Hilbert irreducibility for integral points on punctured linear algebraic groups
Cedric Luger
TL;DR
The paper tackles conjectures of Campana and Corvaja–Zannier by proving that punctured connected linear algebraic groups G \/ Z (with Z of codimension at least 2) satisfy the weak Hilbert property after a finite field extension. It develops the OPS/OPSE framework and leverages a combination of Hilbert irreducibility for algebraic groups, integral strong approximation, and product/descents to assemble a general descent argument. The main result shows that every connected linear algebraic group over a number field exhibits the weak Hilbert property after suitable base changes, supporting the conjectured links between density of integral points and non-thinness under ramified covers. This work integrates and extends results for tori, unipotent groups, and semi-simple simply connected groups to the broader class of linear algebraic groups, contributing a substantial step toward the punctured version of CZHP.
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.