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

Let X→ C be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve C in characteristic p≥ 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 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 SF¯p intersects special divisors of SF¯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 SF¯p have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.