Group schemes with additional structures and Weyl group cosets

Let Y be an abelian variety of dimension g over an algebraically closed eld k of characteristic p To Y we can associate its pkernel Y p which is a nite commutative kgroup scheme of rank pg In the unpublished manuscript Kraft showed that xing g there are only nitely many such group schemes up to isomorphism As we shall discuss later Kraft also gave a very nice description of all possible types Some years later this result was reobtained independently by Oort Together with Ekedahl he used it to de ne and study a strati cation of the moduli space Agk of principally polarized abelian varieties over k The strata correspond to the pairs Y such that the pkernel is of a xed isomorphism type Their results can be found in and see also related work by van der Geer i