Embeddings of Certain Exceptional Shimura Varieties into Siegel Modular Varieties

TL;DR

This paper introduces a new class of local Shimura varieties related to exceptional groups, constructs a functor to p-divisible groups, and embeds certain exceptional Shimura varieties into Siegel modular varieties, proving their perfectoid nature at infinite level.

Contribution

It defines a novel class of local Shimura varieties for exceptional groups, constructs a functor to p-divisible groups, and embeds certain exceptional Shimura varieties into Siegel modular varieties.

Findings

Proved local Shimura varieties in this class are representable and perfectoid at infinite level.

Constructed a functor from (G, μ)-displays to p-divisible groups.

Embedded certain exceptional Shimura varieties into Siegel modular varieties and proved their perfectoid property.

Abstract

We define a class of local Shimura varieties that contains some local Shimura varieties for exceptional groups, and for this class, we construct a functor from $\left(G, \mu\right)$-displays to $p$-divisible groups. As an application, we prove that for this class, the local Shimura variety is representable and perfectoid at the infinite level. Considering the global counterpart of this class, we embed certain exceptional Shimura varieties into Siegel modular varieties. In particular, we prove that they are perfectoid at the infinite level.