Perfectoidness of Proper Shimura Varieties With Limpid Integral Models

Source abstract

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.