Divisibility conditions on the order of the reductions of algebraic numbers

TL;DR

This paper establishes asymptotic formulas and density results for primes in number fields where the order of reductions of a finitely generated subgroup is divisible by a fixed integer, without assuming GRH.

Contribution

It provides the first unconditional asymptotic and density results for divisibility conditions on orders of reductions of algebraic numbers in number fields.

Findings

Asymptotic formula for primes with order divisible by a fixed integer

Rational expression for the natural density of such primes

Results on primes with k-free order and prescribed -adic valuations

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 $(G\bmod\mathfrak p)$ 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.