Intersections of the Ekedahl-Oort and Newton Strata 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 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 completely determine which of these intersections are non-empty in dimension five. As a consequence, we give an explicit description of the induced Ekedahl-Oort stratification on the supersingular locus ${\mathcal{S}}_5$.

significant improvement of main result, change of title, improved exposition, comments welcome!