A course in p-adic analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2000
|
| Schriftenreihe: | Graduate texts in mathematics
198 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 437 S. graph. Darst. |
| ISBN: | 0387986693 |
Internformat
MARC
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| 100 | 1 | |a Robert, Alain M. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A course in p-adic analysis |c Alain M. Robert |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2000 | |
| 300 | |a XV, 437 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 198 | |
| 650 | 4 | |a p-adische Analysis | |
| 650 | 4 | |a p-adic analysis | |
| 650 | 0 | 7 | |a p-adische Analysis |0 (DE-588)4252360-6 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-008953527 | ||
Datensatz im Suchindex
| _version_ | 1804127841033388032 |
|---|---|
| adam_text | Contents
Preface v
1 p adic Numbers 1
1. The Ring Zp of /7 adic Integers 1
1.1 Definition 1
1.2 Addition of p adic Integers 2
1.3 The Ring of p adic Integers 3
1.4 The Order of a p adic Integer 4
1.5 Reduction mod p 5
1.6 The Ring of p adic Integers is a Principal Ideal Domain .... 6
2. The Compact Space Zp 7
2.1 Product Topology on Zp 7
2.2 The Cantor Set 8
2.3 Linear Models of Zp 9
2.4 Free Monoids and Balls of Zp 11
*2.5 Euclidean Models 12
*2.6 An Exotic Example 16
3. Topological Algebra 17
3.1 Topological Groups 17
3.2 Closed Subgroups of Topological Groups 19
3.3 Quotients of Topological Groups 20
3.4 Closed Subgroups of the Additive Real Line 22
3.5 Closed Subgroups of the Additive Group of/j adic Integers . . 23
3.6 Topological Rings 24
3.7 Topological Fields, Valued Fields 25
x Contents
4. Projective Limits 26
4.1 Introduction 26
4.2 Definition 28
4.3 Existence 28
4.4 Projective Limits of Topological Spaces 30
4.5 Projective Limits of Topological Groups 31
4.6 Projective Limits of Topological Rings 32
4.7 Back to the p adic Integers 33
*4.8 Formal Power Series and p adic Integers 34
5. The Field Qp of p adic Numbers 36
5.1 The Fraction Field of Zp 36
5.2 Ultrametric Structure on Qp 37
*5.3 Characterization of Rational Numbers Among
p adic Ones 39
5.4 Fractional and Integral Parts of p adic Numbers 40
5.5 Additive Structure of Qp andZp 43
*5.6 Euclidean Models of Qp 44
6. Hensel s Philosophy 45
6.1 First Principle 45
6.2 Algebraic Preliminaries 46
6.3 Second Principle 46
6.4 The Newtonian Algorithm 47
6.5 First Application: Invertible Elements in Zp 49
6.6 Second Application: Square Roots in Qp 49
6.7 Third Application: nth Roots of Unity in Zp 51
Table: Units, Squares, Roots of Unity 53
*6.8 Fourth Application: Field Automorphisms of Qp 53
Appendix to Chapter I: The p adic Solenoid 54
*A.l Definition and First Properties 55
* A.2 Torsion of the Solenoid 55
*A.3 Embeddings of R and Qp in the Solenoid 56
*A.4 The Solenoid as a Quotient 57
* A.5 Closed Subgroups of the Solenoid 60
*A.6 Topological Properties of the Solenoid 61
Exercises for Chapter I 63
2 Finite Extensions of the Field of p adic Numbers 69
1. Ultrametric Spaces 69
1.1 Ultrametric Distances 69
Table: Properties of Ultrametric Distances 73
1.2 Ultrametric Principles in Abelian Groups 73
Table: Basic Principles of Ultrametric Analysis 77
1.3 Absolute Values on Fields 77
1.4 Ultrametric Fields: The Representation Theorem 79
1.5 General Form of Hensel s Lemma 80
Contents xi
1.6 Characterization of Ultrametric Absolute Values 82
1.7 Equivalent Absolute Values 83
2. Absolute Values on the Field Q 85
*2.1 Ultrametric Absolute Values on Q 85
*2.2 Generalized Absolute Values 86
*2.3 Ultrametric Among Generalized Absolute Values 88
*2.4 Generalized Absolute Values on the Rational Field 88
3. Finite Dimensional Vector Spaces 90
3.1 Normed Spaces over Qp 90
3.2 Locally Compact Vector Spaces over Qp 93
3.3 Uniqueness of Extension of Absolute Values 94
3.4 Existence of Extension of Absolute Values 95
3.5 Locally Compact Ultrametric Fields 96
4. Structure of p adic Fields 97
4.1 Degree and Residue Degree 97
4.2 Totally Ramified Extensions 101
4.3 Roots of Unity and Unramified Extensions 104
4.4 Ramification and Roots of Unity 107
*4.5 Example 1: The Field of Gaussian 2 adic Numbers Ill
