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

• 成果类型: Article
• 署名作者: Maulik, Davesh; Shankar, Ananth N.; Tang, Yunqing
• 署名单位: Massachusetts Institute of Technology (MIT); University of Wisconsin System; University of Wisconsin Madison; Princeton University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01097-x
• 发表日期: 2022
• 页码: 1075-1143
• 关键词: Shimura varieties; integral models; reductions; cycles

摘要

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 S-(F) over barp intersects special divisors of S-(F) over barp 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 S-(F) over barp have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.