Equality of orders of a set of integers modulo a prime

Abstract

For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods for determining if there exists infinitely many primes $p$ such that the index of the reduction of $W_i$ modulo $p$ divides $k_i$ and such that the Artin symbol of $p$ on $F$ is contained in $C$. The results are a multivariable generalization of H.W. Lenstra's work. As an application, we determine all integers $a_1, \ldots , a_n$ such that $\text{ord}_p(a_1) = \ldots = \text{ord}_p(a_n)$ for infinitely many primes $p$. We also discuss the set of those $p$ for which $\text{ord}_p(a_1) > \ldots > \text{ord}_p(a_n)$. The obtained results are conditional to a generalization of the Riemann hypothesis.