Supersingular Ekedahl-Oort strata and Oort's conjecture

This paper confirms Oort's conjecture by proving that for even $g$ and $p \geq 5$, every geometric generic member of the maximal supersingular Ekedahl-Oort stratum in the moduli space of principally polarized abelian varieties has an automorphism group consisting only of $\{\pm 1\}$, while also establishing the result for the specific case of $g=4$ for any prime $p$.

The Gist

Imagine you are an architect exploring a vast, mysterious city called $A_g$. This city is built entirely out of mathematical shapes called Abelian Varieties. Think of these shapes as complex, multi-dimensional donuts (or tori) that have a special "polarization" attached to them—a bit like a unique magnetic field that gives them a specific orientation and structure.

The city has different neighborhoods. Some are ordinary, but there is a special, chaotic district called the Supersingular Locus ($S_g$). In this district, the shapes behave in a very wild, "supersingular" way. They are so complex that their internal machinery (their "endomorphism rings") is huge and tangled.

The Big Question: Oort's Conjecture

For a long time, mathematicians have been asking a simple question about the "average" or "generic" shape you would find if you wandered randomly through this supersingular district:

"If I pick a random shape here, how many ways can I rotate or flip it so that it looks exactly the same?"

This is asking about the Automorphism Group.

• If a shape is very special, it might have many symmetries (like a perfect sphere has infinite symmetries).
• If a shape is "generic" (random), it usually has very few symmetries.

Oort's Conjecture states that for most of these shapes in this wild district, the only ways to rotate or flip them to look the same are the "do nothing" move and the "turn it upside down" move. In math-speak, the group is just $\{ \pm 1 \}$.

The authors of this paper, Karemaker and Yu, set out to prove this conjecture is true, but with a few specific rules about the size of the shapes ($g$) and the type of math used ($p$).

The Tools: The "Relative Endomorphism Algebra"

To solve this, the authors invented a new way of looking at these shapes. They used a tool called the Relative Endomorphism Algebra.

The Analogy:
Imagine you have a large, transparent box (the vector space $V$) and inside it, you place a smaller, floating cloud of dust (the subspace $W$).

• The "Endomorphism Algebra" is the set of all possible ways you can shake the box so that the dust cloud stays inside the box.
• Usually, if the dust cloud is in a weird, random position, the only way to shake the box without spilling the dust is to do nothing or shake it in a very specific, limited way.
• The authors proved that in the "maximal" part of the supersingular district, the dust clouds are almost always in such a random, "generic" position that the only allowed shakes are the trivial ones ($\pm 1$).

The Strategy: Mapping the City

The paper is divided into two main missions:

Mission 1: The Even Dimensions ($g$ is even) and Large Primes ($p \ge 5$)

The authors focused on the "Maximal Supersingular EO Stratum." Think of this as the largest, most central neighborhood in the supersingular district.

• They showed that this neighborhood can be mapped to a Lagrangian Variety.
• The Metaphor: Imagine the neighborhood is a giant, smooth hill. The authors proved that if you stand on the very top of this hill (the "generic" point), the view is so clear and unobstructed that the shape has no hidden symmetries.
• They used a clever trick involving Hecke Correspondences. Imagine these as magical teleportation gates that connect different parts of the city. They proved that if you can teleport from any part of the city to this "generic" hilltop, and the hilltop has no extra symmetries, then the whole city (or at least the main components of it) shares this property.
• Result: They confirmed Oort's Conjecture for all even dimensions ($g=2, 4, 6...$) as long as the prime number $p$ is 5 or larger.

Mission 2: The Special Case of Dimension 4 ($g=4$)

Dimension 4 is tricky. It's like a puzzle piece that doesn't fit the general pattern easily.

• The authors didn't just rely on the general theory; they got their hands dirty with explicit computations.
• They used Dieudonné Modules.
  • The Metaphor: Think of a Dieudonné Module as the "blueprint" or the "DNA" of the shape. It's a detailed list of instructions on how the shape is built using numbers.
  • For $g=4$, they wrote out these blueprints explicitly. They calculated exactly how the "dust cloud" (the subspace) sits inside the "box" for a generic 4-dimensional shape.
  • They proved that even for $p=2$ and $p=3$ (which usually cause trouble in math), the blueprint forces the symmetry group to be just $\{ \pm 1 \}$.

Why Does This Matter?

1. Solving a Mystery: Oort's Conjecture was a long-standing open problem. Proving it gives mathematicians a solid foundation for understanding the "average" behavior of these complex shapes.
2. The "Generic" Rule: It tells us that in the chaotic world of supersingular shapes, the "normal" state is actually very simple (only two symmetries). The complex symmetries only appear in rare, special cases.
3. New Tools: The "Relative Endomorphism Algebra" they developed is a new tool that other mathematicians can use to study similar problems in different areas of math, like cryptography or number theory.

Summary in One Sentence

Karemaker and Yu proved that if you pick a random, high-dimensional "supersingular donut" (with specific size and math rules), it is so uniquely shaped that the only way to rotate it to look the same is to turn it upside down—confirming a decades-old guess about the simplicity hidden within mathematical chaos.

Technical Summary

Here is a detailed technical summary of the paper "Supersingular Ekedaahl-Oort Strata and Oort's Conjecture" by Valentijn Karemaker and Chia-Fu Yu.

1. Problem Statement

The paper addresses Oort's Conjecture regarding the automorphism groups of generic supersingular abelian varieties.

