Product of local points of subvarieties of almost isotrivial semi-abelian varieties over a global function field

TL;DR

This paper investigates the structure of local points on subvarieties of semi-abelian varieties over global function fields, revealing how global points are characterized by local conditions and the Brauer-Manin obstruction.

Contribution

It establishes a precise description of the closure of finitely generated subgroups in local points for certain semi-abelian varieties over global function fields, linking local-global principles.

Findings

The topological closure of a finitely generated subgroup matches the global points in the subvariety.

The Brauer-Manin condition exactly characterizes rational points on certain super-singular curves.

Results apply to varieties isogenous to isotrivial semi-abelian varieties over global function fields.

Abstract

For a semi-abelian variety over a global function field which is isogenous to an isotrivial one, we show that on the product of local points of a subvariety satisfying a minor condition, the topological closure of a finitely generated subgroup of %the group of its global points cuts out exactly the global points of the subvariety lying in this subgroup. As a corollary, on every non-isotrivial super-singular curve of genus two over a global function field, we conclude that the Brauer-Manin condition cuts out exactly the set of its rational points.