| Dwork, Bernard |
| Preface |
| Introduction |
| List of symbols |
| Valued Fields / Chapter I： |
| Valuations / 1： |
| Complete Valued Fields / 2： |
| Normed Vector Spaces / 3： |
| Hensel's Lemma / 4： |
| Extensions of Valuations / 5： |
| Newton Polygons / 6： |
| The y-intercept Method / 7： |
| Ramification Theory / 8： |
| Totally Ramified Extensions / 9： |
| Zeta Functions / Chapter II： |
| Logarithms |
| Newton Polygons for Power Series |
| Newton Polygons for Laurent Series |
| The Binomial and Exponential Series |
| Dieudonne's Theorem |
| Analytic Representation of Additive Characters |
| Meromorphy of the Zeta Function of a Variety |
| Condition for Rationality |
| Rationality of the Zeta Function |
| Appendix to Chapter II |
| Differential Equations / Chapter III： |
| Differential Equations in Characteristic p |
| Nilpotent Differential Operators. Katz-Honda Theorem |
| Differential Systems |
| The Theorem of the Cyclic Vector |
| The Generic Disk. Radius of Convergence |
| Global Nilpotence. Katz's Theorem |
| Regular Singularities. Fuchs' Theorem |
| Formal Fuchsian Theory |
| Effective Bounds. Ordinary Disks / Chapter IV： |
| p-adic Analytic Functions |
| Effective Bounds. The Dwork-Robba Theorem |
| Effective Bounds for Systems |
| Analytic Elements |
| Some Transfer Theorems |
| The Binomial Series |
| The Hypergeometric Function of Euler and Gauss |
| Effective Bounds. Singular Disks / Chapter V： |
| The Dwork-Frobenius Theorem |
| Effective Bounds for Solutions in a Singular Disk: the Case of Nilpotent Monodromy. The Christol-Dwork Theorem: Outline of the Proof |
| Proof of Step V |
| Proof of Step IV. The Shearing Transformation |
| Proof of Step III. Removing Apparent Singularities |
| The Operators (CHARACTER O w/ slash through it) and (CHARACTER U w/ slash through it) |
| Proof of Step I. Construction of Frobenius |
| Proof of Step II. Effective Form of the Cyclic Vector |
| Effective Bounds. The Case of Unipotent Monodromy |
| Transfer Theorems into Disks with One Singularity / Chapter VI： |
| The Type of a Number |
| Transfer into Disks with One Singularity: a First Estimate |
| The Theorem of Transfer of Radii of Convergence |
| Differential Equations of Arithmetic Type / Chapter VII： |
| The Height |
| The Theorem of Bombieri-Andre |
| Transfer Theorems for Differential Equations of Arithmetic Type |
| Size of Local Solution Bounded by its Global Inverse Radius |
| Generic Global Inverse Radius Bounded by the Global Inverse Radius of a Local Solution Matrix |
| G-Series. The Theorem of Chudnovsky / Chapter VIII： |
| Definition of G-Series- Statement of Chudnovsky's Theorem |
| Preparatory Results |
| Siegel's Lemma |
| Conclusion of the Proof of Chudnovsky's Theorem |
| Appendix to Chapter VIII |
| Convergence Polygon for Differential Equations / Appendix I： |
| Archimedean Estimates / Appendix II： |
| Cauchy's Theorem / Appendix III： |
| Bibliography |
| Index |
図書
