Oort's conjecture on automorphisms of generic supersingular abelian varieties

This paper proves Oort's conjecture that the automorphism group of a generic principally polarized supersingular abelian variety in characteristic $p$ is $\{\pm 1\}$, except for the specific cases of genus 2 or 3 with $p=2$, while also providing an explicit description of the $a=1$-locus in the corresponding Rapoport-Zink space and establishing analogous results for supersingular $p$-divisible groups.

The Gist

Here is an explanation of Eva Viehmann's paper, "Oort's Conjecture on Automorphisms of Generic Supersingular Abelian Varieties," translated into everyday language with creative analogies.

The Big Picture: The "Generic" Rule

Imagine you have a massive, infinite library of complex geometric shapes called Abelian Varieties. These are like multi-dimensional donuts (tori) that have very specific rules about how they are stretched and twisted.

Mathematicians are interested in a special section of this library called the Supersingular Locus. Think of this as the "extreme sports" section of the library. The shapes here are built in a very specific, chaotic way using a special type of math called "characteristic $p$" (where $p$ is a prime number like 2, 3, 5, etc.).

The Question:
If you pick a shape at random from this "extreme sports" section, how many ways can you rotate or flip it so that it looks exactly the same?

• The Obvious Moves: You can always flip it upside down (multiply by -1) or leave it alone (multiply by 1).
• The Question: Are there any other secret moves? Can you twist it in a weird way that makes it look identical to the start?

The Conjecture (Oort's Guess):
In 2001, a mathematician named Frans Oort guessed that for almost all shapes in this section, the answer is NO. The only moves that work are the obvious ones: 1 and -1.

There are a few exceptions (specifically when the shape is 2-dimensional and $p=2$, or 3-dimensional and $p=2$), but for everything else, the shapes are "rigid." They don't have hidden symmetries.

The Problem: Why is this hard?

Imagine trying to prove that a specific, unique snowflake has no hidden symmetry. But instead of looking at one snowflake, you have to look at a whole cloud of them.

The problem is that in this "extreme sports" section, the shapes are incredibly complex. They are built from tiny building blocks called $p$-divisible groups. To understand the whole shape, you have to understand these tiny blocks.

For a long time, mathematicians had proven Oort's guess for small sizes (like 2D or 3D shapes) or for specific prime numbers. But they couldn't prove it for all sizes and all prime numbers. It was like proving a rule about traffic in New York City, but you couldn't prove it for traffic in Tokyo or London.

The Solution: Breaking it Down

Eva Viehmann's paper solves this puzzle by breaking the problem into three manageable steps, using a method that feels like reverse engineering a lock.

Step 1: Zooming In (The Microscope)

Instead of looking at the whole giant shape, Viehmann zooms in on the tiniest possible version of the shape where the "chaos" is at its minimum. She calls this the $a=1$ locus.

• Analogy: Imagine trying to find a specific grain of sand on a beach. Instead of scanning the whole beach, she finds the one specific spot where the sand is most organized. If you can prove the rule holds for this "most organized" spot, it holds for the whole beach.

Step 2: The Blueprint (Dieudonné Modules)

She translates the geometric shapes into algebraic blueprints called Dieudonné modules.

• Analogy: Think of the geometric shape as a physical sculpture. The Dieudonné module is the CAD drawing or the instruction manual for building that sculpture.
• The paper focuses on a specific type of instruction manual where the instructions are "self-dual" (the instructions for the left side perfectly mirror the right side).
• She creates a detailed list of rules (coordinates) that these instruction manuals must follow to be valid.

Step 3: The "Generic" Test

Now, she asks: "If I follow these rules randomly, will I accidentally create a shape with extra symmetry?"

• She uses a technique called counting solutions. She sets up a massive system of equations (like a giant Sudoku puzzle) that represents the rules for the shape to have extra symmetry.
• The Discovery: She proves that for almost all random choices of numbers (coordinates), this system of equations has no solution.
• The Metaphor: It's like trying to find a specific combination of keys that opens a secret door. She proves that if you pick your keys at random from a huge pile, the odds of finding the secret combination are zero. The only keys that work are the standard "1" and "-1" keys.

The Exceptions (The "Special Cases")

The paper confirms that there are two specific scenarios where the rule breaks:

1. Dimension 2, Prime 2: Like a 2D shape made with a very specific, chaotic material.
2. Dimension 3, Prime 2: A 3D shape with the same material.
   In these rare cases, the shapes do have extra symmetries (like a group of 8 moves instead of just 2). Viehmann's proof explicitly accounts for these exceptions, showing that everywhere else, the "Generic Rule" holds true.

