Notes on Rapoport--Zink spaces of Hodge type with parahoric level structure

In this article, we treat two questions on Rapoport–Zink spaces of Hodge type constructed by Hamacher and Kim. One of which is their singularities, and the other is $p$-adic uniformization of Shimura varieties. More precisely, we prove that the singularity of a Rapoport–Zink space is controlled by its asssociated local model, and the basic locus of a Kisin–Pappas integral model of a Shimura variety is uniformized by the corresponding Rapoport–Zink space. These results extend the known facts by Rapoport and Zink in the case of PEL type.