Both modular forms and Dirichlet series are central topics in number theory, and the connection between them is at the heart of many important recent results. Goro Shimura is a master of this material, so one welcomes anything he writes, while also noting that he is notoriously hard to read. When I was a graduate student, Shimura's Introduction to the Arithmetic Theory of Automorphic Functions was already a classic, out of print but eagerly sought out. (It is now back in print, and a good thing too!) This new book may well become similarly important.
Despite the title, this is not an introductory book. The tone is set very clearly in the author's preface. "A book on any mathematical subject above textbook level is not of much value unless it contains new ideas and new perspectives," he says. And he means it: what this book contains is a new account of some important ideas about modular forms and the L-functions corresponding to them. It contains both new methods and new results. "The essential points of this new books have never been presented publicly or privately," Shimura explains. So, "elementary" or not, this should be viewed as a research monograph, an extended research article published in book form. The writing is dense, full of formulas, and many details are either assumed known or dealt with by giving a reference, as one might expect in a research article.
At times one suspects that the author doesn't quite realize how difficult this material actually is. (Or, perhaps equivalently, how little beginners actually know.) My favorite is a remark that the fact that the integral from minus infinity to infinity of exp(–π x2) is equal to the square root of π "can be found in any elementary calculus book."
The book has a few production glitches. The first page of the introduction already signals that the proofreading was less than complete, with a reference to "functional equaltions". Typos of this kind appear throughout ("subsect" for "subject" on page 6, "aregular" for "a regular" on page 13, etc.); most of them can be figured out by the attentive reader. Every so often one gets a strange turn of phrase; for example, "In this section, or rather in this whole book..." A good copyeditor would have caught most of these.
Some production decisions are also to be lamented. The pages are far too full, resulting in lines that are too long and margins that are too small. Fermat wouldn't even be able to state his result in these margins!
Despite the disappointing production values, this is still a book many will want to read. Beginners will want to fortify themselves first, perhaps by reading the introductory account by Diamond and Shurman, Shimura's own classic treatment , or Miyake's book . But they should spend some time with this book at some point. After all, Shimura is a master, and braving the thickets of notation and the long lines will lead to treasure in the end.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.