数域中的傅里叶分析

PREFACEINDEX OF NOTATION1 TOPOLOGICAL GROUPS 1.1 Basic Notions 1.2 Haar Measure 1.3 Profinite Groups 1.4 Pro-p-Groups Exercises2 SOME REPRESENTATION THEORY 2.1 Representations of Locally Compact Groups 2.2 Banach Algebras and the Gelfand Transform 2.3 The Spectral Theorems 2.4 Unitary Representations Exercises3 DUALITY FOR LOCALLY COMPACT ABELIAN GROUPS

PREFACEINDEX OF NOTATION1 TOPOLOGICAL GROUPS 1.1 Basic Notions 1.2 Haar Measure 1.3 Profinite Groups 1.4 Pro-p-Groups Exercises2 SOME REPRESENTATION THEORY 2.1 Representations of Locally Compact Groups 2.2 Banach Algebras and the Gelfand Transform 2.3 The Spectral Theorems 2.4 Unitary Representations Exercises3 DUALITY FOR LOCALLY COMPACT ABELIAN GROUPS 3.1 The Pontryagin Dual 3.2 Functions of Positive Type 3.3 The Fourier Inversion Formula 3.4 Pontryagin Duality Exercises4 THE STRUCTURE OF ARITHMETIC FIELDS 4.1 The Module of an Antomorphism 4.2 The Classification of Locally Compact Fields 4.3 Extensions of Local Fields 4.4 Places and Completions of Global Fields 4.5 Ramification and Bases Exercises5 ADELES, IDELES, AND THE CLASS GROUPS 5.1 Restricted Direct Products,Characters,and Measures 5.2 Adeles,Ideles,and the Approximation Theorem 5.3 The Geometry of AKIK 5.4 The Class Groups Exercises6 A QUICK TOUR OF CLASS FIELD THEORY7 TATE’S THESIS AND APPLICATIONSAPPENDICESREFERENCESINDEX

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.