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Non-thin rational points for elliptic K3 surfaces

Damián Gvirtz-Chen, Giacomo Mezzedimi

TL;DR

The paper proves that elliptic K3 surfaces over a number field with two distinct elliptic fibrations satisfy the potential Hilbert property: after a finite base change, their rational points form a non-thin set. The authors leverage a descent framework via K3 covers and a universal-cover finiteness result for open K3 surfaces to transfer the property from a carefully constructed Z with two fibrations and a small over-exceptional lattice to X, with a uniformly bounded extension degree. A key component is a complete lattice-theoretic classification of K3 surfaces with a unique elliptic fibration, which isolates 93 explicit cases that remain to be handled by future methods. Together, these results advance the broader conjecture that all K3 surfaces over number fields have the potential Hilbert property and provide a near-complete program contingent on resolving the remaining 93 classes.

Abstract

We prove that elliptic K3 surfaces over a number field which admit a second elliptic fibration satisfy the potential Hilbert property. Equivalently, the set of their rational points is not thin after a finite extension of the base field. Furthermore, we classify those families of elliptic K3 surfaces over an algebraically closed field which do not admit a second elliptic fibration.

Non-thin rational points for elliptic K3 surfaces

TL;DR

The paper proves that elliptic K3 surfaces over a number field with two distinct elliptic fibrations satisfy the potential Hilbert property: after a finite base change, their rational points form a non-thin set. The authors leverage a descent framework via K3 covers and a universal-cover finiteness result for open K3 surfaces to transfer the property from a carefully constructed Z with two fibrations and a small over-exceptional lattice to X, with a uniformly bounded extension degree. A key component is a complete lattice-theoretic classification of K3 surfaces with a unique elliptic fibration, which isolates 93 explicit cases that remain to be handled by future methods. Together, these results advance the broader conjecture that all K3 surfaces over number fields have the potential Hilbert property and provide a near-complete program contingent on resolving the remaining 93 classes.

Abstract

We prove that elliptic K3 surfaces over a number field which admit a second elliptic fibration satisfy the potential Hilbert property. Equivalently, the set of their rational points is not thin after a finite extension of the base field. Furthermore, we classify those families of elliptic K3 surfaces over an algebraically closed field which do not admit a second elliptic fibration.
Paper Structure (4 sections, 9 theorems, 15 equations)

This paper contains 4 sections, 9 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Elliptic K3 surfaces
  3. Proof of \ref{['thm:main']}
  4. K3 surfaces with a unique elliptic fibration

Key Result

Theorem 1.1

Let $X$ be a K3 surface over a number field $k$ admitting two distinct elliptic fibrations. Then, after a finite field extension $k'/k$, the set of rational points $X(k')$ is not thin.

Theorems & Definitions (21)

  • Theorem 1.1
  • Proposition 2.1: Shioda-Tate formula
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • proof : Proof of the main theorem
  • ...and 11 more