数域上的傅里叶分析

PREFACE

INDEX OF NOTATION

TOPOLOGICAL GROUPS

1.1 Basic Notions

1.2 Haar Measure

1.3 Profinite Groups

1.4 Pro-p-Groups

Exercises

2 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

Exercises

3 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

Exercises

4 THE STRUCTURE OF ARITHMETIC FIELDS

4. I The Module of an Automorphism

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

Exercises

5 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 Ar/K

5.4 The Class Groups

Exercises

6 A QUICK TOUR OF CLASS FIELD THEORY

6.1 Frobenius Elements

6.2 The Tchebotarev Density Theorem

6.3 The Transfer Map

6.4 Artin's Reciprocity Law

6.5 Abelian Extensions of Q and Qp

Exercises

7 TATE'S THESIS AND APPLICATIONS

7.1 Local （-Functions

7.2 The Riemann-Roch Theorem

7.3 The Global Functional Equation

7.4 Hecke L-Functions .

7.5 The Volume of C and the Regulator

7.6 Dirichlet's Class Number Formula

7.7 Nonvanishing on the Line Re（s）——I

7.8 Comparison of Hecke L-Functions

Exercises

APPENDICES

Appendix A: Normed Linear Spaces

A. 1 Finite-Dimensional Normed Linear Spaces

A.2 The Weak Topology

A.3 The Weak-Slat Topology

A.4 A Review of LP-Spaces and Duality

Appendix B: Dedekind Domains

B.1 Basic Properties

B.2 Extensions of Dedekind Domains

REFERENCES

INDEX

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 John 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.