A Dieudonn\'e theory for analytic p-divisible groups and applications to Shimura varieties

TL;DR

This paper develops a new Dieudonné theory for analytic p-divisible groups over adic spaces, establishing equivalences with Hodge-Tate structures and local shtukas, and applies these results to Shimura varieties and period maps.

Contribution

It generalizes Fargues' theorem to families over adic spaces, constructs a functor to the Fargues--Fontaine curve, and links p-divisible groups with local shtukas and prismatic Dieudonné theory.

Findings

Equivalence between families of analytic p-divisible groups and Hodge-Tate triples.

Construction of a functor associating p-divisible groups to coherent sheaves on the Fargues--Fontaine curve.

Identification of certain Shimura varieties as moduli spaces of analytic p-divisible groups.

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 Dieudonn\'e theory of Ansch\"utz--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.