The "Non-Polarized" Twist

In the second half of the paper, she removes a specific constraint called "polarization" (which is like a specific type of glue holding the shape together).

• Without the glue: The shapes are even more flexible.
• The Result: She finds that while the "1 and -1" rule mostly still holds, there is a slightly larger set of "standard moves" allowed (scalars in a specific number system). It's like saying, "Without the glue, you can't twist the shape, but you can stretch it slightly in a uniform way."

Why Does This Matter?

This paper is a "final boss" victory for a long-standing conjecture.

1. Completeness: It closes the book on Oort's conjecture. We now know the rule for every dimension and every prime number.
2. Rigidity: It tells us that in the world of these complex geometric shapes, "generic" means "rigid." Most shapes are unique and don't have hidden symmetries. This is a fundamental truth about the structure of the universe of mathematics.
3. Methodology: The tools she developed to analyze these "instruction manuals" (Dieudonné modules) can now be used by other mathematicians to solve different, even harder problems in geometry and number theory.

In short: Eva Viehmann proved that if you pick a complex, multi-dimensional donut from the "chaos zone" of mathematics, it is almost certainly a unique, rigid object with no hidden tricks, unless it is a very specific, small, and weird exception.

Technical Summary

Here is a detailed technical summary of Eva Viehmann's paper, "Oort's Conjecture on Automorphisms of Generic Supersingular Abelian Varieties."

1. Problem Statement

The paper addresses Oort's Conjecture regarding the automorphism groups of principally polarized abelian varieties (PPAVs) within the supersingular locus of the moduli space $\mathcal{A}_g$ over a field of characteristic $p$.

• Context: For a generic point $x$ in the moduli space $\mathcal{A}_g$, the automorphism group of the corresponding abelian variety $(A_x, \lambda_x)$ is known to be $\{\pm 1\}$. However, on the supersingular Newton stratum $S_g$ (where the $p$-divisible group is isogenous to a product of supersingular elliptic curves), the structure is more complex.
• The Conjecture: Oort (2001) conjectured that for $g \geq 2$, the automorphism group of a generic point in the supersingular locus is also $\{\pm 1\}$, with specific exceptions for low dimensions and small primes.
• Prior Status: The conjecture was previously proven for:
  • $g=2, p>2$ (Ibukiyama; Karemaker & Pries).
  • $g=3, p>2$ (Karemaker, Yobuko, Yu).
  • $g=4$ (all $p$) and all even $g$ with $p \geq 5$ (Karemaker & Yu; Dragutinović).
  • Exceptions: The cases $g=2, p=2$ and $g=3, p=2$ were known to be false (the automorphism group is larger).
• Goal: Prove the conjecture for all remaining cases (specifically $g \geq 4$ for $p=2$, and $g \geq 5$ for all $p$, though the proof covers all $g \geq 4$ generally).

2. Methodology

Viehmann employs a reduction strategy combining moduli theory, Dieudonné theory, and explicit coordinate analysis on Rapoport-Zink spaces.

A. Reduction to Dieudonné Modules

1. Uniformization: Using the Rapoport-Zink uniformization theorem, the problem is reduced from the global moduli space $\mathcal{A}_g$ to the local moduli space $\mathcal{M}_g$ of polarized supersingular $p$-divisible groups.
2. Reduction to $a=1$ Locus: The author focuses on the locus $\mathcal{M}_g^\circ$ where the $a$-number (defined as $\dim \text{Hom}(\alpha_p, X)$) is 1. This locus is open and dense in the supersingular stratum.
3. Dieudonné Lattices: The problem is translated into the language of Dieudonné modules. A point in $\mathcal{M}_g^\circ$ corresponds to a Dieudonné lattice $M \subset \breve{N}$ (where $\breve{N}$ is the rational Dieudonné module of a superspecial group) such that:
   • $M$ is generated by a single element $v$ (since $a(M)=1$).
   • $M$ is self-dual with respect to the induced symplectic pairing.
   • $M$ is contained in a specific $\tau$-stable lattice $\Lambda_0$ (where $\tau = p^{-1}F^2$).
4. Symmetry Reduction: The group $J(\mathbb{Q}_p)$ of self-quasi-isogenies acts transitively on the irreducible components of $\mathcal{M}_g^\circ$. It suffices to prove the result for a single fixed irreducible component $C_{\Lambda_0}$ associated with a specific lattice $\Lambda_0$.

