Divisibility conditions on the order of the reductions of algebraic numbers

Abstract

Let $K$ be a number field, and let $G$ be a finitely generated subgroup of $K^{\times }$. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes $\mathfrak{p}$ of $K$ such that the order of $(G\mathrm{mod}\mathfrak{p})$ is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes $\mathfrak{p}$ for which the order is $k$-free, and those for which the order has a prescribed $\ell$-adic valuation for finitely many primes $\ell$. An additional condition on the Frobenius conjugacy class of $\mathfrak{p}$ may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.

arXiv comment24 pages, comments are welcome