Adelic Mordell-Lang and the Brauer-Manin obstruction

TL;DR

This paper proves that under certain conditions, adelic points on a subvariety of an abelian variety over a global function field are limits of rational points, linking to the Brauer-Manin obstruction and Mordell-Lang conjecture.

Contribution

It establishes a new adelic Mordell-Lang type result over function fields and number fields, connecting rational points, adelic limits, and the Brauer-Manin obstruction.

Findings

Adelic points on subvarieties are limits of rational points under specified conditions.

Rational points are dense in the Brauer set assuming Tate-Shafarevich group finiteness.

Conditional results over number fields depend on an adelic Mordell-Lang conjecture.

Abstract

Let $X$ be a closed subvariety of an abelian variety $A$ over a global function field $k$ such that the base change of $A$ to an algebraic closure does not have any positive dimensional isotrivial quotient. We prove that every adelic point on $X$ which is the limit of a sequence of $k$-rational points on $A$ is a limit of $k$-rational points on $X$. Assuming finiteness of the Tate-Shafarevich group of $A$, this implies that the rational points on $X$ are dense in the Brauer set of $X$. Similar results are obtained over totally imaginary number fields, conditionally on an adelic Mordell-Lang conjecture.