Arithmetic and Geometry over Local Fields: VIASM 2018

Preface
	Contents of This Volume
Acknowledgments
Contents
Contributors
1 Some Elements on Berthelot's Arithmetic D-Modules
	1.1 Introduction
	Introduction
	1.2 Ring of Differential Operators
		1.2.1 Kähler Differential
		1.2.2 Ring of Differential Operators
		1.2.3 Smooth Differential Case
		1.2.4 Left D A/R-Modules
	1.3 Berthelot's Ring of Differential Operators of Finite Level and Infinite Order
		1.3.1 Formal Affine Case, Standard Ring of Differential Operators
		1.3.2 Formal Affine Case, Berthelot's Ring of Differential Operators of Level m
		1.3.3 Left D (m) A/R-modules
		1.3.4 Weak p-Adic Completion
		1.3.5 Sheafification, Coherence
	Appendix: Further Reading
	Appendix: Further Reading
	References
2 Difference Galois Theory for the ``Applied'' Mathematician
	2.1 Introduction
	2.2 Glossary of Difference Algebra
	2.3 Picard-Vessiot Rings
	2.4 The Galois Group
		2.4.1 A Short Digression on Group Schemes
		2.4.2 The Galois Group of a Linear Difference System
		2.4.3 Transcendence Degree of the Picard-Vessiot Extension
	2.5 The Galois Correspondence (First Part)
	2.6 Application to Transcendence and Differential Transcendence
		2.6.1 General Statements
		2.6.2 Differential Algebraicity and D-Finiteness
	2.7 Applications to Special Cases
		2.7.1 Finite Difference Equations and Hölder Theorem
		2.7.2 Linear Inhomogeneous q-Difference Equations of the First Order
		2.7.3 A Particular Case of the Ishizaki-Ogawara's Theorem
	2.8 The Galois Correspondence (Second Part)
	Appendix: Behavior of the Galois Group with Respect to the Iteration of τ
	Appendix: Behavior of the Galois Group with Respect to the Iteration of τ
	References
3 Igusa's Conjecture on Exponential Sums Modulo pm and the Local-Global Principle
	3.1 Introduction
	3.2 Igusa Local Zeta Functions and Exponential Sums Modulo pm
		3.2.1 Local Fields
		3.2.2 Embedded Resolutions
		3.2.3 Igusa Local Zeta Functions and the MonodromyConjecture
		3.2.4 Exponential Sums and Fiber Integration
	3.3 Igusa's Conjecture on Exponential Sums Modulo pm
		3.3.1 Adèles
		3.3.2 Poisson Formulas and Formulas of Siegel Type
		3.3.3 Some Expected Results
	3.4 Progress on Igusa's Conjecture
		3.4.1 The Non-degenerate Case
		3.4.2 Beyond the Non-degenerate Case
	3.5 A Long History of the Local-Global Principle
		3.5.1 The Local-Global Principle
		3.5.2 Progress on the Local-Global Principle
		3.5.3 Igusa's Approach
	References
4 From the Carlitz Exponential to Drinfeld Modular Forms
	4.1 Introduction
	4.2 Rings and Fields
		4.2.1 Local Compactness, Local Fields
		4.2.2 Valued Rings and Fields for Modular Forms
		4.2.3 Algebraic Extensions
		4.2.4 Analytic Functions on Disks
		4.2.5 Further Properties of the Field C∞
	4.3 Drinfeld Modules and Uniformisation
		4.3.1 Drinfeld A-Modules and A-Lattices
		4.3.2 From Drinfeld Modules to Exponential Functions
	4.4 The Carlitz Module and Its Exponential
		4.4.1 A Formula for ν
		4.4.2 A Factorization Property for the Carlitz Exponential
		4.4.3 The Function expA and Local Class Field Theory
	4.5 Topology of the Drinfeld Upper-Half Plane
		4.5.1 Rigid Analytic Spaces
