On Primitive Roots of One-Dimensional Tori

Abstract

We exploit an analogue of Artin's primitive roots conjecture for one-dimensional tori over Q. Let T be a one-dimensional torus over Q, and let P∈T(Q) be a nontorsion point. Under Generalized Riemann Hypothesis, we derive an explicit density formula for the set of rational primes ℓ such that P modulo ℓ generates T(Fℓ).