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

Format:

Book

Author/Creator:

Dwork, Bernard, author.

Gerotto, Giovanni, author.

Sullivan, Francis J., author.

Series:

Annals of mathematics studies ; no. 133.

Annals of Mathematics Studies ; 316

Language:

English

Subjects (All):

H-functions.

p-adic analysis.

Physical Description:

1 online resource (349 pages) : illustrations.

Place of Publication:

Princeton, NJ : Princeton University Press, [2016]

Language Note:

In English.

Summary:

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.

Contents:

Frontmatter

CONTENTS

PREFACE / Dwork, B.

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

Notes:

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

ISBN:

9781400882540

1400882540

OCLC:

979968812