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

Zachary Gardner, Keerthi Madapusi

TL;DR

The work advances Drinfeld’s algebraicity conjecture by building a uniform stacky, prismatic framework for $p$-divisible groups with $G$-structure, via smooth stacks $ ext{BT}^{G,μ}_{n}$ attached to a smooth affine group scheme $G$ and a $1$-bounded cocharacter $μ$. It develops the animated higher-frame machinery and a Grothendieck–Messing–type deformation theory to prove representability and smoothness, and shows the GL$_h$ case recovers truncated $p$-divisible groups, generalizing prior linear-algebra classifications. The construction interfaces with syntomic and prismatic cohomology, Brieul–Kisin-type frames, and $F$-gauges, enabling applications to Shimura varieties and integral canonical models through formal étale maps to the $G^c$-modified moduli of $p$-divisible groups. These results yield a robust, functorial, group-theoretic encoding of $p$-divisible group data across very general $p$-adic bases, with potential to realize spaces of isogenies and Hecke correspondences in a framework not restricted to classical representations. The techniques unify and extend prior approaches (Bültel–Pappas, Anschütz–Le Bras) and provide algebraicity results for stacks of perfect $F$-gauges, highlighting the deep role of syntomic/prismatic bounds in moduli theory and Shimura-variety applications.

Abstract

We use the newly developed stacky prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks $\mathrm{BT}^{G,μ}_{n}$ attached to a smooth affine group scheme $G$ over $\mathbb{Z}_p$ and $1$-bounded cocharacter $μ$, verifying a recent conjecture of Drinfeld. This can be viewed as a refinement of results of Bültel-Pappas, who gave a related construction using $(G,μ)$-displays defined via rings of Witt vectors. We show that, when $G = \mathrm{GL}_h$ and $μ$ 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 $μ$). This gives a generalization of results of Anschütz-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 $n$.

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

TL;DR

The work advances Drinfeld’s algebraicity conjecture by building a uniform stacky, prismatic framework for -divisible groups with -structure, via smooth stacks attached to a smooth affine group scheme and a -bounded cocharacter . It develops the animated higher-frame machinery and a Grothendieck–Messing–type deformation theory to prove representability and smoothness, and shows the GL case recovers truncated -divisible groups, generalizing prior linear-algebra classifications. The construction interfaces with syntomic and prismatic cohomology, Brieul–Kisin-type frames, and -gauges, enabling applications to Shimura varieties and integral canonical models through formal étale maps to the -modified moduli of -divisible groups. These results yield a robust, functorial, group-theoretic encoding of -divisible group data across very general -adic bases, with potential to realize spaces of isogenies and Hecke correspondences in a framework not restricted to classical representations. The techniques unify and extend prior approaches (Bültel–Pappas, Anschütz–Le Bras) and provide algebraicity results for stacks of perfect -gauges, highlighting the deep role of syntomic/prismatic bounds in moduli theory and Shimura-variety applications.

Abstract

We use the newly developed stacky prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks attached to a smooth affine group scheme over and -bounded cocharacter , verifying a recent conjecture of Drinfeld. This can be viewed as a refinement of results of Bültel-Pappas, who gave a related construction using -displays defined via rings of Witt vectors. We show that, when and is a minuscule cocharacter, these stacks are isomorphic to the stack of truncated -divisible groups of height and dimension (the latter depending on ). This gives a generalization of results of Anschütz-Le Bras, yielding a linear algebraic classification of -divisible groups over very general -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 -bounded in a suitable sense. As an immediate application, we prove algebraicity for the stack of perfect -gauges of Hodge-Tate weights and level .

Paper Structure

This paper contains 92 sections, 120 theorems, 618 equations.

Table of Contents

  1. Introduction
  2. Method of proof
  3. Truncated $(G,\mu)$-apertures
  4. Higher animated frames
  5. Further remarks on the proofs
  6. Application to Shimura varieties
  7. A note on the terminology
  8. Structure of the paper
  9. Notation and other conventions
  10. Stacks and other preliminaries
  11. Square-zero extensions and differential conditions
  12. Derived (pre)stacks
  13. Derived vector stacks
  14. $p$-adic formal stacks
  15. Weil restrictions
  16. ...and 77 more sections

Key Result

Theorem A

Let $\mathcal{BT}_n(X)$ be the category of $n$-truncated Barsotti-Tate groups over $X$MR0801922, and let $\mathrm{Vect}_{\{0,1\}}(X^{\mathrm{syn}})$ be the category of vector bundles on $X^{\mathrm{syn}}\otimes\mathbb{Z}/p^n\mathbb{Z}$ with Hodge-Tate weights in $\{0,1\}$. Then there is a canonical compatible with Cartier duality.

Theorems & Definitions (449)

  • Theorem A
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem B
  • Theorem C
  • Remark 6
  • Remark 7
  • ...and 439 more