Local–Global Principle of Affine Varieties Over a Subgroup of Units in a Function Field

Abstract

Over a large class of function fields, we show that for many linear varieties in an affine space, the set of their points over the topological closure of a certain subgroup of the group of units of the function field is exactly the topological closure of the set of their points over this subgroup. This provides some evidence on the split-algebraic-torus analog of a conjecture for abelian varieties by Poonen and Voloch, as well as the function-field analog of an old conjecture by Skolem.