**Mathematical Surveys and Monographs**

Volume: 222;
2017;
344 pp;
Hardcover

MSC: Primary 11; 14; 53;

**Print ISBN: 978-1-4704-3623-0
Product Code: SURV/222**

List Price: $124.00

AMS Member Price: $99.20

MAA Member Price: $111.60

**Electronic ISBN: 978-1-4704-4089-3
Product Code: SURV/222.E**

List Price: $124.00

AMS Member Price: $99.20

MAA Member Price: $111.60

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#### Supplemental Materials

# Foundations of Arithmetic Differential Geometry

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*Alexandru Buium*

The aim of this book is to introduce and develop an arithmetic
analogue of classical differential geometry. In this new geometry the
ring of integers plays the role of a ring of functions on an infinite
dimensional manifold. The role of coordinate functions on this
manifold is played by the prime numbers. The role of partial
derivatives of functions with respect to the coordinates is played by
the Fermat quotients of integers with respect to the primes. The role
of metrics is played by symmetric matrices with integer
coefficients. The role of connections (respectively curvature)
attached to metrics is played by certain adelic (respectively global)
objects attached to the corresponding matrices.

One of the main conclusions of the theory is that the spectrum of
the integers is “intrinsically curved”; the study of this
curvature is then the main task of the theory. The book follows, and
builds upon, a series of recent research papers. A significant part of
the material has never been published before.

#### Readership

Graduate students and researchers interested in algebraic geometry, number theory, and algebraic groups.

#### Table of Contents

# Table of Contents

## Foundations of Arithmetic Differential Geometry

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Introduction 112
- Chapter 1. Algebraic background 2738
- Chapter 2. Classical differential geometry revisited 3950
- 2.1. Connections in principal bundles and curvature 3950
- 2.2. Lie algebra and classical groups 5364
- 2.3. Involutions and symmetric spaces 5869
- 2.4. Logarithmic derivative and differential Galois groups 6374
- 2.5. Chern connections: the symmetric/anti-symmetric case 6475
- 2.6. Chern connections: the hermitian case 7081
- 2.7. Levi-Cività connection and Fedosov connection 7283
- 2.8. Locally symmetric connections 7889
- 2.9. Ehresmann connections attached to inner involutions 7990
- 2.10. Connections in vector bundles 8091
- 2.11. Lax connections 8293
- 2.12. Hamiltonian connections 8596
- 2.13. Cartan connection 92103
- 2.14. Weierstrass and Riccati connections 94105
- 2.15. Differential groups: Cassidy and Painlevé 95106

- Chapter 3. Arithmetic differential geometry: generalities 99110
- 3.1. Global connections and their curvature 99110
- 3.2. Adelic connections 112123
- 3.3. Semiglobal connections and their curvature; Galois connections 115126
- 3.4. Curvature via analytic continuation between primes 119130
- 3.5. Curvature via algebraization by correspondences 125136
- 3.6. Arithmetic jet spaces and the Cartan connection 138149
- 3.7. Arithmetic Lie algebras and arithmetic logarithmic derivative 147158
- 3.8. Compatibility with translations and involutions 152163
- 3.9. Arithmetic Lie brackets and exponential 159170
- 3.10. Hamiltonian formalism and Painlevé 161172
- 3.11. 𝑝-adic connections on curves: Weierstrass and Riccati 166177

- Chapter 4. Arithmetic differential geometry: the case of 𝐺𝐿_{𝑛} 169180
- 4.1. Arithmetic logarithmic derivative and Ehresmann connections 169180
- 4.2. Existence of Chern connections 178189
- 4.3. Existence of Levi-Cività connections 193204
- 4.4. Existence/non-existence of Fedosov connections 198209
- 4.5. Existence/non-existence of Lax-type connections 202213
- 4.6. Existence of special linear connections 214225
- 4.7. Existence of Euler connections 215226
- 4.8. Curvature formalism and gauge action on 𝐺𝐿_{𝑛} 218229
- 4.9. Non-existence of classical 𝛿-cocycles on 𝐺𝐿_{𝑛} 228239
- 4.10. Non-existence of 𝛿-subgroups of simple groups 236247
- 4.11. Non-existence of invariant adelic connections on 𝐺𝐿_{𝑛} 242253

- Chapter 5. Curvature and Galois groups of Ehresmann connections 245256
- Chapter 6. Curvature of Chern connections 267278
- Chapter 7. Curvature of Levi-Cività connections 317328
- Chapter 8. Curvature of Lax connections 325336
- Chapter 9. Open problems 333344
- Bibliography 339350
- Index 343354
- Back Cover Back Cover1357