Partial canonic sub-groups

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 >= 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 show that there exists a strict neighbourhood of the tube on which this subgroup extends in a finite flat way. On the ordinary stratum and its neighbourhood, we recover the usuel canonical subgroup studied by Abbes and Mokrane, and Andreatta and Gasbarri.