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An Introduction to G-Functions. (AM-133)
Bernard Dwork, Giovanni Gerotto, & Francis J. Sullivan

Book Description | Reviews

TABLE OF CONTENTS:

PREFACE
INTRODUCTION xiii
LIST OF SYMBOLS xix
CHAPTER I Valued Fields
1. Valuations 3
2. Complete Valued Fields 6
3. Normed Vector Spaces 8
4. Hensel's Lemma 10
5. Extensions of Valuations 17
6. Newton Polygons 24
7. The y-intercept Method 28
8. Ramification Theory 30
9. Totally Ramified Extensions 33
CHAPTER II Zeta Functions
1. Logarithms 38
2. Newton Polygons for Power Series 41
3. Newton Polygons for Laurent Series 46
4. The Binomial and Exponential Series 49
5. Dieudonne's Theorem 53
6. Analytic Representation of Additive Characters 56
7. Meromorphy of the Zeta Function of a Variety 61
8. Condition for Rationality 71
9. Rationality of the Zeta Function 74
Appendix to Chapter II 76
CHAPTER III Differential Equations
1. Differential Equations in Characteristic p 77
2. Nilpotent Differential Operators. Katz-Honda Theorem 81
3. Differential Systems 86
4. The Theorem of the Cyclic Vector 89
5. The Generic Disk. Radius of Convergence 92
6. Global Nilpotence. Katz's Theorem 98
7. Regular Singularities. Fuchs' Theorem 100
8. Formal Fuchsian Theory 102
CHAPTER IV Effective Bounds. Ordinary Disks
1. p-adic Analytic Functions 114
2. Effective Bounds. The Dwork-Robba Theorem 119
3. Effective Bounds for Systems 126
4. Analytic Elements 128
5. Some Transfer Theorems 133
6. Logarithms 138
7. The Binomial Series 140
8. The Hypergeometric Function of Euler and Gauss 150
CHAPTER V Effective Bounds. Singular Disks
1. The Dwork-Frobenius Theorem 155
2. Effective Bounds for Solutions in a Singular Disk: the Case of Nilpotent Monodromy. The Christol-Dwork Theorem: Outline of the Proof 159
3. Proof of Step V 168
4. Proof of Step IV. The Shearing Transformation 169
5. Proof of Step III. Removing Apparent Singularities 170
6. The Operators (CHARACTER O w/ slash through it) and (CHARACTER U w/ slash through it) 173
7. Proof of Step I. Construction of Frobenius 176
8. Proof of Step II. Effective Form of the Cyclic Vector 180
9. Effective Bounds. The Case of Unipotent Monodromy 189
CHAPTER VI Transfer Theorems into Disks with One Singularity
1. The Type of a Number 199
2. Transfer into Disks with One Singularity: a First Estimate 203
3. The Theorem of Transfer of Radii of Convergence 212
CHAPTER VII Differential Equations of Arithmetic Type
1. The Height 222
2. The Theorem of Bombieri-Andre 226
3. Transfer Theorems for Differential Equations of Arithmetic Type 234
4. Size of Local Solution Bounded by its Global Inverse Radius 243
5. Generic Global Inverse Radius Bounded by the Global Inverse Radius of a Local Solution Matrix 254
CHAPTER VIII G-Series. The Theorem of Chudnovsky
1. Definition of G-Series- Statement of Chudnovsky's Theorem 263
2. Preparatory Results 267
3. Siegel's Lemma 284
4. Conclusion of the Proof of Chudnovsky's Theorem 289
Appendix to Chapter VIII 300
APPENDIX I Convergence Polygon for Differential Equations 301
APPENDIX II Archimedean Estimates 307
APPENDIX III Cauchy's Theorem 310
BIBLIOGRAPHY 317
INDEX 321

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File created: 11/11/2014

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