On Artin's conjecture for rank one Drinfeld modules

Let k be a global function field with a chosen degree one prime divisor inf inity, and O subset of k is the subring consisting of all functions regular away from infinity. Let phi be a sgn-normalized rank one Drinfeld O-module defined over O ', the integral closure of O in the Hilbert class field of O. We prove an analogue of the classical Artin's primitive roots conjecture for phi. Given any a not equal0 in O ', we show that the density of the se t consisting of all prime ideals B ' in O ' such that a (mod B ') is a gene rator of phi (O ' /B') is always positive, provided the constant field of k has more than two elements. (C) 2001 Academic Press.