The distribution of the multiplicative index of algebraic numbers over residue classes

Abstract

Let K be a number field and G a finitely generated torsion-free subgroup of \(K^\times \). Given a prime \(\mathfrak {p}\) of K we denote by \({{\,\textrm{ind}\,}}_\mathfrak {p}(G)\) the index of the subgroup \((G\bmod \mathfrak {p})\) of the multiplicative group of the residue field at \(\mathfrak {p}\). Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.