A Local-Global Equality on Every Affine Variety Admitting Points in an Arbitrary Rank-One Subgroup of a Global Function Field

TL;DR

This paper proves a topological density result for points on affine varieties over global function fields, relating points with coordinates in a rank-one subgroup to their closure in local completions.

Contribution

It establishes a local-global equality for points on affine varieties over global function fields with coordinates in a rank-one subgroup, extending previous density results.

Findings

Points with coordinates in the subgroup are dense in the closure.

The result applies to all affine varieties over global function fields.

It generalizes known density theorems to a broader setting.

Abstract

For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field is topologically dense in the set of its points with coordinates in the topological closure of this subgroup in the product of the multiplicative group of those local completions of this function field over all but finitely many places.