Kurt Hensel (1861-1941) discovered the p-adic numbers around the turn of the century. These exotic numbers (or so they appeared at first) are now well-established in the mathematical world and used more and more by physicists as well. This book offers a self-contained presentation of basic p-adic analysis. The author is especially interested in the analytical topics in this field. Some of the features which are not treated in other introductory p-adic analysis texts are topological models of p-adic spaces inside Euclidean space, a construction of spherically complete fields, a p-adic mean value theorem and some consequences, a special case of Hazewinkel's functional equation lemma, a remainder formula for the Mahler expansion, and most importantly a treatment of analytic elements.
Author(s): Alain M. Robert
Series: Graduate Texts in Mathematics
Edition: 1
Publisher: Springer
Year: 2000
Language: English
Pages: 454
Contents......Page 10
Preface......Page 6
1.1 Definition......Page 18
1.2 Addition of p-adic Integers......Page 19
1.3 The Ring of p-adic Integers......Page 20
1.4 The Order of a p-adic Integer......Page 21
1.5 Reduction mod p......Page 22
1.6 The Ring of p-adic Integers is a Principal Ideal Domain......Page 23
2.1 Product Topology on Z_p......Page 24
2.2 The Cantor Set......Page 25
2.3 Linear Models of Z_p......Page 26
2.4 Free Monoids and Balls of Z_p......Page 28
2.5 Euclidean Models......Page 29
2.6 An Exotic Example......Page 33
3.1 Topological Groups......Page 34
3.2 Closed Subgroups of Topological Groups......Page 36
3.3 Quotients of Topological Groups......Page 37
3.4 Closed Subgroups of the Additive Real Line......Page 39
3.5 Closed Subgroups of the Additive Group of p-adic Integers......Page 40
3.6 Topological Rings......Page 41
3.7 Topological Fields, Valued Fields......Page 42
4.1 Introduction......Page 43
4.3 Existence......Page 45
4.4 Projective Limits of Topological Spaces......Page 47
4.5 Projective Limits of Topological Groups......Page 48
4.6 Projective Limits of Topological Rings......Page 49
4.7 Back to the p-adic Integers......Page 50
4.8 Formal Power Series and p-adic Integers......Page 51
5.1 The Fraction Field of Z_p......Page 53
5.2 Ultrametric Structure on Q_p......Page 54
5.3 Characterization of Rational Numbers Among p-adic Ones......Page 56
5.4 Fractional and Integral Parts of p-adic Numbers......Page 57
5.5 Additive Structure of Q_p and Z_p......Page 60
5.6 Euclidean Models of Q_p......Page 61
6.1 First Principle......Page 62
6.3 Second Principle......Page 63
6.4 The Newtonian Algorithm......Page 64
6.6 Second Application: Square Roots in Q_p......Page 66
6.7 Third Application: nth Roots of Unity in Z_p......Page 68
6.8 Fourth Application: Field Automorphisms of Q_p......Page 70
Appendix to Chapter I: The p-adic Solenoid......Page 71
A.2 Torsion of the Solenoid......Page 72
A.3 Embeddings of Rand Q_p in the Solenoid......Page 73
A.4 The Solenoid as a Quotient......Page 74
A.5 Closed Subgroups of the Solenoid......Page 77
A.6 Topological Properties of the Solenoid......Page 78
Exercises for Chapter I......Page 80
1.1 Ultrametric Distances......Page 86
1.2 Ultrametric Principles in Abelian Groups......Page 90
1.3 Absolute Values on Fields......Page 94
1.4 Ultrametric Fields: The Representation Theorem......Page 96
1.5 General Form of Hensel's Lemma......Page 97
1.6 Characterization of Ultrametric Absolute Values......Page 99
1.7 Equivalent Absolute Values......Page 100
2.1 Ultrametric Absolute Values on Q......Page 102
2.2 Generalized Absolute Values......Page 103
2.4 Generalized Absolute Values on the Rational Field......Page 105
3.1 Normed Spaces over Q_p......Page 107
