Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non-zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s. After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, Andre, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G -series is again a G -series. This book will be indispensable for those wishing to study the work of Bombieri and Andre on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.

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