$G$-displays of Hodge type and formal $p$-divisible groups

TL;DR

This paper constructs a functor linking nilpotent $(G,)$-displays to formal $p$-divisible groups with crystalline tensors in the Hodge-type setting, establishing an equivalence under certain conditions and comparing Rapoport-Zink functors.

Contribution

It introduces a new functor that connects $(G,)$-displays to formal $p$-divisible groups, extending Dieudonne9 theory to a broader context and comparing key moduli functors.

Findings

Established an equivalence of categories under specific conditions.

Constructed a $G$-crystal extending Dieudonne9 theory.

Provided an explicit comparison of Rapoport-Zink functors.

Abstract

Let $G$ be a reductive group scheme over the $p$-adic integers, and let $\mu$ be a minuscule cocharacter for $G$. In the Hodge-type case, we construct a functor from nilpotent $(G,\mu)$-displays over $p$-nilpotent rings $R$ to formal $p$-divisible groups over $R$ equipped with crystalline Tate tensors. When $R/pR$ has a $p$-basis \'etale locally, we show that this defines an equivalence between the two categories. The definition of the functor relies on the construction of a $G$-crystal associated with any adjoint nilpotent $(G,\mu)$-display, which extends the construction of the Dieudonn\'e crystal associated with a nilpotent Zink display. As an application, we obtain an explicit comparison between the Rapoport-Zink functors of Hodge type defined by Kim and by B\"ultel and Pappas.