Dimensions of Group Schemes of Automorphisms of Truncated Barsotti-Tate Groups

Let D be a p-divisible group over an algebraically closed field k of characteristic p > 0. Let n(D) is an element of N be the smallest nonnegative integer such that D is determined by D[p(nD)] within the class of p-divisible groups over k of the same codimension c and dimension d as D. We study n(D), lifts of D[p(m)] to truncated Barsotti-Tate groups of level m + 1 over k, and the numbers gamma(D)(i) := dim(Aut(D[p(i)])). We show that n(D) <= cd, (gamma(D)(i + 1) - gamma(D)(i))(i is an element of N) is a decreasing sequence in N, for cd > 0 we have gamma(D)(1) < gamma(D)(2) < ... < gamma(D)(n(D)), and for m is an element of {1, ..., n(D) - 1} there exists an infinite set of truncated Barsotti-Tate groups of level m + 1 which are pairwise nonisomorphic and lift D[p(m)]. Different generalizations to p-divisible groups with a smooth integral group scheme in the crystalline context are also proved.