Reductions of one-dimensional tori

Abstract

Consider a non-split one-dimensional torus defined over a number field K. For a finitely generated group G of rational points and for a prime number l, we investigate for how many primes p of K the size of the reduction of G modulo p is coprime to l. We provide closed formulas for the corresponding Dirichlet density in terms of finitely many computable parameters. To achieve this, we determine in general which torsion fields and Kummer extensions contain the splitting field