Table of Contents
Introduction
0. Motivation
0.1. The Fuchsian Uniformization
0.2. Reformulation in Terms of Metrics
0.3. Reformulation in Terms of Indigenous Bundles
0.4. Frobenius Invariance and Integrality
0.5. The Canonical Real Analytic Trivialization of the Schwarz Torsor
0.6. The Frobenius Action on the Schwarz Torsor at the Infinite Prime
0.7. Review of the Case of Abelian Varieties
0.8. Arithmetic Frobenius Venues
0.9. The Classical Ordinary Theory
0.10. Intrinsic Hodge Theory
1. Overview of the Contents of the Present Book
1.1. Major Themes
1.2. Atoms, Molecules, and Nilcurves
1.3. The MTv-Object Point of View
1.4. The Generalized Notion of a Frobenius Invariant Indigenous Bundle
1.5. The Generalized Ordinary Theory
1.6. Geometrization
1.7. The Canonical Galois Representation
1.8. Ordinary Stable Bundles
2. Open Problems
2.1. Basic Questions
2.2. Canonical Curves and Hyperbolic Geometry
2.2.1. Review of Kleinian Groups
2.2.2. Review of Three-Dimensional Hyperbolic Geometry
2.2.3. Rigidity and Density Results
2.2.4. QF-Canonical Curves
2.2.5. The Case of CM Elliptic Curves
2.2.6. The Third Real Dimension as the Probenius Dimension
2.3. Towards an Arithmetic Kodaira-Spencer Theory
2.3.1. The Schwarz Torsor as Dual to the Kodaira-Spencer Morphism
2.3.2. Arithmetic Resolutions of the Schwarz Torsor


Chapter I: Crys-Stable Bundles
0. Introduction
1. Definitions and First Properties
1.1. Notation Concerning the Underlying Curve
1.2. Definition of a Crys-Stable Bundle
1.3. Isomorphisms
1.4. De Rham Cohomology
2. Moduli
2.1. Boundedness
2.2. Definition of Various Functors
2.3. Representability
2.4. Radimmersions
3. Further Structure
3.1. Crystal in Algebraic Spaces
3.2. Hodge Morphisms
3.3. Clutching Behavior 4. Torally Indigenous Bundles
4.1. Definitions
4.2. Explicit Computation of Monodromy
4.3. Moduli and de Rham Cohomology
4.4. Clutching Morphisms
5. The Universal Torsor of Torally Indigenous Bundles
5.1. Notation
5.2. Computation
5.3. The Case of Dimension One


Chapter II: Torally Crys-Stable Bundles in Positive Characteristic
0. Introduction
1. The p-Curvature of a Torally Crys-Stable Bundle
1.1. Terminology
1.2. The p-Curvature at a Marked Point
1.3. The Verschiebung Morphism
1.4. Torally Crys-Stable Bundles of Arbitrary Positive Level
1.5. The Geometric Connectedness of ,
1.6. Degenerations of Torally Crys-Stable Bundles of Positive Level
2. Nilpotent Connections of Higher Order
2.1. Higher Order Connections
2.2. De Rham Cohomology Computations
2.3. Versal Families at Infinity
3. Mildly Spiked Bundles
3.1. Definition and First Properties
3.2. De Rham Cohomology Computations
3.3. Deformation Theory


Chapter III: VF-Patterns
0. Introduction
1. The Moduli Stack Associated to a VF-Pattern
1.1. Definition of a VF-Pattern
1.2. Construction of Link Stacks
1.3. The Stack Associated to a VF-Pattern
2. Atfineness Properties
2.1. A Trivialization of a Certain Line Bundle on n
2.2. Some Ampleness Results
2.3. Affine Stacks
2.4. Absolute Affineness
2.5. The Connectedness of the Moduli Stack of Curves


Chapter IV: Construction of Examples
0. Introduction
1. Explicit Computation in the Case
1.1. Irreducible Components of Degree Two
1.2. The Case of Radius
1.3. Conclusions
2. Higher Order Connections and Lubin-Tate Stacks
2.1. The Projective Line Minus Three Points
2.2. Elliptic Curves
2.3. Lubin-Tate Stacks
3. Anabelian Stacks
3.1. Basic Definitions3.2. Nondormant Bundles on the Projective Line Minus Three Points
3.3. Explicit Construction of Spiked Data
Pictorial Appendix


