The canonical subgroup for families of abelian varieties

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