Products of varieties with many integral points

2024/01/10 by Cedric Luger, Luger, Cedric · 3 citations
Computer Science · Mathematics · #14G40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Primary: 14G99. Secondary: 14G05

paper · pdf · doi:10.48550/arxiv.2401.05203

openalex publication_date 2024/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Corvaja and Zannier asked whether a smooth projective integral variety with a dense set of rational points over a number field satisfies the weak Hilbert property. We introduce an extension of the weak Hilbert property for schemes over arithmetic base rings by considering near-integral points, extending Corvaja-Zannier's question beyond the projective case. Building on work of Bary-Soroker-Fehm-Petersen and Corvaja-Demeio-Javanpeykar-Lombardo-Zannier, we prove several properties of this more general notion, in particular its persistence under products. We also answer positively Corvaja-Zannier's question for all algebraic groups over finitely generated fields of characteristic zero.

Cited by

• Hilbert properties of varieties
• Fibration theorems for varieties with the weak Hilbert property
• Hilbert irreducibility for abelian varieties over function fields of characteristic zero

Related

• The Hilbert property for arithmetic schemes
• Fibration theorems for varieties with the weak Hilbert property
• Hilbert irreducibility for integral points on punctured linear algebraic groups
• Rational points and ramified covers of products of two elliptic curves
• On the Hilbert Property and the Fundamental Group of Algebraic Varieties