Overconvergent Hilbert modular forms via perfectoid modular varieties

We give a new construction of p-adic overconvergent Hilbert modular forms by using Scholze’s perfectoid Shimura varieties at infinite level and the Hodge–Tate period map. The definition is analytic, closely resembling that complex as holomorphic functions satisfying transformation property under congruence subgroups. As special case, we first revisit case elliptic forms, extending recent work Chojecki, Hansen Johansson. then construct sheaves geometric well subsheaves integral vary our definitions in families. show resulting spaces are isomorphic Hecke modules to earlier constructions Andreatta, Iovita Pilloni. Finally, direct arithmetic compare this via descent from case.