An algebraicity conjecture of Drinfeld and the moduli of $p$-divisible groups

TL;DR

This paper employs advanced stacky prismatic techniques to construct and analyze moduli stacks of $p$-divisible groups, confirming conjectures of Drinfeld and extending classification results over general $p$-adic bases.

Contribution

It provides a uniform, group-theoretic construction of stacks related to $p$-divisible groups, verifying conjectures and generalizing previous classification results.

Findings

Constructed smooth stacks $ ext{BT}^{G,}_{n}$ using stacky prismatic technology.

Proved isomorphism of these stacks with truncated $p$-divisible groups for specific $G$ and $$.

Established algebraicity of the stack of perfect $F$-gauges with certain Hodge-Tate weights.

Abstract

We use the newly developed stacky prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks $\mathrm{BT}^{G,\mu}_{n}$ attached to a smooth affine group scheme $G$ over $\mathbb{Z}_p$ and $1$-bounded cocharacter $\mu$, verifying a recent conjecture of Drinfeld. This can be viewed as a refinement of results of B\"ultel-Pappas, who gave a related construction using $(G,\mu)$-displays defined via rings of Witt vectors. We show that, when $G = \mathrm{GL}_h$ and $\mu$ is a minuscule cocharacter, these stacks are isomorphic to the stack of truncated $p$-divisible groups of height $h$ and dimension $d$ (the latter depending on $\mu$). This gives a generalization of results of Ansch\"utz-Le Bras, yielding a linear algebraic classification of $p$-divisible groups over very general $p$-adic bases, and verifying another conjecture of Drinfeld. The proofs use deformation techniques from derived algebraic geometry, combined with an animated variant of Lau's theory of higher frames and displays, and -- with a view towards applications to the study of local and global Shimura varieties -- actually prove representability results for a wide range of stacks whose tangent complexes are $1$-bounded in a suitable sense. As an immediate application, we prove algebraicity for the stack of perfect $F$-gauges of Hodge-Tate weights $0,1$ and level $n$.