Supersingular Ekedahl-Oort strata and Oort's conjecture
Valentijn Karemaker, Chia-Fu Yu
TL;DR
The paper proves that for even dimension $g$ and $p\ge 5$, the geometric generic point in the maximal supersingular EO stratum of ${\mathcal A}_g$ has automorphism group ${\\{\\pm 1\\}}$, verifying Oort's conjecture in this setting, and separately establishes the conjecture for $g=4$ and arbitrary $p$. It develops a robust algebraic framework of relative endomorphism algebras End$(V,W)$ for pairs of vector spaces, and uses stratifications of Grassmannians and Lagrangian varieties to track endomorphism rings across EO strata. Central tools include explicit GL and symplectic endomorphism-descriptions, a stratification refined by End$(V,W)$ data, and $\ell$-adic Hecke correspondences on the supersingular locus, which yield transitivity of irreducible components and enable spreading automorphism-control from maximal strata to entire supersingular components. The work also connects endomorphism-algebra data to endomorphism rings of abelian varieties via Dieudonné module theory, and provides mass formulae for supersingular strata, together with explicit Dieudonné-module computations in the critical $g=4$ case. Collectively, these results illuminate how arithmetic invariants vary on supersingular EO strata and establish key cases of Oort's conjecture while offering a versatile toolkit for future investigations in EO stratifications and supersingular moduli.
Abstract
Let $\mathcal{A}_g$ be the moduli space over $\overline{\mathbb{F}}_p$ of $g$-dimensional principally polarised abelian varieties, where $p$ is a prime. We show that if $g$ is even and $p\geq 5$, then every geometric generic member in the maximal supersingular Ekedahl-Oort stratum in $\mathcal{A}_g$ has automorphism group $\{ \pm 1\}$. This confirms Oort's conjecture in the case of $p\geq 5$ and even $g$. We also separately prove Oort's conjecture for $g=4$ and any prime $p$.