3.2 Locally Compact Vector Spaces over Q_p......Page 110
3.3 Uniqueness of Extension of Absolute Values......Page 111
3.4 Existence of Extension of Absolute Values......Page 112
3.5 Locally Compact Ultrametric Fields......Page 113
4.1 Degree and Residue Degree......Page 114
4.2 Totally Ramified Extensions......Page 118
4.3 Roots of Unity and Unramified Extensions......Page 121
4.4 Ramification and Roots of Unity......Page 124
4.5 Example 1: The Field of Gaussian 2-adic Numbers......Page 128
4.6 Example 2: The Hexagonal Field of 3-adic Numbers......Page 129
4.7 Example 3: A Composite of Totally Ramified Extensions......Page 131
A.1 Haar Measures......Page 132
A.3 Closed Balls are Compact......Page 133
A.5 Classification......Page 135
A.6 Finite-Dimensional Topological Vector Spaces......Page 136
A.7 Locally Compact Vector Spaces Revisited......Page 138
A.8 Final Comments on Regularity of Haar Measures......Page 139
Exercises for Chapter II......Page 140
1.1 Extension of the Absolute Value......Page 144
1.2 Maximal Unramified Subextension......Page 145
1.4 The Algebraic Closure Q^a_p is not Complete......Page 146
1.5 Krasner's Lemma......Page 147
1.6 A Finiteness Result......Page 149
1.7 Structure of Totally and Tamely Ramified Extensions......Page 150
2.1 More Results on Ultrametric Fields......Page 151
2.2 Construction of a Universal Field \Omega_p......Page 154
2.3 The Field \Omega_p is Algebraically Closed......Page 155
2.4 Spherically Complete Ultrametric Spaces......Page 156
3.1 Definition of C_p......Page 157
3.2 Finite-Dimensional Vector Spaces over a Complete Ultrametric Field......Page 158
3.4 The Field C_p is not Spherically Complete......Page 160
3.5 The Field C_p is Isomorphic to the Complex Field C......Page 161
4.1 Choice of Representatives for the Absolute Value......Page 163
4.2 Roots of Unity......Page 164
4.3 Fundamental Inequalities......Page 165
4.4 Splitting by Roots of Unity of Order Prime top......Page 167
4.5 Divisibility of the Group of Units Congruent to 1......Page 168
A.1 Definition and First Properties......Page 169
A.2 Ultrafilters......Page 170
A.3 Convergence and Compactness......Page 171
Exercises for Chapter III......Page 173
1.1 Integer-Valued Functions on the Natural Integers......Page 177
1.2 Integer-Valued Polynomial Functions......Page 180
1.3 Periodic Functions Taking Values in a Field of Characteristic p......Page 181
1.4 Convolution of Functions of an Integer Variable......Page 183
1.5 Indefinite Sum of Functions of an Integer Variable......Page 184
2.1 Review of Some Classical Results......Page 187
2.3 Mahler Series......Page 189
2.4 The Mahler Theorem......Page 190
2.5 Convolution of Continuous Functions on Z_p......Page 192
3.1 Review of General Properties......Page 195
3.2 Characteristic Functions of Balls of Z_p......Page 196
3.3 The van der Put Theorem......Page 199
4.1 Direct Sums of Banach Spaces......Page 200
4.2 Normal Bases......Page 203
4.3 Reduction of a Banach Space......Page 206
4.5 The Monna-Fleischer Theorem......Page 207
4.6 Spaces of Linear Maps......Page 209
4.7 The p-adic Hahn-Banach Theorem......Page 211
5.1 Delta Operators......Page 212
5.2 The Basic System of Polynomials of a Delta Operator......Page 214
5.3 Composition Operators......Page 215
5.4 The van Hamme Theorem......Page 218
5.5 The Translation Principle......Page 221
6.1 Sheffer Sequences......Page 224
6.2 Generating Functions......Page 226
6.3 The Bell Polynomials......Page 228
Exercises for Chapter IV......Page 229
1.1 Strict Differentiability......Page 234
1.2 Granulations......Page 238
1.3 Second-Order Differentiability......Page 239
1.4 Limited Expansions of the Second Order......Page 241
