tailieunhanh - Đề tài " Minimal p-divisible groups "

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. | Annals of Mathematics Minimal p-divisible groups By Frans Oort Annals of Mathematics 161 2005 1021 1036 Minimal p-divisible groups By Frans Oort 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 see for a definition of minimal see . This is necessary and sufficient in the sense that for any G that is not minimal there exist infinitely many mutually nonisomorphic p-divisible groups with p-kernel isomorphic to G see . Remark motivation . You might wonder why this is interesting. EO. In 7 we defined a natural stratification of the moduli space of polarized abelian varieties in positive characteristic moduli points are in the same stratum if and only if the corresponding p-kernels are geometrically isomorphic. Such strata are called EO-strata. Fol. In 8 we define in the same moduli spaces a foliation Moduli points are in the same leaf if and only if the corresponding p-divisible groups are geometrically isomorphic in this way we obtain a foliation of every open Newton polygon stratum. Fol c EO. The observation X Y X p Y p shows that any leaf in the second sense is contained in precisely one stratum in the first sense the main result of this paper X is minimal if and only if X p is minimal 1022 FRANS OORT shows that a stratum in the first sense and a leaf .