K3 surfaces over number fields with geometric Picard number one

A long-standing question in the theory of rational points of algebraic surfaces is whether a K3 surface X over a number field K acquires a Zariski-dense set of L-rational points over some finite extension L/K. In this case, we say X has potential density of rational points. In case XC has Picard rank greater than 1, Bogomolov and Tschinkel [2] have shown in many cases that X has potential density of rational points, using the existence of elliptic fibrations on X or large automorphism groups of X. By contrast, we do not know a single example of a K3 surface X/K with geometric Picard number 1 which can be shown to have potential density of rational points; nor is there an example which we can show not to have potential density of rational points. In fact, the situation is even worse; the moduli space of polarized K3 surfaces of a given degree contains a countable union of subvarieties, each parametrizing a family of K3 surfaces with geometric Picard number greater than 1. Since Q̄ is countable, it is not a priori obvious that these subvarieties don’t cover the Q̄-points of the moduli space. In other words, it is a non-trivial fact that there exists a K3 surface over any number field with geometric Picard number 1! In this note, we correct this slightly embarrassing situation by proving the following theorem: