In this paper we prove that certain points in the Hecke orbit of a CM point in the Siegel modular variety do not lie in the open Torelli locus under suitable conditions on the field of definition and the CM factors that arise in the corresponding CM abelian variety. The proof is a combination of properties of stable Faltings height and known cases of the Sato–Tate equidistribution, and is motivated by an analogue over function fields proved by Kukulies. We also discuss the relation of our result with questions of Ekedahl–Serre type and refine a previous result showing that certain Shimura subvarieties of unitary type in the Siegel modular variety only meet the open Torelli locus in dimension zero.

中文翻译：

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在远离 Torelli 轨迹的 CM 点上

在本文中，我们证明了 Siegel 模变体中 CM 点的 Hecke 轨道中的某些点在定义领域和相应 CM 阿贝尔变体中出现的 CM 因子的适当条件下不位于开放的 Torelli 轨迹中。证明是稳定法尔廷斯高度的性质和佐藤-泰特等分布的已知情况的组合，并且受到 Kukulies 证明的函数场上的模拟的启发。我们还讨论了我们的结果与 Ekedahl-Serre 类型问题的关系，并改进了先前的结果，该结果表明 Siegel 模变体中某些单一类型的 Shimura 子变体仅满足零维的开放 Torelli 轨迹。