I have a question about the book on p-adic geometry by Scholze and Weinstein.
There are two ‘local theories of Shimura varieties’ written in it.
The one is a local model of a Shimura variety. This is based on the work of Rapoport and others. There is a conjecture by the authors that there is a flat projective scheme over OE (the ring of integers of a p-adic field) with a map to Grassmanian. They made it explicit for PE and PEL data cases using the ‘Drinfeld example’.(Reference is chapter 21 of the book)
The other one is a local Shimura variety. Local Shimura variety is an analogue of global number field Shimura variety conjectured by Rapoport and Viehmann. The authors define it to be a tower of smooth rigid spaces over the product of p-adic fields, associated with local Shimura datum. It is related to a local shtuka by the diamond functor.(Reference is chapter 25 of the book)
Then my questions:
⓪a moduli problem and local Shimura variety
As most (i.e. abelian type) global Shimura variety is a moduli space of abelian motives, are most local Shimura varieties related to moduli problems? For example, when a reductive group is a linear algebraic group, it is the moduli of deformation of p-divisible groups and PE and PEL cases are actually related as shown in the book.
①How they are related?
For example, beyond PE and PEL cases, do we consider the moduli problem determined by the local model that is related to the integral model of a local Shimura? Or do we consider the moduli problem with an isomorphism to a parahoric element of a local Shimura tower?
②How they are similar and different?
For example, it seems both spaces have the Hodge-Tate period map toward a Grassmannian. Does cohomology of local shimura variety realize representations of a p-adic reductive group?
