On Artin's Conjecture for Rank One Drinfeld Modules

Abstract

Let k be a global function field with a chosen degree one prime divisor ∞, and O⊂k is the subring consisting of all functions regular away from ∞. Let φ 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 φ. Given any a≠0 in O′, we show that the density of the set consisting of all prime ideals P′ in O′ such that a (modP′) is a generator of φ(O′/P′) is always positive, provided the constant field of k has more than two elements. © 2001 Academic Press.