Multiplicative subgroups avoiding linear relations in finite fields and a local-global principle

Abstract

We study a local-global principle for polynomial equations with coefficients in a finite field and solutions restricted in a rank-one multiplicative subgroup in a function field over this finite field. We prove such a local-global principle for all sufficiently large characteristics, and we show that the result should hold in full generality under certain reasonable hypothesis related to the existence of large multiplicative subgroups of finite fields avoiding linear relations. We give a method for verifying the latter hypothesis in specific cases, and we show that it is a consequence of the classical Artin primitive root conjecture. In particular, this function field local-global principle is a consequence of GRH. We also discuss the relation of these problems with a finite field version of the Manin-Mumford conjecture.