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

TL;DR

This paper proves that for certain families of K3 surfaces and Shimura varieties over finite fields, the Picard rank and special divisors exhibit infinite jumping and intersection behavior, confirming the Zariski-density of Hecke orbits.

Contribution

It establishes the infinite jumping of Picard ranks for K3 surfaces and the intersection of curves with special divisors in orthogonal Shimura varieties, confirming the Hecke orbit conjecture in these cases.

Findings

Picard rank jumps infinitely often in families of K3 surfaces.

Proper curves intersect infinitely many special divisors in Shimura varieties.

Hecke orbits of ordinary points are Zariski dense in the studied Shimura varieties.

Abstract

Let $\mathscr{X} \rightarrow C$ be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve $C$ in characteristic $p \geq 5$. 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.