recent developments, applications, and cognate topics will be useful for current researchers in the field." *-- Mathematical Reviews* |