			4.5.1.1 Analytification
			4.5.1.2 The Rigid Analytic Variety Ω
		4.5.2 Fundamental Domains for "026E30F Ω
			4.5.2.1 Some Structural Properties of =GL2(A)
			4.5.2.2 Bruhat-Tits Trees and `Good Fundamental Domains'
		4.5.3 An Elementary Result on Translation-Invariant Functions Over Ω
			4.5.3.1 A Digression
	4.6 Some Quotient Spaces
		4.6.1 A-Periodic Functions Over Ω
			4.6.1.1 Analytification and quotients
			4.6.1.2 Proof of Proposition 4.6.1
			4.6.1.3 The Bruhat-Tits Tree and expA
		4.6.2 The Quotient GL2(A)"026E30F Ω
	4.7 Drinfeld Modular Forms
		4.7.1 u-Expansions
			4.7.1.1 Constructing Drinfeld Modular Forms
		4.7.2 Construction of Non-trivial Cusp Forms
			4.7.2.1 Drinfeld Modular Forms and the Bruhat-Tits Tree
	4.8 Eisenstein Series with Values in Banach Algebras
	4.9 Modular Forms with Values in Banach Algebras
		4.9.1 Weak Modular Forms of Weight -1
		4.9.2 Jacobi-Like Forms
	References
5 Berkovich Curves and Schottky Uniformization I: The Berkovich Affine Line
	5.1 Introduction
	5.2 The Underlying Set
	5.3 Classification of Points
	5.4 Topology
	5.5 Analytic Functions
	5.6 Extension of Scalars
	5.7 Connectedness
	5.8 Virtual Discs and Annuli
		5.8.1 Definitions
		5.8.2 The Case of an Algebraically Closed and Maximally Complete Base Field
		5.8.3 The General Case
	5.9 Lengths of Intervals
	5.10 Variation of Rational Functions
	References
6 Berkovich Curves and Schottky Uniformization II: Analytic Uniformization of Mumford Curves
	6.1 Introduction
	6.2 The Berkovich Projective Line and Möbius Transformations
		6.2.1 Affine Berkovich Spaces
		6.2.2 The Berkovich Projective Line
		6.2.3 Möbius Transformations
		6.2.4 Loxodromic Transformations and Koebe Coordinates
	6.3 Berkovich k-Analytic Curves
		6.3.1 Berkovich A1-like Curves
		6.3.2 Arbitrary Smooth Curves
		6.3.3 Mumford Curves
	6.4 Schottky Groups
		6.4.1 Schottky Figures
		6.4.2 Group-Theoretic Version
		6.4.3 Twisted Ford Discs
		6.4.4 Local Fields
	6.5 Uniformization of Mumford Curves
		6.5.1 The Uniformization Theorem
		6.5.2 Automorphisms of Mumford Curves
	Appendix: Further Reading
	Appendix: Further Reading
	A.1 Berkovich Spaces and their Skeleta
	A.2 Non-Archimedean Uniformization in Arithmetic Geometry
	A.3 The Relevance of Mumford Curves
	References
7 On the Stark Units of Drinfeld Modules
	7.1 Introduction
	7.2 Background
		7.2.1 Notation
		7.2.2 Fitting Ideals
		7.2.3 Ratio of Covolumes
	7.3 Drinfeld Modules
		7.3.1 Drinfeld Modules
		7.3.2 Exponential and Logarithm
		7.3.3 The Carlitz Module
	7.4 Stark Units
		7.4.1 Taelman Modules
		7.4.2 The Module of Stark Units
		7.4.3 Link with Anderson's Special Points
	7.5 Class Formulas
		7.5.1 Taelman's Class Formula
		7.5.2 The Equivariant Class Formula
		7.5.3 Examples
	7.6 The Multi-Variable Deformation of a Drinfeld A-Module
		7.6.1 The Multi-Variable Setting
		7.6.2 The Canonical Deformation of the Carlitz Module
	7.7 Applications
		7.7.1 Discrete Greenberg Conjectures
			7.7.1.1 Case n1 8mu(mod6muq-1)
			7.7.1.2 Case n1 8mu(mod6muq-1)
			7.7.1.3 Evaluation at Characters
		7.7.2 On the Bernoulli-Carlitz Numbers
	7.8 Stark Units in More General Settings
	References