On Artinʹs Conjecture for Rank One Drinfeld Modules Original Research Article

Let k be a global function field with a chosen degree one prime divisor ∞, and imagesubset ofk is the subring consisting of all functions regular away from ∞. Let φ be a sgn-normalized rank one Drinfeld image-module defined over image′, the integral closure of image in the Hilbert class field of image. We prove an analogue of the classical Artinʹs primitive roots conjecture for φ. Given any a≠0 in image′, we show that the density of the set consisting of all prime ideals image′ in image′ such that a (mod image′) is a generator of φ(image′/image′) is always positive, provided the constant field of k has more than two elements.