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

Let $\mathcal{X}\to C$ 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 ${\mathcal{S}}_{{\bar{\mathbb{F}}}_p}$ intersects special divisors of ${\mathcal{S}}_{{\bar{\mathbb{F}}}_p}$ 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 ${\mathcal{S}}_{{\bar{\mathbb{F}}}_p}$ have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.