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Modular Forms

Toshitsune Miyake
Springer Verlag
Publication Date: 
Number of Pages: 
Springer Monographs in Mathematics
[Reviewed by
Fernando Q. Gouvêa
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This seems to be an unchanged (except for the cover) reprint of a monograph published in 1989. It has a complex publication history before that, being, as the author tells us in the introduction, an expanded version of a portion of a book written in Japanese. It treats the basic theory of Fuchsian groups and automorphic forms in some generality, including both the usual congruence groups acting on the upper halfplane and the examples coming from quaternion algebras. The book concludes with material on the Eichler-Selberg trace formula and on Eisenstein series.

Miyake writes very much under the influence of Shimura, whose famous (and famously hard to read) Introduction to the Arithmetic Theory of Automorphic Functions is a constant presence in the background. He is particularly good on the computation of the dimensions of the spaces of modular forms.

When this book was first published, there were few readable books on the subject, and it had little competition. Things are tougher now; in particular, Diamond and Shurmer have just published A First Course on Modular Forms, a book that makes a bid to become the standard graduate text on the subject. Still, it is useful to have Miyake's book back in print and available.

Fernando Q. Gouvêa read Miyake's book soon after it first came out in 1989, and was glad to have done so. He thinks he was probably in Brazil at the time. He is now professor of mathematics at Colby College in Waterville, ME.

 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.