Partial canonical subgroups

TL;DR

This paper constructs and studies partial canonical subgroups of p-divisible groups over Siegel varieties, extending known canonical subgroups in a finite flat manner to neighborhoods of certain strata.

Contribution

It introduces a new notion of partial canonical subgroups for Siegel varieties and proves their finite flat extension beyond the ordinary locus.

Findings

Partial canonical subgroups exist and extend in a finite flat way.

On the ordinary stratum, the partial subgroup coincides with the classical canonical subgroup.

The construction generalizes known canonical subgroups to non-ordinary strata.

Abstract

The reduction of Siegel varieties modulo a prime number p is stratified by the multiplicative rank of the p-divisible group of the universal abelian variety. For r\geq 0 the maximal multiplicative subgroup of the restriction of the p-torsion group of the universal abelian variety to the r-th stratum lifts canonically to the tube of this stratum and defines a partial canonical subgroup of rank r. We prove that this subgroup extends in a finite flat way on some strict neighborhood of the tube. On the ordinary stratum and on its neighborhood, we recover the usual canonical subgroup considered by Abbes and Mokrane, and Andreatta and Gasbarri. ----- La reduction des varietes de Siegel modulo un nombre premier p est stratifiee par le rang multiplicatif du groupe p-divisible de la variete abelienne universelle. Pour r\geq 0, le sous-groupe multiplicatif maximal de la restriction du groupe de p-torsion de la variete abelienne universelle a la r-ieme strate se releve canoniquement sur le tube de cette strate et definit un sous-groupe canonique partiel de rang r. Nous montrons qu'il existe un voisinage strict du tube sur lequel ce sous-groupe s'etend de maniere finie et plate. Sur la strate ordinaire et au voisinage de celle-ci, on retrouve le sous-groupe canonique usuel etudie par Abbes et Mokrane d'une part, Andreatta et Gasbarri d'autre part.