Book:Alain M. Robert/A Course in p-adic Analysis

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Alain M. Robert: A Course in p-adic Analysis

Published $\text {2000}$, Springer

ISBN 978-1-4419-53150-4


Subject Matter


Contents

Preface

1$\quad$$p$-adic Numbers

1.$\quad$The Ring $\Z_p$ of $p$-adic Integers
1.1$\quad$Definition
1.2$\quad$Addition of $p$-adic Integers
1.3$\quad$The Ring of $p$-adic Integers
1.4$\quad$The Order of a $p$-adic Integers
1.5$\quad$Reduction mod $p$
1.6$\quad$The Ring of $p$-adic Integers is a Principal Ideal Domain
2.$\quad$The Compact Space $\Z_p$
2.1$\quad$Product Topology on $\Z_p$
2.2$\quad$The Cantor Set
2.3$\quad$Linear Models of $\Z_p$
2.4$\quad$Free Monoids and Balls of $\Z_p$
2.5$\quad$Euclidean Models
2.6$\quad$An Exotic Example
3.$\quad$Topological Algebra
3.1$\quad$Topological Groups
3.2$\quad$Closed Subgroups of Topological Groups
3.3$\quad$Quotients of Topological Groups
3.4$\quad$Closed Subgroups of the Additive Real Line
3.5$\quad$Closed Subgroups of the Additive Group of $p$-adic Integers
3.6$\quad$Topological Rings
3.7$\quad$Topological Fields, Valued Fields
4.$\quad$Projective Limits
4.1$\quad$Introduction
4.2$\quad$Definition
4.3$\quad$Existence
4.4$\quad$Projective Limits of Topological Spaces
4.5$\quad$Projective Limits of Topological Groups
4.6$\quad$Projective Limits of Topological Rings
4.7$\quad$Back to the $p$-adic Integers
4.8$\quad$Formal Power Series and $p$-adic Integers
5.$\quad$The Field $\Q_p$ of $p$-adic Numbers
5.1$\quad$The fraction Field of $\Z_p$
5.2$\quad$Ultrametric Structure on $\Q_p$
5.3$\quad$Characterization of Rational Numbers Amoung $p$-adic Ones
5.4$\quad$Fractional and Integral Parts of $p$-adic Numbers
5.5$\quad$Additive Structure of $\Q_p$ and $\Z_p$
5.6$\quad$Euclidean Models of $\Q_p$
6.$\quad$Hensel’s Philosophy
6.1$\quad$First Principle
6.2$\quad$Algebraic Preliminaries
6.3$\quad$Second Principle
6.4$\quad$The Newtonian Algorithm
6.5$\quad$First Application: Invertible Elements of $\Z_p$
6.6$\quad$Second Application:Square Roots in $\Q_p$
6.7$\quad$Third Application: $n$th Roots of Unity in $\Z_p$
$\quad\quad$Table: Units, Squares, Roots of Unity
6.8$\quad$Fourth Application: Field Automorphisms of $\Q_p$
Appendix to Chapter I: The $p$-adic Solenoid
A.1$\quad$Definition and First Properties
A.2$\quad$Torsion of the Solenoid
A.3$\quad$Embedding of $\R$ and $\Q_p$ in the Solenoid
A.4$\quad$The Solenoid as a Quotient
A.5$\quad$Closed Subgroups of the Solenoid
A.6$\quad$Topological Properties of the Solenoid
Exercises for Chapter I


2$\quad$$p$-adic Numbers

1.$\quad$Ultrametric Spaces
1.1$\quad$Ultrametric Distances
$\quad\quad$Table: Properties of Ultrametric Distances
1.2$\quad$Ultrametric Principles in Abelian Groups
$\quad\quad$Table: Principles of Ultrametric Analysis
1.3$\quad$Absolute Values on Fields
1.4$\quad$Ultrametric Fields: The Representation Theorem
1.5$\quad$General Form of Hensel’s Lemma
1.6$\quad$Characterization of Ultrametric Absolute Values
1.7$\quad$Equivalent Absolute Values
2.$\quad$Absolute Values on the Field of $\Q$
2.1$\quad$Ultrametric Absolute Values on $\Q$
2.2$\quad$Generalised Absolute Values
2.3$\quad$Ultrametric Amoung Generalized Absolute Values
2.4$\quad$Generalized Absolute Values on the Rational Field
3.$\quad$Finite-Dimensional Vector Spaces
3.1$\quad$Normed Spaces over $\Q_p$
3.2$\quad$Locally Compact Vector Spaces over $\Q_p$
3.3$\quad$Uniqueness of Extension of Absolute Values
3.4$\quad$Existence of Extension of Absolute Values
3.5$\quad$Locally Compact Ultrametric Fields
4.$\quad$Structure of $p$-adic Fields
4.1$\quad$Degree and Residue Degree
4.2$\quad$Totally Ramified Extensions
4.3$\quad$Roots of Unity and Unramified Extensions
4.4$\quad$Ramification and Roots of Unity
4.5$\quad$Example 1: The Field of Gaussian $2$-adic Numbers
4.6$\quad$Example 2: The Hexagonal Field of $3$-adic Numbers
4.7$\quad$Example 3: A Composite of Totally Ramified Extensions
Appendix to Chapter II: Classification of Locally Compact Fields
A.1$\quad$Haar Measures
A.2$\quad$Continuity of the Modulus
A.3$\quad$Closed Balls are Compact
A.4$\quad$The Modulus is a Strict Homomorphism
A.5$\quad$Classification
A.6$\quad$Finite-Dimensional Topological Vector Spaces
A.7$\quad$Locally Compact Vector Spaces Revisited
A.8$\quad$Final Comments on Regularity of Haar Measure
Exercises for Chapter II


