Equality of orders of a set of integers modulo a prime

2019/12/05 by Järviniemi, Olli
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.02554

Abstract

For finitely generated subgroups W₁, … , W_t of ℚ^×, integers k₁, … , k_t, a Galois extension F of ℚ and a union of conjugacy classes C ⊂ Gal(F/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 a₁, … , a_n such that ord_p(a₁) = … = ord_p(a_n) for infinitely many primes p. We also discuss the set of those p for which ord_p(a₁) > … > ord_p(a_n). The obtained results are conditional to a generalization of the Riemann hypothesis.

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