Supersingular Ekedahl-Oort strata and Oort's conjecture

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 $p$.

💡 Research Summary

The paper addresses a long‑standing conjecture of Frans Oort concerning the automorphism groups of generic points in the supersingular locus of the moduli space 𝔸_g of principally polarized abelian varieties over an algebraic closure of a finite field of characteristic p. Oort’s conjecture predicts that for any genus g ≥ 2, the geometric generic member of the supersingular locus S_g should have automorphism group {±1}, despite the fact that its endomorphism ring is a large ℤ‑module of rank 4g².

The authors prove the conjecture in two new families: (i) when g is even and p ≥ 5, and (ii) when g = 4 for all primes p. The main technical innovation is the introduction of a “relative endomorphism algebra” End(V,W) attached to a pair (V,W) where V is a finite‑dimensional K‑vector space and W is an L‑subspace of V⊗_K L for a field extension L/K. End(V,W) consists of K‑linear endomorphisms of V that preserve W after base change to L. This construction interpolates between the full endomorphism algebra of V (when W is defined over K) and the scalar field K (when W is sufficiently generic).

The paper proceeds as follows. Section 2 develops the algebraic foundations of relative endomorphism algebras, including criteria for when the evaluation map on polynomials is injective, and a classification of the possible algebras in terms of field extensions. Section 3 studies the Grassmannian Gr(V,r) of r‑dimensional subspaces of V and stratifies it according to the isomorphism type of End(V,W). Theorem 3.5 shows that on an open dense subset of the Grassmannian, End(V,W) ≅ K, except in the degenerate cases where K is algebraically closed or real closed. Section 4 carries out the analogous construction for Lagrangian varieties of isotropic r‑dimensional subspaces in a 2r‑dimensional symplectic space, yielding Theorem 4.6 with the same generic description.

With these tools, the authors turn to the supersingular Ekedahl–Oort (EO) stratification of 𝔸_g. Let S_eog be the union of all supersingular EO strata; each irreducible component of S_eog admits a finite cover by a Lagrangian variety L. By pulling back the relative‑endomorphism stratification from L to S_eog, they obtain a new refinement of the EO stratification that records the jumps of the endomorphism rings of the underlying abelian varieties. There is a unique maximal stratum, open and dense in S_eog, where the relative endomorphism algebra is just K. For even g and p ≥ 5, the authors prove that any geometric point in this maximal stratum corresponds to a supersingular abelian variety whose endomorphism ring is ℤ and whose automorphism group is precisely {±1}.

Theorem A (the main result) states that for even g and p ≥ 5, the generic point of the maximal supersingular EO stratum has automorphism group {±1}. Theorem B shows how this immediately implies Oort’s conjecture for all supersingular components when g is even and p ≥ 5: the ℓ‑adic Hecke correspondences act transitively on the set of irreducible components of S_g, and each component contains a piece of the maximal EO stratum, hence inherits the same automorphism property.

Section 7 treats the remaining case g = 4 and the small primes p = 2, 3, which are not covered by the generic arguments. The authors perform explicit Dieudonné‑module calculations for all possible supersingular 4‑dimensional principally polarized abelian varieties, determining their automorphism groups directly. They also refine the previously defined mass function on the supersingular locus, providing a new mass formula (Theorem 6.19) for each refined stratum.

In summary, the paper introduces a novel relative‑endomorphism framework that yields a fine stratification of supersingular EO loci, resolves Oort’s conjecture for even genus with p ≥ 5, and settles the genus‑four case for all primes. The remaining open cases are (g,p) = (2,2) and (3,2), which are known counter‑examples. The methods suggest possible extensions to odd genus and to a deeper understanding of the interaction between Hecke orbits, EO strata, and automorphism groups.