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

Abstract

Let $\alpha_1,\ldots,\alpha_r$ 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 $(\alpha_i\bmod\mathfrak p)$ lies in a given arithmetic progression associated to $\alpha_i$. We also study the primes $\mathfrak p$ for which the index of $(\alpha_i\bmod\mathfrak p)$ 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.