[Paper Review] (G, \mu)-displays and Rapoport-Zink spaces

Oliver Bueltel, G. Pappas|arXiv (Cornell University)|Feb 1, 2017

Advanced Algebra and Geometry20 references3 citations

TL;DR

This paper introduces $(G,\mu)$-displays, a group-theoretic generalization of Zink's Witt vector displays, to construct Rapoport-Zink formal schemes purely from reductive groups and cocharacters, without relying on $p$-divisible groups. The key contribution is a conjectural and partially proven representability of these formal schemes as integral models of $p$-adic local Shimura varieties, especially in the Hodge-type case.

ABSTRACT

Let (G, \\mu) be a pair of a reductive group G over the p-adic integers and a minuscule cocharacter {\\mu} of G defined over an unramified extension. We introduce and study "(G, \\mu)-displays" which generalize Zink's Witt vector displays. We use these to define certain Rapoport-Zink formal schemes purely group theoretically, i.e. without p-divisible groups.

Motivation & Objective

• To develop a purely group-theoretic construction of Rapoport-Zink formal schemes using reductive groups $G$ and minuscule cocharacters $\mu$, without relying on $p$-divisible groups.
• To generalize Zink's theory of displays to the setting of reductive group schemes over $\mathbb{Z}_p$ via $G$-torsors and loop group structures.
• To define a functor ${\mathrm{RZ}}_{G,\mu,b}$ on $p$-nilpotent algebras that conjecturally represents the integral model of a local Shimura variety at hyperspecial level.
• To prove the representability of this functor as a formal scheme when the local Shimura datum is of Hodge type, i.e., embeddable into $\mathrm{GL}_n$.

Proposed method

• Introduce the divided Frobenius map $\Phi_{G,\mu}: H^\mu \to L^+G_{W(k_0)}$, defined via conjugation by $\mu^\sigma(p)$ and the Frobenius on Witt vectors.
• Define a $(G,\mu)$-display as a triple $(P, Q, u)$, where $P$ is an $L^+G$-torsor, $Q$ is an $H^\mu$-torsor, and $u: Q \to P$ is a morphism compatible with $\Phi_{G,\mu}$.
• Use the quotient stack $[L^+G_{W(k_0)} /_{\Phi_{G,\mu}} H^\mu]$ to parametrize $(G,\mu)$-displays, generalizing the moduli of displays.
• Construct the functor ${\mathrm{RZ}}_{G,\mu,b}$ on $p$-nilpotent algebras via the moduli of $(G,\mu)$-displays with additional structure related to a $\sigma$-conjugacy class $[b]$.
• Apply deformation theory and lifting techniques to study the local structure of the moduli functor.
• Prove representability of ${\mathrm{RZ}}_{G,\mu,b}$ as a formal scheme when the local Shimura datum is of Hodge type, using a reduction to the $\mathrm{GL}_n$ case.

Experimental results

Research questions

• RQ1 Can Rapoport-Zink formal schemes be defined purely group-theoretically, without reference to $p$-divisible groups?
• RQ2 How can Zink's theory of displays be generalized to reductive group schemes via $G$-torsors and loop group structures?
• RQ3 Is the functor ${\mathrm{RZ}}_{G,\mu,b}$, defined via $(G,\mu)$-displays and a $\sigma$-conjugacy class $[b]$, representable by a formal scheme?
• RQ4 What conditions ensure that the moduli space of $(G,\mu)$-displays with additional structure is a formal scheme?
• RQ5 Does the construction of ${\mathrm{RZ}}_{G,\mu,b}$ yield an integral model of a $p$-adic local Shimura variety in the Hodge-type case?

Key findings

• The paper constructs a group-theoretic definition of Rapoport-Zink formal schemes via $(G,\mu)$-displays, bypassing the need for $p$-divisible groups.
• The divided Frobenius $\Phi_{G,\mu}$ is defined as $\Phi_{G,\mu}(h) = \mu^\sigma(p) \cdot F(h) \cdot \mu^\sigma(p)^{-1}$ in $G(W(R)[1/p])$, providing a key structure for the display theory.
• The moduli functor ${\mathrm{RZ}}_{G,\mu,b}$ is conjectured to be representable by a formal scheme that serves as an integral model of a local Shimura variety at hyperspecial level.
• In the Hodge-type case, the representability of ${\mathrm{RZ}}_{G,\mu,b}$ is proven by reducing to the $\mathrm{GL}_n$ case using embedding techniques.
• The theory establishes a direct link between the geometry of Rapoport-Zink spaces and the representation theory of reductive groups via $G$-torsors over Witt rings.
• The paper proves that idempotent, bounded nilpotent ideals in almost Frobenius separated algebras are trivial, a technical result used in the representability argument.