目錄
PREFACE
INDEX OF NOTATION
1 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
4.1 The Module Of an Automorphism
4.2 The Classification ofLocally 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 Ak/K
5.4 The Class Groups
Exercises
6 A QUICK TOUR 0F CLASS FIELD THEORY
6.1 Frobenius Eiements
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 tile Line Re(s)=1
7.8 Comparison of Hecke L-Functions
Exercises
APPENDICES
Appendix A: Normed Linear Spaces
A.1 Finite-Dimensional Nor med Linear Spaces
A.2 The Weak Topology
A.3 The Weak-Star 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
[