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A Dieudonné theory for analytic p-divisible groups and applications to Shimura varieties

Lucas Gerth

TL;DR

This work develops a Dieudonné theory for families of analytic $p$-divisible groups over adic spaces defined over $\mathbb{Q}_p$, establishing an equivalence with $Hodge$-$Tate$ triples and extending Fargues’ framework to a relative setting. A central construction assigns to an analytic $p$-divisible group a coherent sheaf on the relative Fargues–Fontaine curve, with dualizability characterized by vector-bundle structure and a natural correspondence to minuscule local shtukas, compatible with prismatic Dieudonné theory. The authors then apply the theory to moduli problems for local Shimura varieties (EL/PEL types) and reinterpret the Hodge–Tate period map in terms of topologically $p$-torsion subgroups of abeloid varieties, providing a new geometric lens on $p$-adic uniformization and period maps. Together, these results connect analytic $p$-divisible groups, $v$-descent, vector bundles on the FF-curve, and prismatic/dieudonné frameworks to give moduli descriptions and period-map reinterpretations with potential for new insights in Shimura varieties and $p$-adic Hodge theory.

Abstract

We study families of analytic $p$-divisible groups over adic spaces $S$ defined over $\mathbb{Q}_p$. We prove an equivalence between such families and Hodge-Tate triples, generalizing a theorem of Fargues. For a perfectoid space $S$, we construct a functor associating to an analytic $p$-divisible group $\mathcal{G} \rightarrow S$ a coherent sheaf $\mathcal{E}(\mathcal{G})$ on the relative Fargues--Fontaine curve $X_S$. Restricting to analytic $p$-divisible groups admitting a Cartier dual, we obtain an equivalence of categories with local shtukas satisfying a minuscule condition, compatible with the prismatic Dieudonné theory of Anschütz--Le Bras. We conclude with applications to moduli spaces: we show that the local Shimura varieties of EL and PEL types of Scholze--Weinstein are moduli spaces of analytic $p$-divisible groups with extra structure, and we give a reinterpretation of the Hodge--Tate period map of Scholze in terms of topologically $p$-torsion subgroups of abelian varieties.

A Dieudonné theory for analytic p-divisible groups and applications to Shimura varieties

TL;DR

This work develops a Dieudonné theory for families of analytic -divisible groups over adic spaces defined over , establishing an equivalence with - triples and extending Fargues’ framework to a relative setting. A central construction assigns to an analytic -divisible group a coherent sheaf on the relative Fargues–Fontaine curve, with dualizability characterized by vector-bundle structure and a natural correspondence to minuscule local shtukas, compatible with prismatic Dieudonné theory. The authors then apply the theory to moduli problems for local Shimura varieties (EL/PEL types) and reinterpret the Hodge–Tate period map in terms of topologically -torsion subgroups of abeloid varieties, providing a new geometric lens on -adic uniformization and period maps. Together, these results connect analytic -divisible groups, -descent, vector bundles on the FF-curve, and prismatic/dieudonné frameworks to give moduli descriptions and period-map reinterpretations with potential for new insights in Shimura varieties and -adic Hodge theory.

Abstract

We study families of analytic -divisible groups over adic spaces defined over . We prove an equivalence between such families and Hodge-Tate triples, generalizing a theorem of Fargues. For a perfectoid space , we construct a functor associating to an analytic -divisible group a coherent sheaf on the relative Fargues--Fontaine curve . Restricting to analytic -divisible groups admitting a Cartier dual, we obtain an equivalence of categories with local shtukas satisfying a minuscule condition, compatible with the prismatic Dieudonné theory of Anschütz--Le Bras. We conclude with applications to moduli spaces: we show that the local Shimura varieties of EL and PEL types of Scholze--Weinstein are moduli spaces of analytic -divisible groups with extra structure, and we give a reinterpretation of the Hodge--Tate period map of Scholze in terms of topologically -torsion subgroups of abelian varieties.
Paper Structure (34 sections, 84 theorems, 336 equations)

This paper contains 34 sections, 84 theorems, 336 equations.

Table of Contents

  1. Introduction
  2. Analytic $p$-divisible groups
  3. Dieudonné theory
  4. Applications to local and global Shimura varieties
  5. Limitations and Generalizations
  6. Plan of the paper
  7. Preliminaries
  8. $p$-adic local systems
  9. Proétale vector bundles
  10. Hodge--Tate and de Rham local systems
  11. Smooth commutative adic groups
  12. Analytic $p$-divisible groups
  13. Definitions
  14. Analytic Weil pairings
  15. Analytic $p$-divisible groups and Hodge--Tate triples
  16. ...and 19 more sections

Key Result

Theorem 1.1

(Theorem thm: extending Fargues' equivalence of categories) Let $S$ be a good adic space over $\mathbb{Q}_p$. Then the following categories are canonically equivalent If $\mathcal{G}$ and $(\mathbb{L},E,f)$ correspond to each-other, we have $\mathbb{L} = T_p\mathcal{G}$, $E=\mathop{\mathrm{Lie}}\nolimits(\mathcal{G})$ and the map $f=f_{\mathcal{G}}$ fits in the following commutative diagram of gr

Theorems & Definitions (226)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 216 more