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

Subject Matter

 * Analysis
 * $p$-adic Numbers

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