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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