Fourier Analysis on Number Fields

1. Front Matter

   Pages i-xxi

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2. Topological Groups

   • Dinakar Ramakrishnan, Robert J. Valenza

   Pages 1-45

3. Some Representation Theory

   • Dinakar Ramakrishnan, Robert J. Valenza

   Pages 46-85

4. Duality for Locally Compact Abelian Groups

   • Dinakar Ramakrishnan, Robert J. Valenza

   Pages 86-131

5. The Structure of Arithmetic Fields

   • Dinakar Ramakrishnan, Robert J. Valenza

   Pages 132-178

6. Adeles, Ideles, and the Class Groups

   • Dinakar Ramakrishnan, Robert J. Valenza

   Pages 179-212

7. A Quick Tour of Class Field Theory

   • Dinakar Ramakrishnan, Robert J. Valenza

   Pages 213-240

8. Tate’s Thesis and Applications

   • Dinakar Ramakrishnan, Robert J. Valenza

   Pages 241-314

9. Back Matter

   Pages 315-353

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This book grew out of notes from several courses that the first author has taught over the past nine years at the California Institute of Technology, and earlier at the Johns Hopkins University, Cornell University, the University of Chicago, and the University of Crete. Our general aim is to provide a modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasizing harmonic analysis on topological groups. Our more particular goal is to cover Jolm Tate's visionary thesis, giving virtually all of the necessary analytic details and topological preliminaries-technical prereq­ uisites that are often foreign to the typical, more algebraically inclined number theorist. Most of the existing treatments of Tate's thesis, including Tate's own, range from terse to cryptic; our intent is to be more leisurely, more comprehen­ sive, and more comprehensible. To this end we have assembled material that has admittedly been treated elsewhere, but not in a single volume with so much detail and not with our particular focus. We address our text to students who have taken a year of graduate-level courses in algebra, analysis, and topology. While our choice of objects and methods is naturally guided by the specific mathematical goals of the text, our 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.

• Algebra
• Arithmetic
• Finite
• Topology
• calculus
• harmonic analysis

• Mathematics Department, California Institute of Technology, Pasadena, USA

  Dinakar Ramakrishnan

• Department of Mathematics, Claremont McKenna College, Claremont, USA

  Robert J. Valenza

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