Ekedahl-Oort strata and the supersingular locus in the GU(q-2,2) Shimura variety

Abstract: This paper concerns the characteristic-$p$ fibers of $\mathsf{GU}(q-2,2)$ Shimura varieties, which classify abelian varieties with additional structure. These Shimura varieties admit two stratifications of interest: the Ekedahl-Oort stratification, based on the isomorphism class of the $p$-torsion subgroup scheme, and the Newton stratification, based on the isogeny class of the $p$-divisible group. It is natural to ask which Ekedahl-Oort strata intersect the unique closed Newton stratum, called the supersingular locus. In this paper, we present several novel techniques that give information about the interaction between the two stratifications for general signature $(q-2,2)$, and as an application, we completely answer this question for the signature $(3,2)$.