Reductions of subgroups of the multiplicative group

2013/12/23 by Christophe Debry, Antonella Perucca, Debry, Christophe +1
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Finite Group Theory Research #math.NT

paper · pdf · doi:10.48550/arxiv.1312.6620

v2: improved results, notations simplified, added heuristics

arxiv created 2014/09/17 · arxiv updated 2014/09/18

Abstract

Let K be a number field, and let G⊂ K^× be a finitely generated subgroup. Fix some prime number ℓ, and consider the set of primes \mathfrakp of K satisfying the following property: the reduction of G modulo \mathfrakp is well-defined and has size coprime to ℓ. We give a closed--form expression for the density of this set.

Related

• Divisibility conditions on the order of the reductions of algebraic numbers
• The order of the reductions of an algebraic integer
• Unconditional results for Artin-type problems over number fields
• Multiplicative subgroups are not restricted sumsets
• Arithmetic progressions in multiplicative groups of finite fields