1.5 Differentiability of Mahler Series......Page 243
1.6 Strict Differentiability of Mahler Series......Page 249
2.1 A Completion of the Polynomial Algebra......Page 250
2.2 Numerical Evaluation of Products......Page 252
2.3 Equicontinuity of Restricted Formal Power Series......Page 253
2.4 Differentiability of Power Series......Page 255
2.5 Vector-Valued Restricted Series......Page 257
3.1 The p-adic Valuation of a Factorial......Page 258
3.2 First Form of the Theorem......Page 259
3.3 Application to Classical Estimates......Page 262
3.4 Second Form of the Theorem......Page 264
3.5 A Fixed-Point Theorem......Page 265
3.6 Second-Order Estimates......Page 266
4.1 Convergence of the Defining Series......Page 268
4.2 Properties of the Exponential and Logarithm......Page 269
4.3 Derivative of the Exponential and Logarithm......Page 274
4.4 Continuation of the Exponential......Page 275
4.5 Continuation of the Logarithm......Page 276
5.1 Definition via Riemann Sums......Page 280
5.2 Computation via Mahler Series......Page 282
5.3 Integrals and Shift......Page 283
5.4 Relation to Bernoulli Numbers......Page 286
5.5 Sums of Powers......Page 289
5.6 Bernoulli Polynomials as an Appell System......Page 292
Exercises for Chapter V......Page 293
1.1 Formal Power Series......Page 297
1.2 Convergent Power Series......Page 300
1.3 Formal Substitutions......Page 303
1.4 The Growth Modulus......Page 307
1.5 Substitution of Convergent Power Series......Page 311
1.6 The valuation Polygon and its Dual......Page 314
1.7 Laurent Series......Page 320
2.1 Finiteness of Zeros on Spheres......Page 322
2.2 Existence of Zeros......Page 324
2.3 Entire Functions......Page 330
2.4 Rolle's Theorem......Page 332
2.5 The Maximum Principle......Page 334
2.6 Extension to Laurent Series......Page 335
3.1 Linear Fractional Transformations......Page 338
3.2 Rational Functions......Page 340
3.3 The Growth Modulus for Rational Functions......Page 343
3.4 Rational Mittag-Leffler Decompositions......Page 347
3.5 Rational Motzkin Factorizations......Page 350
3.6 Multiplicative Norms on K(X)......Page 354
4.1 Enveloping Balls and Infraconnected Sets......Page 356
4.2 Analytic Elements......Page 359
4.3 Back to the Tate Algebra......Page 361
4.4 The Amice-Fresnel Theorem......Page 364
4.5 The p-adic Mittag-Leffler Theorem......Page 365
4.6 The Christol-Robba Theorem......Page 367
4.7 Analyticity of Mahler Series......Page 371
4.8 The Motzkin Theorem......Page 374
Exercises for Chapter VI......Page 376
1. The Gamma Function \Gamma_p......Page 383
1.1 Definition......Page 384
1.2 Basic Properties......Page 385
1.3 The Gauss Multiplication Formula......Page 388
1.4 The Mahler Expansion......Page 391
1.5 The Power Series Expansion of log \Gamma_p......Page 392
1.6 The Kazandzidis Congruences......Page 397
1.7 About \Gamma_2......Page 399
2. The Artin-Hasse Exponential......Page 402
2.1 Definition and Basic Properties......Page 403
2.2 Integrality of the Artin-Hasse Exponential......Page 405
2.3 The Dieudonne-Dwork Criterion......Page 408
2.4 The Dwork Exponential......Page 410
2.5 Gauss Sums......Page 414
2.6 The Gross-Koblitz Formula......Page 418
3.1 Additive Version of the Dieudonne-Dwork Quotient......Page 420
3.2 The Hazewinkel Maps......Page 421
3.3 The Hazewinkel Theorem......Page 425
3.4 Applications to Classical Sequences......Page 427
3.5 Applications to Legendre Polynomials......Page 428
3.6 Applications to Appell Systems of Polynomials......Page 429
Exercises for Chapter VII......Page 431
Specific References for the Text......Page 436
Bibliography......Page 440
Basic Principles of Ultrametric Analysis......Page 446
Conventions, Notation, Terminology......Page 448
Index......Page 452
