具有清澈整数模型的适当 Shimura 簇的完美体性
Perfectoidness of Proper Shimura Varieties With Limpid Integral Models
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中文总结 AI 辅助
本文证明任何具有清澈整数模型的适当 Shimura 簇在无限层级上成为完美体空间，提出绕过通用阿贝尔概形的新几何方法，并推广至所有适当 Shimura 簇。
中文摘要 AI 辅助
我们证明了任何在 Madapusi Pera 意义下具有清澈整数模型的适当 Shimura 簇，在 $p$ 处无限层级上成为完美体空间。我们的方法提供了一种新的几何途径，绕过了对通用阿贝尔概形的需求，使其适用于 Hodge 类型的 Shimura 簇之外。受 Scholze 对 Siegel 模簇完美体性证明的启发，我们在普通轨迹的严格邻域内构造了典范子群和反典范塔的类似物。为实现这一目标，我们利用了清澈整数模型上的通用 $G$-孔径的几何。由于每个 Shimura 簇对于足够大的素数都具有清澈整数模型，我们的结果意味着所有适当的 Shimura 簇在足够大的素数下，在无限层级上都是全局完美体的。
英文摘要
We prove that any proper Shimura variety admitting a limpid integral model in the sense of Madapusi Pera becomes a perfectoid space at infinite level at $p$. Our method provides a new geometric approach that circumvents the need for a universal abelian scheme, making it applicable beyond Shimura varieties of Hodge type. Inspired by Scholze's proof of the perfectoidness of the Siegel modular variety, we construct analogues of the canonical subgroup and the anticanonical tower within a strict neighborhood of the ordinary locus. To achieve this we utilize the geometry of the universal $G$-aperture over the limpid integral model. Because every Shimura variety admits a limpid integral model for sufficiently large primes, our result implies that all proper Shimura varieties are globally perfectoid at infinite level for big enough primes.
