Treffer: On primitive roots of tori: The case of function fields: On primitive roots of tori: the case of function fields

Artin's conjecture on primitive roots (1927) states that for any integer \(g\not=-1\) or a square, there are infinitely many primes \(p\) for which \(g\) is a primitive root. An analogous conjecture has been proposed for polynomials, namely that for \(p\) a prime and \(a(x)\) a non-constant polynomial mod \(p\) there are infinitely many irreducible polynomials \(P(x)\) such that \(a(x)\) generates the classes of \((\mathbb {F}_p[x]/(P(x)))^*\). The latter conjecture was proved by \textit{H. Bilharz} [Math. Ann. 114, 476--492 (1937; Zbl 0016.34301)] under the assumption of the Riemann hypothesis for curves; Bilharz' theorem subsequently became unconditional as a consequence of work of A. Weil. The authors generalize the theorem of Bilharz to all one-dimensional tori over global function fields of a finite constant field. To be more precise, let \(\mathbb F\) be a global function field having a finite field \(k\) as field of constants and let \({\mathbb T}(\mathbb F)\) be a torus in dimension one and given a non-torsion rational point \(P_0\) in this torus. Let \(M_{P_0}\) be the set consisting of prime divisors \(v\) of \(\mathbb F\) where \({\mathbb T}\) has good reduction \({\tilde {\mathbb T}}\) and \(P_0\) modulo \(v\) generates the abelian group \({\tilde {\mathbb T}}(\mathbb F(v))\), where \(\mathbb F(v)\) is the finite residue field of \(\mathbb F\) at \(v\). The main result of this paper states that if \(P_0\) in \({\mathbb T}(\mathbb F)\) is a non-torsion rational point, then the set \(M_{P_0}\) has a positive density if and only if \(P_0\) is not in \({\mathbb T}(\mathbb F)^q\) for every prime \(q\) dividing the cardinality of the number of torsion points in \({\mathbb T}(\mathbb F)\). As an application the authors derive an analogue of a theorem of Chen-Kitaoka-Yu and, independently, Roskam, on the distribution of fundamental units modulo primes. The work goes into the algebraic part; the basic analytic tool used is due to Bilharz and Clark-Kuwata and gives conditions under which the density involved exists and expresses it into Galois theoretic terms from which the positivity of the density is not directly clear.