Dieudonné theory via cohomology of classifying stacks

We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the crystalline cohomology of the classifying stack of a p-divisible group is a symmetric algebra (in degree $2$ ) on its Dieudonné module. We also prove mixed-characteristic analogues of some of these results using prismatic cohomology.

Local Shtukas and Divisible Local Anderson Modules

We develop the analog of crystalline Dieudonné theory for$p$-divisible groups in the arithmetic of function fields. In our theory$p$-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian$t$-modules and$t$-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.

Dieudonné theory over semiperfect rings and perfectoid rings

The Dieudonné crystal of a $p$-divisible group over a semiperfect ring $R$ can be endowed with a window structure. If $R$ satisfies a boundedness condition, this construction gives an equivalence of categories. As an application we obtain a classification of $p$-divisible groups and commutative finite locally free $p$-group schemes over perfectoid rings by Breuil–Kisin–Fargues modules if $p\geqslant 3$.

GROUP ACTION ON THE DEFORMATIONS OF A FORMAL GROUP OVER THE RING OF WITT VECTORS

A recent result by the authors gives an explicit construction for a universal deformation of a formal group $\unicode[STIX]{x1D6F7}$ of finite height over a finite field $k$ . This provides in particular a parametrization of the set of deformations of $\unicode[STIX]{x1D6F7}$ over the ring ${\mathcal{O}}$ of Witt vectors over $k$ . Another parametrization of the same set can be obtained through the Dieudonné theory. We find an explicit relation between these parameterizations. As a consequence, we obtain an explicit expression for the action of $\text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ on the set of ${\mathcal{O}}$ -deformations of $\unicode[STIX]{x1D6F7}$ in the coordinate system defined by the universal deformation. This generalizes a formula of Gross and Hopkins and the authors’ result for one-dimensional formal groups.

The Supersingular Locus of the Shimura Variety for GU(1, s)

In this paper we study the supersingular locus of the reduction modulopof the Shimura variety for GU(1,s) in the case of an inert primep. Using Dieudonné theory we define a stratification of the corresponding moduli space ofp-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat–Tits building of a unitary group.In the case of GU(1, 2), we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.