Alain M. Robert

A Course in /7-adic Analysis

With 27 Figures

Springer

Contents

Preface v

1 /»-adic Numbers 1 1. The Ring Z p of p-adic Integers 1

1.1 Definition 1 1.2 Addition of p-adic Integers 2 1.3 The Ring of/7-adic Integers 3 1.4 The Order of a/7-adic Integer 4 1.5 Reduction mod p 5 1.6 The Ring of /7-adic Integers is a Principal Ideal Domain . . . . 6

2. The Compact Space Zp 7 2.1 Product Topology on Z p 7 2.2 TheCantor Set . 8 2.3 Linear Models ofZp 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/7-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 /?-adic Integers 34 5. The Field Qp of /?-adic Numbers 36

5.1 The Fraction Field of Z p 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 and Zp 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 Z p 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 Q p 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 Existenceof 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 111 *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 HaarMeasures 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

Constitution of Universal p-adic Fields 127 1. The Algebraic Closure Q° ofQ p 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 ofCp 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 RootsofUnity 147 4.3 Fundamental Inequalities 148 4.4 Splitting by RootsofUnity of Order Prime t o p 150 4.5 Divisibility of the Group of Units Congruentto 1 151

Appendix to Chapter III: Filters and Ultrafilters 152 A.l Definition and First Properties 152 A.2 Ultrafilters 153 A.3 Convergence and Compactness 154

*A.4 Circular Filters 156 Exercises for Chapter HI 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 Z p 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 Z p 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 ofa 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 ACompletion 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

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5.3 Integrals and Shift 266 5.4 Relation to BemouUi Numbers 269 5.5 Sums of Powers 272 5.6 BemouUi 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. Zerosof Power Series 305 2.1 Finitenessof 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 Täte 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 Vp 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 Fp 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, Terminoiogy 431

Index 435