On the distribution of the order and index for the reductions of algebraic numbers

Abstract

Let ${\alpha }_1,\dots ,{\alpha }_r$ be algebraic numbers in a number field $K$ generating a subgroup of rank $r$ in $K^{\times }$. We investigate under GRH the number of primes $\mathfrak{p}$ of $K$ such that each of the orders of $({\alpha }_i\mathrm{mod}\mathfrak{p})$ lies in a given arithmetic progression associated to ${\alpha }_i$. We also study the primes $\mathfrak{p}$ for which the index of $({\alpha }_i\mathrm{mod}\mathfrak{p})$ is a fixed integer or lies in a given set of integers for each $i$. An additional condition on the Frobenius conjugacy class of $\mathfrak{p}$ may be considered. Such results are generalizations of a theorem of Ziegler from 2006, which concerns the case $r=1$ of this problem.