**Mathematical Surveys and Monographs**

Volume: 118;
2005;
310 pp;
Hardcover

MSC: Primary 14; 30;
Secondary 53; 37

**Print ISBN: 978-0-8218-3862-4
Product Code: SURV/118**

List Price: $104.00

AMS Member Price: $83.20

MAA Member Price: $93.60

**Electronic ISBN: 978-1-4704-1345-3
Product Code: SURV/118.E**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

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#### Supplemental Materials

# Arithmetic Differential Equations

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*Alexandru Buium*

This research monograph develops an arithmetic analogue of the theory of ordinary differential equations: functions are replaced here by integer numbers, the derivative operator is replaced by a “Fermat quotient operator”, and differential equations (viewed as functions on jet spaces) are replaced by “arithmetic differential equations”. The main application of this theory concerns the construction and study of quotients of algebraic curves by correspondences with infinite orbits. Any such quotient reduces to a point in usual algebraic geometry. But many quotients as above cease to be trivial (and become quite interesting) if one enlarges algebraic geometry by using arithmetic differential equations in place of algebraic equations. The book partly follows a series of papers written by the author; however, a substantial part of the material presented here has never been published before. For most of the book the only prerequisites are the basic facts of algebraic geometry and number theory.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry and number theory.

#### Reviews & Endorsements

The book is written very clearly and organized beautifully.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Arithmetic Differential Equations

- Contents v6 free
- Preface vii8 free
- Introduction xi12 free
- Part 1. Main concepts and results 134 free
- Part 2. General theory 69102
- Chapter 3. Global theory 71104
- Chapter 4. Local theory 107140
- 4.1. Local analogue of the main conjectures 107140
- 4.2. Bounding the rank of the module of δ invariants 119152
- 4.3. δ invariants for the formal multiplicative group 121154
- 4.4. p–jets of formal group laws 123156
- 4.5. Local versus global picture: fixed points and cycles 126159
- 4.6. Some general global converse results 132165

- Chapter 5. Birational theory 141174

- Part 3. Applications 159192
- Chapter 6. Spherical correspondences 161194
- 6.1. Spherical correspondences over R and their cycles 161194
- 6.2. δ sections of bundles on the projective line 165198
- 6.3. Case Γ trivial: δ invariants and δ cohomology 167200
- 6.4. Case Γ non-trivial: δ invariants, δ cohomology and δ fiber 173206
- 6.5. Case 〈Γ,τ〉 solvable 178211
- 6.6. A converse theorem: biquadratic correspondences 181214

- Chapter 7. Flat correspondences 185218
- Chapter 8. Hyperbolic correspondences 227260
- 8.1. Review of Abelian schemes and their crystals 228261
- 8.2. Hecke correspondences over R: δ line bundles and cycles 240273
- 8.3. δ Serre-Tate expansion maps and δ Serre operators 251284
- 8.4. Constructions of δ invariants 260293
- 8.5. δ invariants in the ordinary case 275308
- 8.6. δ invariants in the non-ordinary case 279312
- 8.7. δ cohomology 285318
- 8.8. The relative generic δ fiber 288321
- 8.9. Hecke correspondences with a regular Hecke n–cycle 292325
- 8.10. A converse theorem: non-rational hyperbolic uniformization 295328

- List of Results 299332
- Bibliography 301334
- Index 307340