The canonical subgroup for families of abelian varieties

Abstract

Let V be a complete discrete valuation ring with residue field k of characteristic p > 0 and fraction field K of characteristic zero. Let S be a formal scheme over V and let X S be a locally projective formal abelian scheme. In this paper we prove that, under suitable natural conditions on the Hasse-Witt matrix of X V V/pV , the kernel of the Frobenius morphism on X k can be canonically lifted to a finite and flat subgroup scheme of X over an admissible blow-up of S, called the 'canonical subgroup of X'. This is done by a careful study of torsors under group schemes of order p over X. We also present a filtration on H 1 (X, p ) in the spirit of the Hodge-Tate decomposition.