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