Generalization of Deuring reduction theorem

In this paper we generalize the Deuring theorem on a reduction of an elliptic curve with complex multiplication. More precisely, for an Abelian variety A, arising after reduction of an Abelian variety with complex multiplication by a CM field K over a number field at a place of good reduction. We establish a connection between a decomposition of the first truncated Barsotti-Tate group scheme A[p] and a decomposition of pOK into prime ideals. In particular, we produce these explicit relationships for Abelian varieties of dimensions 1,2 and 3.