Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture.
Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture.
复制标题
K3 曲面在函数场和 Hecke 轨道猜想上的皮卡德排序。
DOI：
10.1007/s00222-022-01097-x
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发表时间：
2022
影响因子：
3.1
通讯作者：
Tang, Yunqing
中科院分区：
文献类型：
--
作者：
Maulik, David;Shankar, Ananth N.;Tang, Yunqing
设是特征曲线Cin上的一类非等平凡的一般K3曲面族。我们证明了几何Picard秩在C的无穷多个闭点处跳跃。更一般地说，假设我们给出了一个正交类型的Shimura变种的正则模型，它与一个自对偶atp的签名格（B，2）相关联。我们证明了任何一般平凡的真曲线Cin与sf的特殊因子在无穷多点相交。作为应用，我们证明了Chai-Oort的平凡Hecke轨道猜想，即证明了中的平凡点具有Zurkiki稠密Hecke轨道.我们还推出了普通的Hecke轨道猜想的某些家庭的酉志村品种。
Letbe a non-isotrivial and generically ordinary family of K3 surfaces over a proper curveCin characteristic. We prove that the geometric Picard rank jumps at infinitely many closed points ofC. More generally, suppose that we are given the canonical model of a Shimura varietyof orthogonal type, associated to a lattice of signature (b, 2) that is self-dual atp. We prove that any generically ordinary proper curveCinintersectsspecial divisorsofat 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 inhave Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.