*4.6 Example 2: The Hexagonal Field of 3 adic Numbers 112
*4.7 Example 3: A Composite of Totally Ramified Extensions ... 114
Appendix to Chapter II: Classification of Locally Compact Fields ... 115
*A.l Haar Measures 115
*A.2 Continuity of the Modulus 116
*A.3 Closed Balls are Compact 116
*A.4 The Modulus is a Strict Homomorphism 118
*A.5 Classification 118
*A.6 Finite Dimensional Topological Vector Spaces 119
*A.7 Locally Compact Vector Spaces Revisited 121
*A.8 Final Comments on Regularity of Haar Measures 122
Exercises for Chapter II 123
3 Construction of Universal p adic Fields 127
1. The Algebraic Closure Q£ of Qp 127
1.1 Extension of the Absolute Value 127
1.2 Maximal Unramified Subextension 128
1.3 Ramified Extensions 129
1.4 The Algebraic Closure Q£ is not Complete 129
1.5 Krasner s Lemma 130
*1.6 A Finiteness Result 132
*1.7 Structure of Totally and Tamely Ramified Extensions 133
2. Definition of a Universal p adic Field 134
2.1 More Results on Ultrametric Fields 134
2.2 Construction of a Universal Field Qp 137
2.3 The Field Qp is Algebraically Closed 138
xii Contents
2.4 Spherically Complete Ultrametric Spaces 139
2.5 The Field Qp is Spherically Complete 140
3. The Completion Cp of the Field Qap 140
3.1 Definition of Cp 140
3.2 Finite Dimensional Vector Spaces over a Complete
Ultrametric Field 141
3.3 The Completion is Algebraically Closed 143
*3.4 The Field Cp is not Spherically Complete 143
*3.5 The Field Cp is Isomorphic to the Complex Field C 144
Table: Notation 145
4. Multiplicative Structure of Cp 146
4.1 Choice of Representatives for the Absolute Value 146
4.2 Roots of Unity 147
4.3 Fundamental Inequalities 148
4.4 Splitting by Roots of Unity of Order Prime top 150
4.5 Divisibility of the Group of Units Congruent to 1 151
Appendix to Chapter III: Filters and Ultrafilters 152
A.I Definition and First Properties 152
A.2 Ultrafilters 153
A.3 Convergence and Compactness 154
*A.4 Circular Filters 156
Exercises for Chapter III 156
4 Continuous Functions on Zp 160
1. Functions of an Integer Variable 160
1.1 Integer Valued Functions on the Natural Integers 160
1.2 Integer Valued Polynomial Functions 163
1.3 Periodic Functions Taking Values in a Field
of Characteristic p 164
1.4 Convolution of Functions of an Integer Variable 166
1.5 Indefinite Sum of Functions of an Integer Variable 167
2. Continuous Functions on Zp 170
2.1 Review of Some Classical Results 170
2.2 Examples of p adic Continuous Functions on Zp 172
2.3 Mahler Series 172
2.4 The Mahler Theorem 173
2.5 Convolution of Continuous Functions on Zp 175
3. Locally Constant Functions on Zp 178
*3.1 Review of General Properties 178
*3.2 Characteristic Functions of Balls of Zp 179
*3.3 The van der Put Theorem 182
4. Ultrametric Banach Spaces 183
4.1 Direct Sums of Banach Spaces 183
4.2 Normal Bases 186
4.3 Reduction of a Banach Space 189
4.4 A Representation Theorem 190
Contents xiii
4.5 The Monna Fleischer Theorem 190
*4.6 Spaces of Linear Maps 192
*4.7 The p adic Hahn Banach Theorem 194
5. Umbral Calculus 195
5.1 Delta Operators 195
5.2 The Basic System of Polynomials of a Delta Operator 197
5.3 Composition Operators 198
5.4 The van Hamme Theorem 201
5.5 The Translation Principle 204
Table: Umbral Calculus 207
6. Generating Functions 207
6.1 Sheffer Sequences 207
6.2 Generating Functions 209
6.3 The Bell Polynomials 211
Exercises for Chapter IV 212
5 Differentiation 217
1. Differentiability 217
1.1 Strict Differentiability 217
*1.2 Granulations 221
1.3 Second Order Differentiability 222
*1.4 Limited Expansions of the Second Order 224
1.5 Differentiability of Mahler Series 226
1.6 Strict Differentiability of Mahler Series 232
2. Restricted Formal Power Series 233
2.1 A Completion of the Polynomial Algebra 233
2.2 Numerical Evaluation of Products 235
2.3 Equicontinuity of Restricted Formal Power Series 236
2.4 Differentiability of Power Series 238
2.5 Vector Valued Restricted Series 240
3. The Mean Value Theorem 241
3.1 The p adic Valuation of a Factorial 241
3.2 First Form of the Theorem 242
3.3 Application to Classical Estimates 245
3.4 Second Form of the Theorem 247
3.5 A Fixed Point Theorem 248
*3.6 Second Order Estimates 249