3$\quad$Construction of Universal $p$-adic Fields

1.$\quad$The Algebraic Closure $\Q^a_p$ of $\Q_p$
1.1$\quad$Extension of the Absolute Value
1.2$\quad$Maximal Unramified Subextension
1.3$\quad$Ramified Extensions
1.4$\quad$The Algebraic Closure $\Q^a_p$ is not Complete
1.5$\quad$Krasner’s Lemma
1.6$\quad$AFiniteness Result
1.7$\quad$Structure of Totally and Tamely Ramified Extensions
2.$\quad$Definition of a Universal $p$-adic Field
2.1$\quad$More Results on Ultrametric Fields
2.2$\quad$Construction of a Universal Field $\Omega_p$
2.3$\quad$The Field $\Omega_p$ is Algebraically Closed
2.4$\quad$Spherically Complete Ultrametric Spaces
2.5$\quad$The Field $\Omega_p$ is Spherically Complete
3.$\quad$The Completion $\C_p$ of the Field $\Q^a_p$
3.1$\quad$Definition of $\C_p$
3.2$\quad$Finite-Dimensional Vector Spaces over a Complete Ultrametric Field
3.3$\quad$The Completion is Algebraically Closed
3.4$\quad$The Field $\C_p$ is not Spherically Complete
3.5$\quad$The Field $\C_p$ is Isomorphic to the Complex Field $\C$
$\quad\quad$Table: Notation
4.$\quad$Multiplicative Structure of $\C_p$
4.1$\quad$Choice of Representatives for the Absolute Value
4.2$\quad$Roots of Unity
4.3$\quad$Fundamental Inequalities
4.4$\quad$Splitting by Roots of Unity of Order Prime to $p$
4.5$\quad$Divisibility of the Group of Units Congruent to $1$
Appendix to Chapter III: Filters and Ultrafilters
A.1$\quad$Definition and First Properties
A.2$\quad$Ultrafilters
A.3$\quad$Convergence and Compactness
A.4$\quad$Circular Filters
Exercises for Chapter III


4$\quad$Continuous Functions on $\Z_p$

1.$\quad$Functions of an Integer Variable
1.1$\quad$Integer-Valued Functions on the Natural Integers
1.2$\quad$Integer-Valued Polynomial Functions
1.3$\quad$Periodic Functions Taking Values in a Field of Characteristic $p$
1.4$\quad$Convolutions of Functions of an Integer Variable
1.5$\quad$Indefinite Sum of Functions of anInteger Variable
2.$\quad$Continuous Functions on $\Z_p$
2.1$\quad$Review of Some Classical Results
2.2$\quad$Examples of $p$-adic Continuous Functions on $\Z_p$
2.3$\quad$Mahler Series
2.4$\quad$The Mahler Theorem
2.5$\quad$Convolution of Continuous Functions on $\Z_p$
3.$\quad$Locally Constant Functions on $\Z_p$
3.1$\quad$Review if General Properties
3.2$\quad$Characteristic Functions of Balls of $\Z_p$
3.3$\quad$The van der Put Theorem
4.$\quad$Ultrametric Banach Spaces
4.1$\quad$Direct Sums of Banach Spaces
4.2$\quad$Normal Bases
4.3$\quad$Reduction of a Banach Spaces
4.4$\quad$A Representation Theorem
4.5$\quad$The Mona-Fleischer Theorem
4.6$\quad$Spaces of Linear Maps
4.7$\quad$The $p$-adic Hahn-Banach Theorem
5.$\quad$Umbral Calculus
5.1$\quad$Delta Operators
5.2$\quad$The Basic System of Polynomials of a Delta Operator
5.3$\quad$Composition Operators
5.4$\quad$The van Hamme Theorem
5.5$\quad$The Translation Principle
$\quad\quad$Table: Umbral Calculus
6.$\quad$Generating Functions
6.1$\quad$Sheffield Sequences
6.2$\quad$Generating Functions
6.3$\quad$The Bell Polynomials
Exercises for Chapter IV


