Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture

Abstract

Let \({\mathscr {X}} \rightarrow C\) be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve C in characteristic \(p \ge 5\). We prove that the geometric Picard rank jumps at infinitely many closed points of C. More generally, suppose that we are given the canonical model of a Shimura variety \({\mathcal {S}}\) of orthogonal type, associated to a lattice of signature (b, 2) that is self-dual at p. We prove that any generically ordinary proper curve C in \({\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}\) intersects special divisors of \({\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}\) at infinitely many points. As an application, we prove the ordinary Hecke orbit conjecture of Chai–Oort in this setting; that is, we show that ordinary points in \({\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}\) have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.