论文标题
功能字段和Hecke Orbit猜想的K3表面的PICARD等级
Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
论文作者
论文摘要
令$ \ mathscr {x} \ rightarrow c $是在适当的曲线$ c $ c $ in特征$ p \ geq 5 $上的非异常和普通的k3表面家族。我们证明，几何PICARD等级以无限的$ C $封闭点跳跃。更普遍地，假设我们获得了正交类型的Shimura品种$ \ Mathcal {s} $的规范模型，该模型与签名$（b，2）$的晶格相关联，该$（b，2）$是$ p $。我们证明，任何一般普通的正确曲线$ c $ in $ \ mathcal {s} _ {\ overline {\ mathbb {f}} _ p} $相交$ \ mathcal {s} _ {s} _ {\ Mathbb {\ Mathbb {\ Mathbb {\ Mathbb {fiend fime copters $ \ Mathcal {s}作为一种应用，我们证明了在这种情况下的柴的普通hecke轨道猜想；也就是说，我们表明$ \ MATHCAL {s} _ {\ overline {\ Mathbb {f}} _ p} $中的普通点具有zariski浓密的hecke orbits。我们还为某些单一shimura品种的家族推论了普通的Hecke轨道猜想。
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.
