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Products of varieties with many integral points

Cedric Luger

TL;DR

This work generalizes the weak Hilbert property to arithmetic schemes via near-integral points and proves that WHP is stable under products, birational maps, and finite étale base changes. It develops arithmetic refinements and a Chevalley–Weil-type lifting for near-integral points to handle non-proper cases, and uses (PB)-covers and HIT to transfer WHP to connected algebraic groups. The main achievement is a positive verification of Corvaja–Zannier's arithmetic conjecture for all connected algebraic groups over finitely generated fields of characteristic zero. Together, these results deepen the understanding of thin sets and rational/near-integral point distribution on products and group schemes, with implications for related conjectures in arithmetic geometry.

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.

Products of varieties with many integral points

TL;DR

This work generalizes the weak Hilbert property to arithmetic schemes via near-integral points and proves that WHP is stable under products, birational maps, and finite étale base changes. It develops arithmetic refinements and a Chevalley–Weil-type lifting for near-integral points to handle non-proper cases, and uses (PB)-covers and HIT to transfer WHP to connected algebraic groups. The main achievement is a positive verification of Corvaja–Zannier's arithmetic conjecture for all connected algebraic groups over finitely generated fields of characteristic zero. Together, these results deepen the understanding of thin sets and rational/near-integral point distribution on products and group schemes, with implications for related conjectures in arithmetic geometry.

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.
Paper Structure (12 sections, 26 theorems, 18 equations)

This paper contains 12 sections, 26 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Vertically ramified covers
  3. Arithmetic refinements
  4. The weak Hilbert property
  5. Product theorem
  6. Birational invariance
  7. Smooth proper images
  8. Finite étale covers and extending the base ring
  9. Conjecture \ref{['Conj:CZ_arithmetic']} for algebraic groups
  10. The property (HIT) for many covers
  11. From (HIT) to the weak Hilbert property
  12. Proof of Theorem \ref{['Thm:AlgGroupsSatisfyConjCZ']}

Key Result

Theorem 1.4

Let $S$ be a regular integral arithmetic scheme and let $\mathcal{X}$ and $\mathcal{Y}$ be quasi-projective arithmetic schemes over $S$. If both $\mathcal{X}$ and $\mathcal{Y}$ satisfy the weak Hilbert property over $S$, then $\mathcal{X} \times_S \mathcal{Y}$ also satisfies the weak Hilbert propert

Theorems & Definitions (60)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Theorem 3.2
  • ...and 50 more