Group Schemes with Additional Structures and Weyl Group Cosets

Abstract

Let Y be an abelian variety of dimensiongover an algebraically closed fieldkof characteristicp >0. To Y we can associate its p-kernel Y[p], which is a finite commutative k-group scheme of rank p^{2 g}g. In the unpublished manuscript [8], Kraft showed that, fixinggthere are only finitely many such group schemes, up to isomorphism. (As we shall discuss later, Kraft also gave a very nice description of all possible types.) About 20 years later, this result was re-obtained, independently, by Oort. Together with Ekedahl he used it to define and study a stratification of the moduli space akof principally polarized abelian varieties overk.The strata correspond to the pairs (Y, such that the p-kernel is of a fixed isomorphism type. Their results can be found in [11] and [12]; see also related work by van der Geer in [, [17].