5$\quad$Differentiation

1.$\quad$Differentiability
1.1$\quad$Strict Differentiability
1.2$\quad$Granulations
1.3$\quad$Second-Order Differentiability
1.4$\quad$Limited Expansions of the Second Order
1.5$\quad$Differentiabilty of Mahler Series
1.6$\quad$Strict Differentiability of Mahler Series
2.$\quad$Restricted Formal Power Series
2.1$\quad$A Completion of the Polynomial Algebra
2.2$\quad$Numerical Evaluation of Products
2.3$\quad$Equicontinuity of Restricted Formal Power Series
2.4$\quad$Differentiability of Power Series
2.5$\quad$Vector-Value Restricted Series
3.$\quad$The Mean Value Theorem
3.1$\quad$The $p$-adic Valuation of a Factorial
3.2$\quad$First Form of the Theorem
3.3$\quad$Application to Classical Estimates
3.4$\quad$Second Form of the Theorem
3.5$\quad$A Fixed-Point Theorem
3.6$\quad$Second-Order Estimates
4.$\quad$The Exponentiel and Logarithm
4.1$\quad$Convergence of the Defining Series
4.2$\quad$Properties of the Exponential and Logarithm
4.3$\quad$Derivative of the Exponential and Logarithm
4.4$\quad$Continuation of the Exponential
4.5$\quad$Continuation of the Logarithm
5.$\quad$The Volkenborn Integral
5.1$\quad$Definition via Riemann Sums
5.2$\quad$Computation via Mahler Series
5.3$\quad$Integrals and Shift
5.4$\quad$Relation to Bernoulli Numbers
5.5$\quad$Sums of Powers
5.6$\quad$Bernoulli Polynomials as an Appell System
Exercises for Chapter V


6$\quad$Analytic Functions and Elements

1.$\quad$Power Series
1.1$\quad$Formal Power Series
1.2$\quad$Convergent Power Series
1.3$\quad$Formal Substitutions
1.4$\quad$The Growth Modulus
1.5$\quad$Substitution of Convergent Power Series
1.6$\quad$The valuation Polygon and its Dual
1.7$\quad$Laurent Series
2.$\quad$Zeros of Power Series
2.1$\quad$Finiteness of Zeros on Spheres
2.2$\quad$Existence of Zeros
2.3$\quad$Entire Functions
2.4$\quad$Rolle’s Theorem
2.5$\quad$The Maximum Principle
2.6$\quad$Extension to Laurent Series
3.$\quad$Rational Functions
3.1$\quad$Linear Fractional Transformations
3.2$\quad$Rational Functions
3.3$\quad$The Growth Modulus for Rational Functions
3.4$\quad$Rational Mittag-Leffler Decompositions
3.5$\quad$Rational Motzkin Factorizations
3.6$\quad$Multiplicative Norms on $\map K X$
4.$\quad$Analytic Elements
4.1$\quad$Enveloping Balls and Infraconnected Sets
4.2$\quad$Analytic Elements
4.3$\quad$Back to the Tate Algebra
4.4$\quad$The Amice-Fresnel Theorem
4.5$\quad$The $p$-adic Mittag-Leffler Theorem
4.6$\quad$The Christos-Robby Theorem
$\quad\quad$Table: Analytic Elements
4.7$\quad$Analyticity of Mahler Series
4.8$\quad$The Motzkin Theorem
Exercises for Chapter VI


7$\quad$Special Functions, Congruences

1.$\quad$The Gamma Function $\Gamma_p$
1.1$\quad$Definition
1.2$\quad$Basic Properties
1.3$\quad$The Gauss Multiplication Formula
1.4$\quad$The Mahler Expansion
1.5$\quad$The Power Series Expansion of $\log \Gamma_p$
1.6$\quad$The Kazandzidis Congruences
1.7$\quad$About $\Gamma_2$
2.$\quad$The Artin-Hasse Exponential
2.1$\quad$Definition and Basic Properties
2.2$\quad$Integrality of the Artin-Hasse Exponential
2.3$\quad$The Dieudonné-Dwork Criterion
2.4$\quad$The Dwork Exponential
2.5$\quad$Gauss Sums
2.6$\quad$The Gross-Koblitz Formula
3.$\quad$The Hazewinkel Theorem and Honda Congruences
3.1$\quad$Additive Version of the Dieudonné-Dwork Quotient
3.2$\quad$The Hazewinkel Maps
3.3$\quad$The Hazewinkel Theorem
3.4$\quad$Applications to Classical Sequences
3.5$\quad$Applications to Legendre Polynomials
3.6$\quad$SApplications to Appell Systems of Polynomials
Exercises for Chapter VII


Specific References for the Text

Bibliography

Tables

Basic Principles of Ultrametric Analysis

Conventions, Notation, Terminology

Index