Rational points in the Noether-Lefschetz locus of moduli spaces of K3 surfaces

TL;DR

This paper investigates the structure of rational points in the Noether-Lefschetz locus of moduli spaces of lattice-polarized K3 surfaces, linking geometric maps to number-theoretic conjectures and providing new insights into rational point distribution.

Contribution

It introduces a method using lattice-induced maps to identify rational points in the Noether-Lefschetz locus and connects these findings to the Bombieri-Lang and Shafarevich conjectures.

Findings

Maps between moduli spaces help detect rational points in the Noether-Lefschetz locus.

Under the Bombieri-Lang conjecture, rational points are non-dense in the locus.

Supports conjectural links between geometric properties and arithmetic distribution.

Abstract

In this paper, we study maps between moduli spaces of lattice-polarized K3 surfaces induced by sublattices of prime index. We show that these maps can be used to determine if a rational point of the moduli space belongs to the Noether-Lefschetz locus. As an application, we prove that the Bombieri-Lang conjecture implies non-density statements for the rational points in the Noether-Lefschetz locus, as predicted by a conjecture of Shafarevich.