# An Introduction to G-functions

Princeton University Press, 1994 - Mathematics - 323 pages

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, Andr , 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 Andr on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.

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### Contents

 Valued Fields 1 Valuations 3 Complete Valued Fields 6 Normed Vector Spaces 8 Hensels Lemma 10 Extensions of Valuations 17 Newton Polygons 24 The intercept Method 28 Ramification Theory 30
 Effective Bounds The Dwork Robba Theorem 119 Effective Bounds for Systems 126 Analytic Elements 128 Some Transfer Theorems 133 Logarithms 138 The Binomial Series 140 The Hypergeometric Function of Euler and Gauss 150 Effective Bounds Singular Disks 1 The DworkFrobenius Theorem 155

 Totally Ramified Extensions 33 Zeta Functions 1 Logarithms 38 Newton Polygons for Power Series 41 Newton Polygons for Laurent Series 46 The Binomial and Exponential Series 49 Diendonnes Theorem 53 S Analytic Representation of Additive Characters 56 Meromorphy of the Zeta Function of a Variety 61 Condition for Rationality 71 Rationality of the Zeta Function 74 Appendix to Chapter II 76 Differential Equations 1 Differential Equations in Charateristic p 77 Nilpotent Differential Operators KatzHonda Theorem 81 Differential Systems 86 The Theorem of the Cyclic Vector 89 The Generic Disk Radius of Convergence 92 Global Nilpotence Katzs Theorem 98 Regular Singularities Fuchs Theorem 100 Formal Fuchsian Theory 102 Effective Bounds Ordinary Disks 1 padic Analytic Functions 114
 the Case of Nilpotent Monodromy The Christol Dwork Theorem Outline of the Proof 159 Proof of Step V 168 Proof of Step IV The Shearing Transformation 169 Proof of Step III Removing Apparent Singularities 170 The Operators 4 and tl 173 Proof of Step I Construction of Frobenius 176 Proof of Step II Effective Form of the Cyclic Vector 180 Effective Bounds The Case of Unipotent Monodromy 189 Transfer Theorems into Disks with One Singularity 1 The Type of a Number 199 a First Estimate 203 The Theorem of Transfer of Radii of Convergence 212 Differential Equations of Arithmetic Type 1 The Height 222 The Theorem of BombieriAndre 226 Transfer Theorems for Differential Equations of Arithmetic Type 234 GSeries The Theorem of Chudnovsky 263 Convergence Polygon for Differential 301 Archimedean Estimates 307 bibliography 317 Copyright