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
