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

Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level ofp-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 fieldK. These series satisfy a linear differential equationLy=0withLIK(x) [d/dx]and have non-zero radii of convergence for each imbedding ofKinto the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the indexs.

After presenting a review of valuation theory and elementaryp-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of thep-adic properties of formal power series solutions of linear differential equations. In particular, thep-adic radii of convergence and thep-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 aG-series is again aG-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.

Bernard Dworkis Professor of Mathematics at Princeton University.Giovanni GerottoandFrancis J. Sullivanare Associate Professors of Mathematics at the University of Padova. [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.