Chapter V: Combinatorialization at Infinity of the Stack of Nilcurves
0. Introduction
1. Statement of Main Results
2. The Main Theorem
2.1. The Aphilial Case
2.2. Grafting on Dormant Atoms I: Virtual p-Curvatures
2.3. Grafting on Dormant Atoms II: Deformation Theory
2.4. Proof of the Main Theorem
3. Examples
3.1. Consequences in the Case (g,r)= (1,1)
3.2. Explicit Computations
Pictorial Appendix


Chapter VI: The Stack of Quasi-Analytic Self-Isogenies
0. Introduction
1. Definition of the Stacks Qg
1.1. Epiperfect Schemes
1.2. The Epiperfect Category
1.3. Epiperfect Log Schemes
1.4. The Definition of the Stack of Quasi-Analytic Self-Isogenies
2. Deformation and Degeneration Properties of the Stacks Qr
2.1. Lifting Properties of Q,,.
2.2. Representability and Affineness Properties of Qg,r
2.3. Embeddings of Qg,
2.4. The Lattice of Subobjects of Sw


Chapter VII: The Generalized Ordinary Theory
0. Introduction
1. The H-Ordinary Locus
1.1. The Frobenius Action on the Crystalline Cohomology
1.2. Interpretation of the Condition of H-Ordinariness
1.3. Systems of Canonical Modular Frobenius Liftings
1.4. The Case of Elliptic Curves
2. The Closure of the Binary Ordinary Locus
2.1. The Deperfection of the Closure
2.2. The Differentials of the Deperfection
2.3. The w-Closedness of the Binary Ordinary Locus
3. Existence Results
3.1. The Binary Case
3.2. The Spiked Case
3.3. Frobenius Liftings in the Very Ordinary Case
Pictorial Appendix


Chapter VIII: The Geometrization of Binary-Ordinary Frobenius Liftings
0. Introduction
1. The General Framework
1.1. Canonical Points
1.2. The Meaning of "Geometrization"
2. The Binary Case
2.1. The Associated Differential Formal Group2.2. The Canonical Uniformizing p-divisible Group
2.3. Multi-Uniformization by the Group 6A
2.4. Canonical Affine Coordinates
2.5. Lubin-Tate Geometries
2.6. Anabelian Geometries
2.7. Deformation of the System of Frobenius Liftings
3. Application to Curves and their Moduli
3.1. Frobenius Liftings on the Moduli Stack
3.2. Frobenius Liftings on the Universal Curve
Pictorial Appendix


Chapter IX: The Geometrization of Spiked Frobenius Liftlngs
0. Introduction
1. The Formal Uniformizing A//jrv-Object
1.1. The Objects in Question
1.2. The Strong Portion of the Uniformization
1.3. The Strong Portion of the Mantle
1.4. The Renormalized Frobenius Pull-back of the Mantle
1.5. Hodge Subspaces
2. Associated Galois Representations
2.1. The Strictly Weak Pair of Frobenius Liftings over the Strong Perfection
2.2. The Associated Non-affine Geometry
2.3. Construction of the Galois Mantle: The Spiked Case
2.4. Discussion of the Resulting Spiked Geometry
2.5. Construction of the Galois Mantle: The Binary-Ordinary Case
3. Application to Curves and their Moduli
3.1. Frobenius Liftings on the Moduli Stack
3.2. Frobenius Liftings on the Universal Curve
Pictorial Appendix


Chapter X: Representations of the Fundamental Group of the Curve
0. Introduction
1. The Binary-Ordinary Case
1.1. The Formal Mv-Object
1.2. The Crystalline Induced Representation
1.3. The Lubin-Tate Case
1.4. Relation to the Profinite Teichmfiller Group
2. The Very Ordinary Spiked Case
2.1. The Formal MS-v-Object
2.2. The Crystalline Induced Representation
2.3. Relation to the Profinite Teichmfiller Group
3. Conclusion


Appendix: Ordinary Stable Bundles on a Curve
0. Introduction
1. The Algebraic Theory
1.1. Basic Definitions
1.2. Moduli
2. The Complex Theory
2.1. Unitary Representations of the Fundamental Group
2.2. The K/ihler Approach
3. The Ordinary p-adic Theory
3.1. Crystals of Bundles with Connection3.2. Frobenius Actions
3.3. The Ordinary Case
3.4. Canonical Coordinates via the Weil Conjectures


Bibliography
Index