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Integral points of bounded height on quintic del Pezzo surfaces over number fields

Christian Bernert, Ulrich Derenthal

TL;DR

The paper proves an asymptotic formula for the number of integral points of bounded log-anticanonical height on split smooth quintic del Pezzo surfaces over number fields, with respect to a boundary line. It develops a universal torsor parameterization, imposes precise region restrictions, and employs Möbius inversion and o-minimal lattice-counting to extract the main term, while computing archimedean and p-adic densities to identify the leading constant. The resulting leading constant factors into a product involving the residue of the Dedekind zeta function, local densities, and the global height constant α(X), matching the predictions of Chambert-Loir–Tschinkel and related works. The method generalizes the universal torsor approach to integral points over number fields and provides a concrete pathway to extend similar results to other Fano varieties. Overall, the work validates the conjectural leading-term structure in this non-toric, non-quadratic del Pezzo setting and highlights the role of global densities in arithmetic counting on rational surfaces.

Abstract

We prove an asymptotic formula for the number of integral points of bounded log-anticanonical height on split smooth quintic del Pezzo surfaces over number fields, with respect to one of the lines as the boundary divisor.

Integral points of bounded height on quintic del Pezzo surfaces over number fields

TL;DR

The paper proves an asymptotic formula for the number of integral points of bounded log-anticanonical height on split smooth quintic del Pezzo surfaces over number fields, with respect to a boundary line. It develops a universal torsor parameterization, imposes precise region restrictions, and employs Möbius inversion and o-minimal lattice-counting to extract the main term, while computing archimedean and p-adic densities to identify the leading constant. The resulting leading constant factors into a product involving the residue of the Dedekind zeta function, local densities, and the global height constant α(X), matching the predictions of Chambert-Loir–Tschinkel and related works. The method generalizes the universal torsor approach to integral points over number fields and provides a concrete pathway to extend similar results to other Fano varieties. Overall, the work validates the conjectural leading-term structure in this non-toric, non-quadratic del Pezzo setting and highlights the role of global densities in arithmetic counting on rational surfaces.

Abstract

We prove an asymptotic formula for the number of integral points of bounded log-anticanonical height on split smooth quintic del Pezzo surfaces over number fields, with respect to one of the lines as the boundary divisor.
Paper Structure (25 sections, 29 theorems, 154 equations)

This paper contains 25 sections, 29 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Heights
  3. The main result
  4. Overview of the proof
  5. Notation and conventions
  6. Passage to universal torsors
  7. Parameterization
  8. Cox rings, universal torsors, models
  9. Heights
  10. Dependent coordinates
  11. Construction of a fundamental domain
  12. Restrictions of the counting problem
  13. Counting units
  14. Symmetry
  15. Restricting the set of $\mathbf{a}"$
  16. ...and 10 more sections

Key Result

Theorem 1

We have with where $\rho_K$ as in eq:def_rho_K is the residue of the Dedekind zeta function $\zeta_K$ at $s=1$, and

Theorems & Definitions (62)

  • Remark
  • Theorem
  • Remark
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark
  • ...and 52 more