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

• Superior document: Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
• VerfasserIn: Dwork, Bernard,; Gerotto, Giovanni,; Sullivan, Francis J.,
• MitwirkendeR: Dwork, B.,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2016]; ©1994
• Year of Publication: 2016
• Language: English
• Series: Annals of Mathematics Studies ; 133
• Physical Description: 1 online resource (352 p.) :; 22 figs.

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

Table of Contents

Frontmatter -- CONTENTS -- PREFACE -- INTRODUCTION -- LIST OF SYMBOLS -- CHAPTER I. VALUED FIELDS -- CHAPTER II. ZETA FUNCTIONS -- CHAPTER III. DIFFERENTIAL EQUATIONS -- CHAPTER IV. EFFECTIVE BOUNDS. ORDINARY DISKS -- CHAPTER V. EFFECTIVE BOUNDS. SINGULAR DISKS -- CHAPTER VI. TRANSFER THEOREMS INTO DISKS WITH ONE SINGULARITY -- CHAPTER VII. DIFFERENTIAL EQUATIONS OF ARITHMETIC TYPE -- CHAPTER VIII. G-SERIES. THE THEOREM OF CHUDNOVSKY -- APPENDIX I. CONVERGENCE POLYGON FOR DIFFERENTIAL EQUATIONS -- APPENDIX II. ARCHIMEDEAN ESTIMATES -- APPENDIX III. CAUCHY'S THEOREM -- BIBLIOGRAPHY -- INDEX