Hilbert irreducibility for integral points on punctured linear algebraic groups

2024/10/17 by Cedric Luger, Luger, Cedric · 2 citations
Mathematics · #14G40 #20G30 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Primary: 14G99. Secondary: 14G05

paper · pdf · doi:10.48550/arxiv.2410.13403

openalex publication_date 2024/10/17 · openalex created_date 2024/10/21 · openalex updated_date 2026/07/28

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∖ 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.

Cited by

• Hilbert properties of varieties
• Fibration theorems for varieties with the weak Hilbert property

Related

• Products of varieties with many integral points
• The arithmetic puncturing problem and integral points
• An Hilbert Irreducibility Theorem for integral points of del Pezzo surfaces
• The Hilbert property for arithmetic schemes
• Arithmetic purity of strong approximation for homogeneous spaces