The Ekedahl-Oort and Newton stratification of the $\mathsf{GU}(3,2)$ Shimura variety

TL;DR

This work delivers a complete description of how the Ekedahl-Oort and Newton stratifications interact on the characteristic-$p$ fiber of the unitary Shimura variety $\mathcal{M}(3,2)$ with inert $p$ in $\mathcal{O}_K$. By combining $p$-rank arguments, a forgetful map to the Siegel moduli variety $\mathcal{A}_5$, generic-slope analysis, and an explicit Dieudonné–Rapoport–Zink construction, the authors classify which EO strata intersect which Newton strata. They identify five EO strata contained in the supersingular locus, one in the $\beta_1$-Newton stratum, one stratum intersecting both $\mathrm{ss}$ and $\beta_1$, two strata in $\beta_2$, and one equal to the $\mu$-ordinary Newton stratum, and they provide explicit geometric and group-theoretic realizations of these intersections. The results offer a concrete, usable map of EO–Newton intersections in this signature and lay groundwork for extending the analysis to other unitary signatures via the same toolkit.

Abstract

This paper concerns the characteristic-$p$ fibers of $\mathsf{GU}(3,2)$ Shimura varieties. Such Shimura varieties parametrize abelian varieties in characteristic $p$ of dimension $5$ with an action of signature $(3,2)$ by an order in an imaginary quadratic field in which $p$ is inert. We completely describe the interaction of two stratifications of these Shimura varieties: 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. We identify which Ekedahl-Oort and Newton strata intersect.

Theorems & Definitions (28)

• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Definition 4.3
• Definition 4.4: Harashita_slope, Definition 3.1
• Lemma 4.5
• proof