Fourier Analysis on Number Fields (Graduate Texts in Mathematics) (v. 186) (Hardcover)

by Dinakar Ramakrishnan (Author), Robert J. Valenza (Author) "Our work begins with the development of a topological framework for the key elements of our subject..." (more)
Key Phrases: first spectral theorem, function modk, idele class characters, Let Ube, Representations of Locally Compact Groups, Use Proposition (more...)

Editorial Reviews

Product Description
The general aim of this book is to provide a modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasizing harmonic analysis on topological groups. The more particular goal is to cover John Tate's visionary thesis, giving virtually all of the necessary analytic details and topological preliminaries--technical prerequisites that are often foreign to the typical, more algebraically inclined number theorist. While most of the existing treatments of Tate's thesis are somewhat terse and less than complete, the authors' intent is to be more leisurely, more comprehensive, and more comprehensible. The text addresses students who have taken a year of graduate-level courses in algebra, analysis, and topology. While the choice of objects and methods is naturally guided by specific mathematical goals, the approach is by no means narrow. In fact, the subject matter at hand is germane not only to budding number theorists, but also to students of harmonic analysis or the representation theory of Lie groups. Moreover, the work should be a good reference for working mathematicians interested in any of these fields. Specific topics include: topological groups, representation theory, duality for locally compact abelian groups, the structure of arithmetic fields, adeles and ideles, an introduction to class field theory, and Tate's thesis and applications.

Product Details

• Hardcover: 350 pages
• Publisher: Springer; 1 edition (December 7, 1998)
• Language: English
• ISBN-10: 0387984364
• ISBN-13: 978-0387984360
• Product Dimensions: 9.5 x 6.3 x 0.9 inches
• Shipping Weight: 1.5 pounds (View shipping rates and policies)
• Average Customer Review: 3.5 out of 5 stars See all reviews (2 customer reviews)
• Amazon.com Sales Rank: #745,004 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

25 of 26 people found the following review helpful:

5.0 out of 5 stars A treat for beginners to some exciting areas of mathematics, July 11, 1999

By A Customer

"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

8 of 20 people found the following review helpful:

2.0 out of 5 stars one more opinion, October 18, 2002

By	Dave Ziegler (USA) - See all my reviews

I don't agree with the previous reviewer about the value of this
book - I think that with several minor exceptions there is nothing in this book which could justify its publication.
Of course, as it is clear to every expert, there is nothing
really new in this book; but sometimes one can rewrite old
things in such a way that a new book is justified.
With the material of this book I know much better expositions
of every chapter of it (including harmonic analysis, number theory and Tate-Iwasawa method) in other sourses.
There are also some mistakes and errors (for example,
the Poisson summation formula is not proven),
some of which may cause the reader
think that there were mistakes on the original works.

This text could have appeared online as lecture notes,
but the publication of it by Springer confirm the well known fact of degradation of their mathematical series.

D. Ziegler