Arithmetic Differential Equations

Contents Preface

vii

Introduction 0.1, Motivation and strategy 0.2. Rough outline of the theory 0.3. Comparison with other theories

xi xi xiv xix

Part 1.

Main concepts and results

1

Chapter 1. Preliminaries from algebraic geometry 1.1. Algebro-geometric terminology 1.2. Categorical quotients in algebraic geometry 1.3. Analytic uniformization and critical

3 3 11 15

finiteness

Chapter 2. Outline of 5—geometry 2.1. Main concepts of i—geometry 2.2. Main conjectures 2.3. Main results: a sample 2.4. Appendix: Axiomatic characterization of δ—geometries

31 31 55 56 61

Part 2.

69

General theory

Chapter 3. Global theory 3.1. p—jet spaces of schemes 3.2. Behavior with respect to etale maps 3.3. Link between p—jet spaces and δ—functions 3.4. Galois covers 3.5. δ—base loci versus postcritical loci 3.6. δ—tangent maps and δ—differentials of δ—functions

71 71 75 78 82 85 88

Chapter 4. Local theory 4.1. Local analogue of the main conjectures 4.2. Bounding the rank of the module of δ—invariants 4.3. δ—invariants for the formal multiplicative group 4.4. p—jets of formal group laws 4.5. Local versus global picture: fixed points and cycles 4.6. Some general global converse results

107 107 119 121 123 126 132

Chapter 5. Birational theory 5.1, The basic graded rings 5.2. δ—Galois groups

141 141 145

vi

CONTENTS

5.3. δ—invariants for subgroups of PGL2(Zp) Part 3. Chapter 6.1. 6.2. 6.3. 6.4. 6.5. 6.6.

Applications 6. Spherical correspondences Spherical correspondences over R and their cycles δ—sections of bundles on the projective line Case Γ trivial: δ—invariants and δ—cohomology Case Γ non-trivial: δ—invariants, δ—cohomology and δ—fiber Case (Γ, τ) solvable A converse theorem: biquadratic correspondences

147 159 161 161 165 167 173 178 181

Chapter 7. Flat correspondences 7.1. Flat correspondences over R: δ—line bundles and cycles 7.2. δ—characters 7.3. δ—invariants 7.4. δ-base loci 7.5. δ—cohomology 7.6. The relative generic δ—fiber 7.7. Converse theorems: quadratic dynamical systems

185 185 190 205 210 212 221 223

Chapter 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10.

227 228 240 251 260 275 279 285 288 292 295

8. Hyperbolic correspondences Review of Abelian schemes and their crystals Hecke correspondences over R: δ—line bundles and cycles δ—Serre-Tate expansion maps and δ—Serre operators Constructions of δ—invariants δ—invariants in the ordinary case δ—invariants in the non-ordinary case δ—cohomology The relative generic £-fiber Hecke correspondences with a regular Hecke n—cycle A converse theorem: non-rational hyperbolic uniformization

List of Results

299

Bibliography

301

Index

307