Very strong approximation for certain algebraic varieties

TL;DR

This paper establishes strong approximation results for certain algebraic varieties over global fields, showing the Brauer-Manin condition precisely characterizes rational points in specific cases and applying these results to conjectures and theorems in number theory.

Contribution

It proves that the Brauer-Manin condition exactly characterizes rational points for subvarieties of tori over certain global fields and applies this to conjectures on integral points and adelic points.

Findings

Brauer-Manin condition characterizes rational points on subvarieties of tori over function fields.

Proves a conjecture of Harari-Voloch for rational hyperbolic curves over global function fields.

Establishes a theorem of Stoll on adelic points of zero-dimensional subvarieties in abelian varieties.

Abstract

Let F be a global field. In this work, we show that the Brauer-Manin condition on adelic points for subvarieties of a torus T over F cuts out exactly the rational points, if either F is a function field or, if F is the field of rational numbers and T is split. As an application, we prove a conjecture of Harari-Voloch over global function fields which states, roughly speaking, that on any rational hyperbolic curve, the local integral points with the Brauer-Manin condition are the global integral points. Finally we prove for tori over number fields a theorem of Stoll on adelic points of zero-dimensional subvarieties in abelian varieties.