A Local-Global Criteria of Affine Varieties Admitting Points in Rank-One Subgroups of a Global Function Field

Abstract

For every affine variety over a finite field, we show that it admits points with coordinates in an arbitrary rank-one multiplicative subgroup of a global function field over this finite field if and only if this variety admits points with coordinates in the topological closure of this subgroup in the product of the multiplicative group of those local completion of this global function field over all but finitely many places. Under the generalized Riemann hypothesis, we also show that the above statement holds for every finite union of affine linear varieties over any global field and for many rank-one multiplicative subgroup. In the case where this finite union is irreducible and defined over a finite field, we moreover show that the topological closure of the set of all such former points is exactly the set of all such latter points.