La filtration canonique des $\mathcal O$-modules $p$-divisibles

TL;DR

This paper constructs a canonical filtration for truncated p-divisible O-modules with a given signature under certain Hasse invariant conditions, generalizing the mu-ordinary case, and applies it to Shimura varieties.

Contribution

It introduces a new filtration for p-divisible O-modules based on crystalline periods, extending the mu-ordinary filtration, with applications to Shimura varieties.

Findings

Established a filtration under explicit Hasse invariant bounds.

Generalized the mu-ordinary filtration for O-modules.

Applied the filtration to neighborhoods in Shimura varieties.

Abstract

English : In this article we associate to $G$, a truncated $p$-divisible $\mathcal O$-module of given signature, where $\mathcal O$ is a finite unramified extension of $\mathbb{Z}_p$, a filtration of $G$ by sub-$\mathcal O$-modules under the conditions that his Hasse $\mu$-invariant is smaller than an explicite bound. This filtration generalise the one given when $G$ is $\mu$-ordinary. The construction of the filtration relies on a precise study of the cristalline periods of a $p$-divisible $\mathcal O$-module. We then apply this result to families of such groups, in particular to stricts neighbourhoods of the $\mu$-ordinary locus inside some PEL Shimura varieties. Fran\c{c}ais : Dans cet article, \`a $G$ un groupe $p$-divisible tronqu\'e muni d'une action d'une extension finie non ramifi\'ee $\mathcal O$ de $\mathbb{Z}_p$, et de signature donn\'ee, on associe sous une condition explicite sur son $\mu$-invariant de Hasse, une filtration de $G$ par des sous-$\mathcal O$-modules qui \'etend la filtration canonique lorsque $G$ est $\mu$-ordinaire. La construction se fait en \'etudiant les p\'eriodes cristallines des groupes $p$-divisibles avec action de $\mathcal O$. On applique ensuite cela aux familles de tels groupes, en particulier des voisinages stricts du lieu $\mu$-ordinaire dans des vari\'et\'es de Shimura PEL.