BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

In preparation for this review, I decided to remind myself how I became interested in automorphic forms as an undergraduate. I seem to remember that Eric Temple Bell [1] had something to do with it. Imagine my surprise when I found the following statement on page 333 of [1]: “The subject, elliptic functions, in which Jacobi did his first great work, has already been given what may seem like its share of space; for after all it is today more or less of a detail in the vaster theory of functions of a complex variable which, in its turn, is fading from the ever-changing scene as a thing of living interest.” After wondering how I could have gone into a dead field, I looked up my review [19] of Serge Lang [10]. I think that review made a pretty good case that the subject of automorphic forms (even the special case of modular forms) was still alive in 1980. Now we can also point to the proof by Andrew Wiles of Fermat’s Last Theorem via the Shimura-Taniyana conjecture (and as mentioned in a recent “Star Trek Deep Space Nine” episode, the subject may still be alive in the twenty-fourth century). So then I looked again at Paul Garrett’s review [3] of my book [18]. The upshot of the review was a complaint that I had not written a book about group representations and further had skipped the hard details. But I will try not to play a similar trick on the author of the book under review. For once more, the Bulletin has chosen a reviewer who would write a completely different book. Unlike [18], the book under review is certainly not full of “extra-mathematical applications and references”, nor can it be criticized for leaving out the hard details. This is a treatise for the specialists. This book does give evidence that E. T. Bell was wrong. We are dealing here with a living field. It is not a subfield of complex analysis. Nor is it a subfield of group representations. Bruggeman’s book uses functional analysis and the theory of several complex variables instead. But there is an intersection with most parts of mathematics. As I have hinted, if you do not know anything about modular forms, this is not the book for you. You should first look perhaps at Svetlana Katok’s beautiful introduction to the subject [9], or at [10], or even at [18, Chapter 3]. A brief summary of the entire subject can be fund in the Japan Mathematical Society Encyclopedia article on automorphic forms [8]. To explain a bit about the subject, let us consider a favorite modular form, the cuspidal Maass wave form. It is an SL(2,Z)-invariant eigenfunction for the nonEuclidean Laplacian on the Poincaré upper half plane H, such that f(z) goes to 0 as z approaches the cusp at infinity. More precisely, a cuspidal Maass wave form is a function f : H → C which satisfies the following three conditions for all z ∈ H: