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

TL;DR

This paper studies the distribution of orders and indices of algebraic numbers modulo primes in a number field, generalizing previous results and assuming GRH to analyze their arithmetic progression properties.

Contribution

It extends Ziegler's 2006 theorem to multiple algebraic numbers, examining their reduction properties modulo primes and the influence of Frobenius conjugacy classes.

Findings

Distribution of prime ideals with prescribed order and index properties.

Generalization of Ziegler's theorem to multiple algebraic numbers.

Conditional results assuming GRH for prime distribution.

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.