Hilbert irreducibility for integral points on punctured linear algebraic groups

TL;DR

This paper proves a conjecture predicting the existence of non-thin sets of integral points on punctured linear algebraic groups, building on recent advances in Hilbert irreducibility and strong approximation.

Contribution

It establishes the predicted property for all connected linear algebraic groups, extending previous work on Hilbert irreducibility and strong approximation.

Findings

Proves the conjecture for all connected linear algebraic groups.

Utilizes recent Hilbert irreducibility results for algebraic groups.

Connects integral strong approximation with the Hilbert property.

Abstract

Let $K$ be a number field, let $X$ be a smooth integral variety over $K$, and assume that there exists a finite set of finite places $S$ of $K$ such that the $S$-integral points on $X$ are dense. Then the combined conjectures of Campana and Corvaja-Zannier predict that, for every closed subscheme $Z$ of $X$ of codimension at least two, there exists a finite extension $L$ of $K$ and a finite set of finite places $T$ of $L$ such that the $T$-integral points on $(X\setminus Z)_L$ are not strongly thin. The main goal of the present paper is to show that this property holds for all connected linear algebraic groups. Our result builds mainly on recent work on a Hilbert irreducibility type theorem for connected algebraic groups, the purity of strong approximation for semi-simple simply connected quasi-split linear algebraic groups, and the relation between integral strong approximation and the Hilbert property.