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An Introduction to

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

Annals of Mathematics Studies (Paperback)


Published: 1st May 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.

"[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

Dwork, Bernard
Introductionp. xiii
List of symbolsp. xix
Valued Fields
Valuationsp. 3
Complete Valued Fieldsp. 6
Normed Vector Spacesp. 8
Hensel's Lemmap. 10
Extensions of Valuationsp. 17
Newton Polygonsp. 24
The y-intercept Methodp. 28
Ramification Theoryp. 30
Totally Ramified Extensionsp. 33
Zeta Functions
Logarithmsp. 38
Newton Polygons for Power Seriesp. 41
Newton Polygons for Laurent Seriesp. 46
The Binomial and Exponential Seriesp. 49
Dieudonne's Theoremp. 53
Analytic Representation of Additive Charactersp. 56
Meromorphy of the Zeta Function of a Varietyp. 61
Condition for Rationalityp. 71
Rationality of the Zeta Functionp. 74
Appendix to Chapter IIp. 76
Differential Equations
Differential Equations in Characteristic pp. 77
Nilpotent Differential Operators. Katz-Honda Theoremp. 81
Differential Systemsp. 86
The Theorem of the Cyclic Vectorp. 89
The Generic Disk. Radius of Convergencep. 92
Global Nilpotence. Katz's Theoremp. 98
Regular Singularities. Fuchs' Theoremp. 100
Formal Fuchsian Theoryp. 102
Effective Bounds. Ordinary Disks
p-adic Analytic Functionsp. 114
Effective Bounds. The Dwork-Robba Theoremp. 119
Effective Bounds for Systemsp. 126
Analytic Elementsp. 128
Some Transfer Theoremsp. 133
Logarithmsp. 138
The Binomial Seriesp. 140
The Hypergeometric Function of Euler and Gaussp. 150
Effective Bounds. Singular Disks
The Dwork-Frobenius Theoremp. 155
Effective Bounds for Solutions in a Singular Disk: the Case of Nilpotent Monodromy. The Christol-Dwork Theorem: Outline of the Proofp. 159
Proof of Step Vp. 168
Proof of Step IV. The Shearing Transformationp. 169
Proof of Step III. Removing Apparent Singularitiesp. 170
The Operators (CHARACTER O w/ slash through it) and (CHARACTER U w/ slash through it)p. 173
Proof of Step I. Construction of Frobeniusp. 176
Proof of Step II. Effective Form of the Cyclic Vectorp. 180
Effective Bounds. The Case of Unipotent Monodromyp. 189
Transfer Theorems into Disks with One Singularity
The Type of a Numberp. 199
Transfer into Disks with One Singularity: a First Estimatep. 203
The Theorem of Transfer of Radii of Convergencep. 212
Differential Equations of Arithmetic Type
The Heightp. 222
The Theorem of Bombieri-Andrep. 226
Transfer Theorems for Differential Equations of Arithmetic Typep. 234
Size of Local Solution Bounded by its Global Inverse Radiusp. 243
Generic Global Inverse Radius Bounded by the Global Inverse Radius of a Local Solution Matrixp. 254
G-Series. The Theorem of Chudnovsky
Definition of G-Series- Statement of Chudnovsky's Theoremp. 263
Preparatory Resultsp. 267
Siegel's Lemmap. 284
Conclusion of the Proof of Chudnovsky's Theoremp. 289
Appendix to Chapter VIIIp. 300
Convergence Polygon for Differential Equationsp. 301
Archimedean Estimatesp. 307
Cauchy's Theoremp. 310
Bibliographyp. 317
Indexp. 321
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691036816
ISBN-10: 0691036810
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 352
Published: 1st May 1994
Country of Publication: US
Dimensions (cm): 22.96 x 15.55  x 2.29
Weight (kg): 0.56