The algebraist’s upper half-plane

Abstract

Introduction. The purpose of this article is to introduce the general mathematical community to some recent developments in algebraic geometry and nonarchimedean analysis. Let r = p n ,p a rational prime. Then these developments center around the beginnings of an "arithmetic" theory of the polynomial ring r [T] over the finite field of r elements. The goal of this theory is to use nonarchimedean analysis to do for Y r [T] what classical analysis does for Z. The theory allows us to find direct analogues of many of the classical functions of arithmetic interest in a situation that, at first glance, seems as nonclassical as possible. In the process much will be learned about the polynomials. Much also will be learned about the unique properties of Z and the classical functions.