Equality of orders of a set of integers modulo a prime

Abstract

For finitely generated subgroups $W_1,\dots ,W_t$ of ${\mathbb{Q}}^{\times }$, integers $k_1,\dots ,k_t$, 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 $a_1,\dots ,a_n$ such that ${\text{ord}}_p(a_1)=\dots ={\text{ord}}_p(a_n)$ for infinitely many primes $p$. We also discuss the set of those $p$ for which ${\text{ord}}_p(a_1)>\dots >{\text{ord}}_p(a_n)$. The obtained results are conditional to a generalization of the Riemann hypothesis.

20 pages. Add section on Kummer-type extensions and improve exposition