Cover

Title page

Contents

Preface

Acknowledgments

Introduction

0.1. Outline of the theory

0.2. Comparison with other theories

Chapter 1. Algebraic background

1.1. Algebra

1.2. Algebraic geometry

1.3. Superalgebra

Chapter 2. Classical differential geometry revisited

2.1. Connections in principal bundles and curvature

2.2. Lie algebra and classical groups

2.3. Involutions and symmetric spaces

2.4. Logarithmic derivative and differential Galois groups

2.5. Chern connections: the symmetric/anti-symmetric case

2.6. Chern connections: the hermitian case

2.7. Levi-Cività connection and Fedosov connection

2.8. Locally symmetric connections

2.9. Ehresmann connections attached to inner involutions

2.10. Connections in vector bundles

2.11. Lax connections

2.12. Hamiltonian connections

2.13. Cartan connection

2.14. Weierstrass and Riccati connections

2.15. Differential groups: Cassidy and Painlevé

Chapter 3. Arithmetic differential geometry: generalities

3.1. Global connections and their curvature

3.2. Adelic connections

3.3. Semiglobal connections and their curvature; Galois connections

3.4. Curvature via analytic continuation between primes

3.5. Curvature via algebraization by correspondences

3.6. Arithmetic jet spaces and the Cartan connection

3.7. Arithmetic Lie algebras and arithmetic logarithmic derivative

3.8. Compatibility with translations and involutions

3.9. Arithmetic Lie brackets and exponential

3.10. Hamiltonian formalism and Painlevé

3.11. 𝑝-adic connections on curves: Weierstrass and Riccati

Chapter 4. Arithmetic differential geometry: the case of 𝐺𝐿_{𝑛}

4.1. Arithmetic logarithmic derivative and Ehresmann connections

4.2. Existence of Chern connections

4.3. Existence of Levi-Cività connections

4.4. Existence/non-existence of Fedosov connections

4.5. Existence/non-existence of Lax-type connections

4.6. Existence of special linear connections

4.7. Existence of Euler connections

4.8. Curvature formalism and gauge action on 𝐺𝐿_{𝑛}

4.9. Non-existence of classical 𝛿-cocycles on 𝐺𝐿_{𝑛}

4.10. Non-existence of 𝛿-subgroups of simple groups

4.11. Non-existence of invariant adelic connections on 𝐺𝐿_{𝑛}

Chapter 5. Curvature and Galois groups of Ehresmann connections

5.1. Gauge and curvature formulas

5.2. Existence, uniqueness, and rationality of solutions

5.3. Galois groups: generalities

5.4. Galois groups: the generic case

Chapter 6. Curvature of Chern connections

6.1. Analytic continuation along tori

6.2. Non-vanishing/vanishing of curvature via analytic continuation

6.3. Convergence estimates

6.4. The cases 𝑛=1 and 𝑚=1

6.5. Non-vanishing/vanishing of curvature via correspondences

Chapter 7. Curvature of Levi-Cività connections

7.1. The case 𝑚=1: non-vanishing of curvature mod 𝑝

7.2. Analytic continuation along the identity

7.3. Mixed curvature

Chapter 8. Curvature of Lax connections

8.1. Analytic continuation along torsion points

8.2. Non-vanishing/vanishing of curvature

Chapter 9. Open problems

9.1. Unifying \hol_{ℚ} and Γ_{ℚ}

9.2. Unifying “∂/∂𝑝” and “∂/∂𝜁_{𝑝^{∞}}”

9.3. Unifying 𝑆ℎ and 𝐺𝐿_{𝑛}

9.4. Further concepts and computations

9.5. What lies at infinity?

Bibliography

Index

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