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A p-divisible group X can be seen as a tower of building blocks, each of which isomorphic to the same finite group scheme Xp. Clearly, if X1 and X2 are isomorphic X1p ∼= X2p; however, conversely X1p ∼= X2p does in general not imply that X1 X2 are isomorphic. Can we give, over an algebraically closed field in characteristic p, a on the p-kernels which ensures this converse? Here are two known examples of such condition: consider the case that X is ordinary, or the case that X is superspecial (X is p-divisible group of a product of supersingular elliptic curves) ; in these cases the p-kernel determines X.
Frans J. Oort (Tue,) studied this question.