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內容簡介

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.

內頁插圖

目錄

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

前言/序言

數域上的傅裏葉分析 [Fourier Analysis on Number Fields] 下載 mobi epub pdf txt 電子書

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這本書非常的不錯，值得閱讀。它從一般意義上的傅裏葉分析推廣到瞭一般數域上，使得我們在一般的數域上可以對其進行分析的運算，這其中也用到瞭拓撲，測度論等知識，如果大傢想真正地讀懂這本書的細節，最好還是要去看看拓撲，代數，測度和錶示的基本知識，希望大傢能夠從這本書中，真正地瞭解代數上的分析學問。祝大傢閱讀愉快！

評分☆☆☆☆☆

有幫助

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數學加英語，一舉兩得

評分☆☆☆☆☆

這是一本數論書，有一定的難度。傅立葉分析幾百年來一直是數學的主流，數論當然也是。建議數學工作者仔細研讀本書，這樣的書不多

評分☆☆☆☆☆

GTM的書太適閤裁對基礎數學熱愛的人瞭

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數學加英語，一舉兩得

評分☆☆☆☆☆

English version is cheaper and looks better

評分☆☆☆☆☆

這本書非常的不錯，值得閱讀。它從一般意義上的傅裏葉分析推廣到瞭一般數域上，使得我們在一般的數域上可以對其進行分析的運算，這其中也用到瞭拓撲，測度論等知識，如果大傢想真正地讀懂這本書的細節，最好還是要去看看拓撲，代數，測度和錶示的基本知識，希望大傢能夠從這本書中，真正地瞭解代數上的分析學問。祝大傢閱讀愉快！

評分☆☆☆☆☆

數學加英語，一舉兩得

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