An Introduction to G-Functions. (AM-133), Volume 133

By: Bernard Dwork, Giovanni Gerotto, Francis J. Sullivan

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Paperback | 2 August 1994

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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.

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"[This book] is well suited for a graduate course, its value as a textbook being enhanced by working out several concrete examples and counterexamples of the phenomena studied in the book."--Mathematical Reviews

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

An Introduction to G-Functions. (AM-133), Volume 133

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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