Minimal p-divisible groups

Introduction. A p-divisible group X can be seen as a tower of building blocks, each of which is isomorphic to the same finite group scheme X[p]. Clearly, if X1 and X2 are isomorphic then X1[p] ∼= X2[p]; however, conversely X1[p] ∼= X2[p] does in general not imply that X1 and X2 are isomorphic. Can we give, over an algebraically closed field in characteristic p, a condition on the p-kernels which ensures this converse? Here are two known examples of such a condition: consider the case that X is ordinary, or the case that X is superspecial (X is the p-divisible group of a product of supersingular elliptic curves); in these cases the p-kernel uniquely determines X. These are special cases of a surprisingly complete and simple answer: if G is “minimal”, then X1[p] ∼= G ∼= X2[p] implies X1 ∼= X2,