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 $n_D\in\mathbb N$ be the smallest nonnegative integer such that D is determined by D[pⁿD] within the class of p-divisible groups over k of the same codimension c and dimension d as D. We study nD, lifts of D[p^m] to truncated Barsotti–Tate groups of level m+1 over k, and the numbers $\gamma_D(i):=\dim({\rm Aut}(D[p^i]))$. We show that nD≤cd, $(\gamma_D(i+1)-\gamma_D(i))_{i\in\mathbb N}$ is a decreasing sequence in $\mathbb N$, for cd>0 we have γD(1)<γD(2)<⋯<γD(nD), and for m∈{1,…,nD−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.