Reductions of subgroups of the multiplicative group

Abstract

Let $K$ be a number field, and let $G\subset K^{\times }$ be a finitely generated subgroup. Fix some prime number $\ell$, and consider the set of primes $\mathfrak{p}$ of $K$ satisfying the following property: the reduction of $G$ modulo $\mathfrak{p}$ is well-defined and has size coprime to $\ell$. We give a closed--form expression for the density of this set.

v2: improved results, notations simplified, added heuristics

References in corpus (2)

• Prescribing valuations of the order of a point in the reductions of abelian varieties and tori
• The order of the reductions of an algebraic integer