Fourier Analysis on Number Fields

✍ Scribed by Dinakar Ramakrishnan, Robert J. Valenza

• Publisher: Springer
• Year: 1998
• Tongue: English
• Leaves: 377
• Series: Graduate Texts in Mathematics v. 186
• Edition: 1
• Category: Library

✦ Synopsis

I agree very much on what Stephen Miller said. This textbook is a very execellent introductory textbook to modern number theory. It does not require any particular math background besides elementary undergraduate maths so that it is suitable to new graduate students. The exercises are very nice and helpful. The level is a little bit challenging. I ever taught courses based on this book twice and both students and I benefit a lot.

For the contents, the textbook provides a thourough treatment on basics of modern NT such as local fields, adeles, ideles, Fourier inverse formula etc. Moreover, I think the textbook might be the best source so far I know for on Tate's thesis as a textbook. It is a perfect starting book for readers who are interested on automorphic forms. Also, just as Miller said, it is also a good reference book to mathematicians with various background, not just merely number theorists.

So I recommend this textbook strongly.

Song Wang, the Morningside Center of Mathematics, AMSS, CAS, China.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 12
Index of Notation......Page 16
1.1 Basic Notions......Page 24
1.2 Haar Measure......Page 32
1.3 Profinite Groups......Page 42
1.4 Pro -p-Groups......Page 59
Exercises......Page 65
2.1 Representations of Locally Compact Groups......Page 69
2.2 Banach Algebras and the Gclfand Transform......Page 73
2.3 The Spectral Theorems......Page 83
2.4 Unitary Representations......Page 96
Exercises......Page 101
3.1 The Pontryagin Dual......Page 109
3.2 Functions of Positive Type......Page 114
3.3 The Fourier Inversion Formula......Page 125
3.4 Pontryagin Duality......Page 141
Exercises......Page 148
4.1 The Module of an Automorphism......Page 155
4.2 The Classification of Locally Compact Fields......Page 163
4.3 Extensions of Local Fields......Page 173
4.4 Places and Completions of Global Fields......Page 177
4.5 Ramification and Bases......Page 188
Exercises......Page 197
5 ADELES, IDELES, AND THE CLASS GROUPS......Page 202
5.1 Restricted Direct Products, Characters, and Measures......Page 203
5.2 Adeles, Ideles, and the Approximation Theorem......Page 212
5.3 The Geometry of AK/K......Page 214
5.4 The Class Groups......Page 219
Exercises......Page 231
6 A QUICK TOUR OF CLASS FIELD THEORY......Page 236
6.1 Frobenius Elements......Page 237
6.2 The Tchebotarev Density Theorem......Page 242
6.3 The Transfer Map......Page 243
6.4 Artin's Reciprocity Law......Page 245
6.5 Abelian Extensions of Q and Qp......Page 249
Exercises......Page 261
7 TATE'S THESIS AND APPLICATIONS......Page 264
7.1 Local -Functions......Page 266
7.2 The Riemann-Roch Theorem......Page 282
7.3 The Global Functional Equation......Page 292
7.4 Hecke L-Functions......Page 299
7.5 The Volume of CK and the Regulator......Page 304
7.6 Dirichlet's Class Number Formula......Page 309
7.7 Nonvanishing on the Line Re(s)=1......Page 312
7.8 Comparison of Hecke L-Functions......Page 318
Exercises......Page 320
A.1 Finite-Dimensional Notmed Linear Spaces......Page 338
A.2 The Weak Topology......Page 340
A.3 The Weak-Star Topology......Page 342
A.4 A Review of LP-Spaces and Duality......Page 346
B.1 Basic Properties......Page 349
B.2 Extensions of Dedekind Domains......Page 357
REFERENCES......Page 362
INDEX......Page 368
Back Cover......Page 377