Let D be a p-divisible group over an algebraically closed field k of
characteristic p>0. Let nDβ be the smallest non-negative integer such that
D is determined by D[pnDβ] within the class of p-divisible groups over
k of the same codimension c and dimension d as D. We study nDβ, lifts
of D[pm] to truncated Barsotti--Tate groups of level m+1 over k, and the
numbers Ξ³Dβ(i):=dim(Aut(D[pi])). We show that nDββ€cd,
(Ξ³Dβ(i+1)βΞ³Dβ(i))iβNβ is a decreasing sequence in N, for cd>0 we have Ξ³Dβ(1)<Ξ³Dβ(2)<...<Ξ³Dβ(nDβ), and for
mβ{1,...,nDββ1} there exists an infinite set of truncated Barsotti--Tate
groups of level m+1 which are pairwise non-isomorphic and lift D[pm].
Different generalizations to p-divisible groups with a smooth integral group
scheme in the crystalline context are also proved.Comment: 52 pages. Final version as close to the galley proofs as possible. To
appear in IMR