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 ∈ N be the smallest nonnegative integer such that D is determined by D[pⁿ_D] 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 γ_D(i):= dim(Aut(D[pⁱ])). We show that n_D ≤ cd, (γ_D(i+1) - γ_D(i))_i∈N is a decreasing sequence in, N, for cd > 0 we have γ_D(1) <γ_D(2)<⋯<γ_D(n_D), and for m ∈ {1,μ,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.