It has always been difficult to start learning about modular forms. When I was in graduate school, it was a "well known fact" that none of the available books were of much help to a beginners — unless they were willing to undertake a long hard slog. Things are much improved now, but we were still lacking a textbook that could be honestly described as both comprehensive and accessible. Diamond and Shurman's *First Course* is a largely successful attempt to provide just such a book.

Of course, since Wiles' work, one of the main reasons people might want to learn about modular forms is to understand what it is that he proved (if not how it was done). Specifically, people might want to understand the modularity conjecture for elliptic curves, which is what Wiles (together with Breuil, Conrad, Diamond, and Taylor) really ended up proving. So the authors of this book have made it turn on the notion of modularity. Throughout the book, they give several equivalent forms of expressing modularity, and they show that all of these statements are equivalent. In the very last section, they give a brief sketch of how modularity is actually proved, and explain why proving modularity is enough to obtain Fermat's Last Theorem.

One of the reasons it is hard to write about modular forms is that there are multiple interpretations of what they are, each of which requires some theoretical background to understand. So modular forms could be a certain kind of holomorphic function on the complex upper half-plane, in which case the author must assume some knowledge of function theory. Or, they could be sections of line bundles on some Riemann surface, which requires understanding of at least some basic geometry of complex manifolds. They can also be thought of as sections of an algebraic line bundle (or an invertible sheaf) on an algebraic curve, which means bringing in algebraic geometry. One can think of modular forms as functions of isomorphism classes of elliptic curves. Or one can relate them to the Jacobian varieties of certain Riemann surfaces (or of certain algebraic curves), or to homology and cohomology groups, or to group representation theory. An honest survey of the subject really has to tackle all of these points of view; that means assuming quite a bit of material.

Diamond and Shurman have decided to assume readers have had "undergraduate semester courses in linear algebra, modern algebra, real analysis, complex analysis, and elementary number theory," and *not* to assume any background in algebraic number theory or algebraic geometry, since these are usually not taught at the undergraduate level. After reading the book, I have come to feel that "undergraduate courses," in the quote above, needs to be replaced either with "first-year graduate courses" or with "very good undergraduate courses." The demands placed on the reader are quite serious. For example, the "modern algebra" they use includes Galois theory and the structure theorem for modules over principal ideal domains.

Not assuming any algebraic number theory turns out to be a minor issue, since what little they need is easily developed. Not assuming any algebraic geometry is a much more serious restriction, since to get to the modularity conjecture (now a theorem, of course) requires using the algebraic geometry point of view in quite a serious way. Because of this, when they get to that point, they end up having to spend quite a lot of time building up the necessary background. At such times, the background work can get close to overwhelming the main topic. This also forces them to restrict their attention to modular forms of weight two beginning about halfway through the book.

Within these limitations, however, *A First Course in Modular Forms* is a success. It offers quite a wide view of the arithmetical aspects of the theory of modular forms. It does not by any means cover everything, but a course taught from this text would be a very good way to lead students into the area. (I think it would be less effective for independent study. There are a lot of trees here, and a student working alone will probably find it hard to see the overall forest.) Students working with this book will need some help from an instructor who is willing to help students distinguish between technicalities and deep insights. In such a setting, I expect that Diamond and Shurman's book would serve very well.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College. He has written several papers about modular forms. He has also written a longer and more detailed review of this book, to appear in the

*American Mathematical Monthly*.