Intersections of the Ekedahl-Oort and Newton Polygon Stratifications of $\mathcal{A}_{5}$

Abstract: The moduli space $\mathcal{A}g$ of principally polarised abelian varieties of dimension $g$, defined over an algebraically closed field of characteristic $p >0$, is studied through various stratifications. The two most prominent ones are the Newton polygon stratification, based on the isogeny class of the $p$-divisible group of an abelian variety, and the Ekedahl-Oort stratification, defined by the isomorphism class of its $p$-torsion group scheme. In general, it is not known how the strata of these two intersect. In this paper we study intersections of the Ekedahl-Oort and Newton polygon stratification in dimension at most $5$. We give an explicit description of the Ekedahl-Oort stratification of the supersingular locus $\mathcal{S}{5}$, and describe other $p$-rank zero intersections of the two stratifications. For intersections of positive $p$-rank, we describe all such intersections in dimensions at most $5$. Our proofs use Oort's minimality of $p$-divisible groups, as well as related results of Chai-Oort and Harashita. In addition to these, a new approach used in this paper is an inductive argument, based on understanding the Ekedahl-Oort type of products of abelian varieties and the known intersections in lower dimensions.