Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Perucca, Antonella; Sgobba, Pietro

doi:10.2478/udt-2020-0004

Abstract

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K^×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓ^e is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓ_e only depends on a through its ℓ-adic valuation.

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Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Abstract

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