p 進 Teichmüller 理論基礎（影印版）
定 價：199 元
- 作者：（日）望月新一（Shinichi Mochizuki）著
- 出版時間：2019/1/1
- ISBN：9787040510089
- 出 版 社：高等教育出版社
- 中圖法分類：O187.1
- 頁碼：548
- 紙張：膠版紙
- 版次：1
- 開本：16K
本書為p 進雙曲曲線及其�？臻g的單值化理論奠定了基礎。一方面，這個理論將復雙曲曲線及其模空間的 Fuchs和Bers單值化推廣到了非阿基米德情形，因此該理論在本書中簡稱為p進 Teichmüller理論。另一方面，該理論可以看作是常阿貝爾簇及其�？臻g的Serre-Tate理論的相當精確的雙曲模擬。 p進雙曲曲線及其�？臻g的單值化理論始于作者先前的一些工作。從某種意義上說，本書是對先前工作的概括和延續(xù)。本書旨在填補所提出的單值化定理與在本科復分析課程中研究的雙曲黎曼曲面的經(jīng)典單值化定理之間的缺口。 ·介紹從p進伽羅瓦表示的角度對曲線�？臻g的一種系統(tǒng)性化處理。 ·給出Serre-Tate理論的雙曲曲線模擬。 ·建立Fuchs和Bers單值化理論的p進模擬。 ·提供 p 進 Hodge理論的一個“非阿貝爾例子”的系統(tǒng)化處理。
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
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