Groupes $p$-divisibles avec condition de Pappas-Rapoport et invariants de Hasse

TL;DR

This paper investigates p-divisible groups with additional structure over schemes of characteristic p, defining invariants that characterize their ordinariness and relate their Hodge and Newton polygons, with applications to Shimura varieties.

Contribution

It introduces Hasse invariants for p-divisible groups with Pappas-Rapoport conditions, linking their Newton and Hodge polygons and characterizing μ-ordinary groups.

Findings

Hodge and Newton polygons are ordered with respect to the Pappas-Rapoport datum.

Total Hasse invariant is non-zero iff the group is μ-ordinary.

Construction applies to special fibers of PEL Shimura varieties.

Abstract

We study $p$-divisible groups $G$ endowed with an action of the ring of integers of a finite (possibly ramified) extension of $\mathbb{Q}_p$ over a scheme of characteristic $p$. We suppose moreover that the $p$-divisible group $G$ satisfies the Pappas-Rapoport condition for a certain datum $\mu$ ; this condition consists in a filtration on the sheaf of differentials $\omega_G$ satisfying certain properties. Over a perfect field, we define the Hodge and Newton polygons for such $p$-divisible groups, normalized with the action. We show that the Newton polygon lies above the Hodge polygon, itself lying above a certain polygon depending on the datum $\mu$. We then construct Hasse invariants for such $p$-divisible groups over an arbitrary base scheme of characteristic $p$. We prove that the total Hasse invariant is non-zero if and only if the $p$-divisible group is $\mu$-ordinary, i.e. if its Newton polygon is minimal. Finally, we study the properties of $\mu$-ordinary $p$-divisible groups. The construction of the Hasse invariants can in particular be applied to special fibers of PEL Shimura varieties models as constructed by Pappas and Rapoport.