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am 12. Juli 1999
"Fourier Analysis on Number Fields" provides a much-needed graduate text for number theorists and group theorists. Though necessarily difficult in parts because of the complicated material it covers, it is very manageable for a student. It includes a number of exercises at the end of each of its seven chapters. At the same time, it is very valuable for a researcher. Perhaps its best feature are the wonderful introductions to each chapter. These give insightful historical overviews, in keeping with the authors' theme of presenting material from disparate sources together in a coherent text. It is obvious that they spent a lot of attention on the beginner's needs.
Indeed, existing texts cover most if not all of the material in this new book. Others, including some new books on automorphic forms, take the reader much further. However, not everyone has the same starting point and all of these can be very frustrating for a beginner. The novelty and utility in this book is that it does not assume the reader comes from some particular background. Off-hand I could name five or six other books I would consult to learn the material "FANF" covers. But each comes from a different community of mathematicians, with their own jargon, in different eras, and are intended for different audiences. "FANF" sacrifices some proofs for clarity, and gives references to the classical sources for further details.
One of the authors' goals was to give explicit background on the structure of the fields involved, particularly the delicate arithmetic structure of number fields which is sometimes frustrating to learn from other sources. They have covered the structure of locally-compact fields very well and clearly. In fact, in one of our graduate courses at Yale University last fall, lectures on p-adic groups and trees were based out of the presentation in "FANF." The book is very concrete, which is especially useful for analysts who aren't used to doing integrals over, say, function fields in finite characteristic. I think it will be a favorite amongst this community - it treats advanced stages of "math phobia."
At the same time this is the natural book for an introductory course on modern automorphic forms. It completely covers the GL(1) theory and leaves the reader in an excellent position to continue on to study the Jacquet-Langlands theory. It has a nice treatment of L-functions, and even includes some analytic results which feature prominently in the recent research of one of the authors. There isn't a book that I know of which fits the nice "FANF" occupies, and better yet, it complements the earlier ones very well.
Let me just mention two examples of recent research which explain why I think a book covering its various topics is so important. Hyman Bass and Alex Lubotzky found a counter-example to the Platonov conjecture. This problem involves the representation theory of profinite groups, and lattices acting on trees. "FANF" has beautiful treatments of these. At the same time, a key ingredient of their proof was understanding the cohomology of discrete subgroups of Lie groups. Ultimately this can be interpreted as a problem in automorphic forms! In fact, they used results of David Vogan and Gregg Zuckerman about cohomological representations in their work. Another example is that the "Selberg Property-Tau" has become very important in p-adic group theory; it originated as a bound on Laplace eigenvalues in modular forms. Fortunately these aspects of algebraic groups are becoming more deeply linked, and "FANF" is a most-recommended book to start learning any of these subjects from.
Stephen D. Miller Department of Mathematics Yale University
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