Seminar : Moduli of $p$-divisible groups with infinite level

In this seminar we will learn about more general moduli problems, period spaces and period maps. On the way we discuss new results of Scholze and Weinstein in \cite{ScholzeWeinstein:2013}. Indeed, Drinfeld's moduli functor can be generalized into moduli problems that are represented by formal schemes that are called Rapoport-Zink spaces, denoted by $\cal{M}$ plus some extra decoration depending on the specific moduli problem at hand. The spaces $\cal{M}$ admit a tower of coverings associated to level structures. Taking the inverse limit we get a space $\cal{M}_\infty$. We have (in a rough form):

$\cal{M}$ is representable by an adic space that is preperfectoid.

One variant of moduli problems of $p$-divisible groups involves so-called EL-data, that we denote by $D$. There is a notion of duality $D \rightarrow \check{D}$ for such data. Various period maps can be defined from such $\cal{M}$ to subsets of flag varieties that we call period spaces. The standard period map is the Grothendieck-Messing period map, constructed using Grothendieck-Messing Dieudonn\'e theory. At infinite level, there is also a Hodge-Tate period map associated to such EL-data, and we have the following duality result:

There is a natural $G(\mathbb{Q}_p) \times \check{G}(\mathbb{Q}_p)$-equivariant isomorphism $\cal{M}_{D,\infty} \simeq \cal{M}_{\check{D},\infty}$ exchanging the GM and HT period maps.

Literature

[RZ] Rapoport, Zink - Period spaces for $p$-divisible groups, 1996
[SW] Scholze, Weinstein - Moduli of $p$-divisible groups, 2013