• Context: Let $A_g$ be the moduli space of $g$-dimensional principally polarized abelian varieties over a field of characteristic $p$. Let $S_g \subset A_g$ be the supersingular locus.
• The Conjecture: Oort conjectured that for $g \ge 2$, the geometric generic member of the supersingular locus $S_g$ has an automorphism group isomorphic to $\{\pm 1\}$, despite its endomorphism ring having a large rank ($4g^2$).
• Previous Status:
  • Proven for $g=2$ and $p>2$ (Ibukiyama; Karemaker & Pries).
  • Proven for $g=3$ and $p>2$ (Karemaker, Yu, & Yobuko).
  • Counterexamples exist for $p=2$ in low dimensions.
  • The case for general even $g$ and $p \ge 5$ remained open, as did the case $g=4$ for all $p$.

2. Methodology

The authors employ a multi-faceted approach combining algebraic geometry, representation theory, and arithmetic geometry.

A. Relative Endomorphism Algebras

A central tool is the study of relative endomorphism algebras for pairs of vector spaces $(V_0, W)$ over a field extension $L/K$.

• Definition: $End(V_0, W) := \{ \alpha \in End_K(V_0) : \alpha(W) \subseteq W \}$, where $V_0$ is a $K$-vector space and $W \subseteq V_0 \otimes_K L$ is an $L$-subspace.
• Stratification: The authors construct a stratification on the Grassmannian $Gr(V_0, r)$ (and the Lagrangian variety $L(V_0)$) based on the isomorphism class of $End(V_0, W)$.
• Key Result (Theorems 3.5 & 4.6): If the field extension $L/K$ is infinite and not "exceptional" (e.g., $K$ is not real closed with $L=K$), there exists a dense open stratum where $End(V_0, W) = K$. This implies that for a generic subspace, the only endomorphisms preserving it are scalars.

B. Geometry of Supersingular EO Strata

The authors analyze the Ekedahl-Oort (EO) stratification of the supersingular locus $S_g$.

• They focus on the maximal supersingular EO stratum, denoted $S^{eo}_g$.
• They establish that irreducible components of $S^{eo}_g$ admit finite covers by Lagrangian varieties $L$ associated with symplectic spaces over $\mathbb{F}_{p^2}$.
• By lifting the stratification of the Lagrangian variety (based on relative endomorphism algebras) to the moduli space, they define a new stratification on $S^{eo}_g$ that tracks the jumps in endomorphism rings.

C. Connection to Dieudonné Modules

The paper bridges the gap between the geometric stratification and the arithmetic invariants of abelian varieties:

• For a supersingular abelian variety $X$, the endomorphism ring $End(X)$ is related to the relative endomorphism algebra of a specific pair of vector spaces derived from its Dieudonné module $M$.
• Specifically, if $X$ is related to a superspecial variety $\tilde{X}$ via a minimal isogeny, $End(X)$ is the preimage of a relative endomorphism algebra under a reduction map.
• The authors utilize $\ell$-adic Hecke correspondences to show that the property of having a small automorphism group on the maximal stratum extends to the entire supersingular locus.

D. Explicit Computations for $g=4$

For the specific case of $g=4$, the authors bypass the general stratification argument to provide a direct proof for all primes $p$ (including $p=2, 3$).

• They utilize the theory of Polarised Flag Type Quotients (PFTQs) developed by Harashita.
• They perform explicit computations on the Dieudonné modules of the flag, analyzing the linear independence of coordinates in the defining equations of the moduli space to prove that the generic endomorphism ring is minimal.

3. Key Contributions and Results

Theorem A (Main Structural Result)

If $g$ is even and $p \ge 5$, every geometric generic member in the maximal supersingular EO stratum has an automorphism group $\{\pm 1\}$.

• Note: The theorem fails if $g$ is odd or $p=2$.

Theorem B (Confirmation of Oort's Conjecture)

Oort's conjecture holds true for even $g$ and $p \ge 5$.

• Logic: The set of irreducible components of $S_g$ is transitive under $\ell$-adic Hecke correspondences. Since the maximal EO stratum intersects every irreducible component, and the generic point of the maximal stratum has $\text{Aut} = \{\pm 1\}$, the generic point of the entire supersingular locus must also have $\text{Aut} = \{\pm 1\}$.

Theorem 7.1 (Dimension $g=4$)

Oort's conjecture holds for $g=4$ for any prime $p$ (including $p=2, 3$).

• This removes the restriction $p \ge 5$ for dimension 4, providing strong evidence that the only counterexamples to Oort's conjecture occur at $(g, p) = (2, 2)$ and $(3, 2)$.

Mass Stratification Refinement

The authors refine the mass stratification (studied in their previous work) using the relative endomorphism algebra. They provide explicit mass formulae for each stratum (Theorem 6.19), showing a direct relationship between the geometry of the strata and the arithmetic mass of the points.

4. Significance

1. Resolution of a Major Conjecture: The paper confirms Oort's conjecture for all even dimensions $g$ in characteristic $p \ge 5$, a significant step in understanding the arithmetic of supersingular abelian varieties.
2. New Geometric Tools: The introduction of stratifications on Grassmannians and Lagrangian varieties based on relative endomorphism algebras provides a powerful new method for studying the variation of endomorphism rings in moduli spaces. This connects abstract algebraic geometry with concrete arithmetic invariants.
3. Handling Small Primes: By solving the $g=4$ case for all $p$, the authors demonstrate that the "exceptional" behavior at small primes is dimension-dependent and not a universal obstruction, refining the understanding of where Oort's conjecture might fail.
4. Interplay of Structures: The work elegantly synthesizes the theory of EO strata, Dieudonné modules, Hecke correspondences, and mass formulas, offering a unified framework for analyzing supersingular loci.

In summary, Karemaker and Yu provide a definitive proof for Oort's conjecture in the even-dimensional case ($p \ge 5$) and the specific case of $g=4$ (all $p$), utilizing a novel stratification technique based on relative endomorphism algebras to control the automorphism groups of generic supersingular abelian varieties.