4. The Exponentiel and Logarithm 251
4.1 Convergence of the Defining Series 251
4.2 Properties of the Exponential and Logarithm 252
4.3 Derivative of the Exponential and Logarithm 257
4.4 Continuation of the Exponential 258
4.5 Continuation of the Logarithm 259
5. The Volkenborn Integral 263
5.1 Definition via Riemann Sums 263
5.2 Computation via Mahler Series 265
xiv Contents
5.3 Integrals and Shift 266
5.4 Relation to Bernoulli Numbers 269
5.5 Sums of Powers 272
5.6 Bernoulli Polynomials as an Appell System 275
Exercises for Chapter V 276
6 Analytic Functions and Elements 280
1. Power Series 280
1.1 Formal Power Series 280
1.2 Convergent Power Series 283
1.3 Formal Substitutions 286
1.4 The Growth Modulus 290
1.5 Substitution of Convergent Power Series 294
1.6 The valuation Polygon and its Dual 297
1.7 Laurent Series 303
2. Zeros of Power Series 305
2.1 Finiteness of Zeros on Spheres 305
2.2 Existence of Zeros 307
2.3 Entire Functions 313
2.4 Rolle s Theorem 315
2.5 The Maximum Principle 317
2.6 Extension to Laurent Series 318
3. Rational Functions 321
3.1 Linear Fractional Transformations 321
3.2 Rational Functions 323
3.3 The Growth Modulus for Rational Functions 326
*3.4 Rational Mittag Leffler Decompositions 330
*3.5 Rational Motzkin Factorizations 333
*3.6 Multiplicative Norms on K(X) 337
4. Analytic Elements 339
*4.1 Enveloping Balls and Infraconnected Sets 339
*4.2 Analytic Elements 342
*4.3 Back to the Tate Algebra 344
*4.4 The Amice Fresnel Theorem 347
*4.5 The p adic Mittag Leffler Theorem 348
*4.6 The Christol Robba Theorem 350
Table: Analytic Elements 354
*4.7 Analyticity of Mahler Series 354
*4.8 The Motzkin Theorem 357
Exercises for Chapter VI 359
7 Special Functions, Congruences 366
1. The Gamma Function Tp 366
1.1 Definition 367
1.2 Basic Properties 368
Contents xv
1.3 The Gauss Multiplication Formula 371
1.4 The Mahler Expansion 374
1.5 The Power Series Expansion of log Vp 375
*1.6 The Kazandzidis Congruences 380
*1.7 About T2 382
2. The Artin Hasse Exponential 385
2.1 Definition and Basic Properties 386
2.2 Integrality of the Artin Hasse Exponential 388
2.3 The Dieudonne Dwork Criterion 391
2.4 The Dwork Exponential 393
*2.5 Gauss Sums 397
*2.6 The Gross Koblitz Formula 401
3. The Hazewinkel Theorem and Honda Congruences 403
3.1 Additive Version of the Dieudonne Dwork Quotient 403
3.2 The Hazewinkel Maps 404
3.3 The Hazewinkel Theorem 408
3.4 Applications to Classical Sequences 410
3.5 Applications to Legendre Polynomials 411
3.6 Applications to Appell Systems of Polynomials 412
Exercises for Chapter VII 414
Specific References for the Text 419
Bibliography 423
Tables 425
Basic Principles of Ultrametric Analysis 429
Conventions, Notation, Terminology 431
Index 435
|
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| isbn | 0387986693 |
| language | English |
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| spelling | Robert, Alain M. Verfasser aut A course in p-adic analysis Alain M. Robert New York [u.a.] Springer 2000 XV, 437 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 198 p-adische Analysis p-adic analysis p-adische Analysis (DE-588)4252360-6 gnd rswk-swf p-adische Analysis (DE-588)4252360-6 s DE-604 Graduate texts in mathematics 198 (DE-604)BV000000067 198 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008953527&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Robert, Alain M. A course in p-adic analysis Graduate texts in mathematics p-adische Analysis p-adic analysis p-adische Analysis (DE-588)4252360-6 gnd |
| subject_GND | (DE-588)4252360-6 |
| title | A course in p-adic analysis |
| title_auth | A course in p-adic analysis |
| title_exact_search | A course in p-adic analysis |
| title_full | A course in p-adic analysis Alain M. Robert |
| title_fullStr | A course in p-adic analysis Alain M. Robert |
| title_full_unstemmed | A course in p-adic analysis Alain M. Robert |
| title_short | A course in p-adic analysis |
| title_sort | a course in p adic analysis |
| topic | p-adische Analysis p-adic analysis p-adische Analysis (DE-588)4252360-6 gnd |
| topic_facet | p-adische Analysis p-adic analysis |
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