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