B. Explicit Coordinate Analysis

The core technical innovation is the explicit parametrization of the generators $v$ of the Dieudonné lattices on the component $C_{\Lambda_0}$.

• Generators: A generator $v$ is expressed as a series $v = \sum a_{i,l} \Pi^l X_i$, where $X_i$ are basis vectors of $\Lambda_0$ and $\Pi$ is a uniformizer in the division algebra $D$.
• Self-Duality Constraints: The condition that the lattice $M = Dv$ is self-dual imposes a system of polynomial equations on the coefficients $a_{i,l}$. Viehmann derives these equations explicitly (Proposition 3.16), showing that:
  • The "leading" coefficients $a_{i,0}$ (for $i \leq \lceil (g+1)/2 \rceil$) can be chosen freely (subject to linear independence over $\mathbb{F}_{p^2}$).
  • The remaining coefficients are determined iteratively by the self-duality conditions.
• Torsion Analysis: The proof relies on Lemma 3.7 (a result by Serre/Karemaker-Yu), which states that certain congruence subgroups of $GL_n(\mathcal{O}_D)$ are torsion-free for specific levels $s$. The goal becomes showing that any finite-order automorphism $j$ must act trivially modulo $\Pi^s$ for a sufficiently large $s$.

C. The Proof Strategy (Inductive on $s$)

The author proves Claim 4.1: For a generic point in the component, any automorphism $j$ stabilizing the lattice $M$ must induce the identity (or $\pm 1$) on the quotient $\Lambda_0 / \Pi^s \Lambda_0$.

• Step $s=1$: Shows that $j$ acts as a scalar on the reduction modulo $\Pi$. This uses the fact that the basis vectors generated by $\tau^k(v)$ span the space and that $j$ must commute with $\tau$.
• Step $s=2$: Analyzes the action on $\Lambda_0 / p\Lambda_0$. By constructing a matrix of coefficients involving $a_{i,0}$ and the perturbation terms $\delta_{ji}$, the author shows that for generic choices of $a_{i,0}$, the determinant of a specific minor is non-zero unless the perturbation vanishes. This forces $j$ to be a scalar.
• Step $s=3$ (for $p=2$): Extends the argument to higher levels, utilizing the specific structure of the equations for $g=4$ and $g \geq 5$ to show that non-scalar automorphisms cannot stabilize the lattice generically.

3. Key Contributions

1. Proof of Oort's Conjecture: The paper provides a complete proof of the conjecture for all $g \geq 4$ and all primes $p$, filling the final gaps left by previous works.
   • Theorem 1.1: For $(g, p) \neq (2, 2), (3, 2)$, there exists a dense open subscheme of the supersingular locus where $\text{Aut}(A_x, \lambda_x) = \{\pm 1\}$.
2. Explicit Description of the $a=1$ Locus: The paper provides a detailed, explicit description of the $a=1$ locus in the Rapoport-Zink space for arbitrary dimension $g$. This includes a precise characterization of the generators of the associated Dieudonné modules and the equations governing their self-duality (Proposition 3.16).
3. Generalization to Non-Polarized Case: The methods are extended to the moduli space of supersingular $p$-divisible groups without polarization (Section 5).
   • Theorem 5.1: For $g \geq 5$, the generic automorphism group consists only of scalars in $\mathbb{Z}_p^\times$.
   • For $g=4$, the generic automorphism group is $\mathbb{Z}_p^\times + p\mathcal{O}_D$ (where $\mathcal{O}_D$ is a maximal order in the division algebra).

4. Results

• Main Theorem: The automorphism group of a generic principally polarized supersingular abelian variety of dimension $g$ in characteristic $p$ is $\{\pm 1\}$, except when $(g,p) = (2,2)$ or $(3,2)$.
• Non-Polarized Result: For generic supersingular $p$-divisible groups of dimension $g$:
  • If $g \geq 5$, $\text{Aut}(X) \cong \mathbb{Z}_p^\times$.
  • If $g = 4$, $\text{Aut}(X) \cong \mathbb{Z}_p^\times + p\mathcal{O}_D$.
• Structural Insight: The proof demonstrates that the "generic" behavior is rigid; the only automorphisms surviving on a dense open set are the trivial ones (scalars), while larger automorphism groups only occur on lower-dimensional closed subschemes defined by specific algebraic relations among the coefficients of the Dieudonné generators.

5. Significance

• Resolution of a Long-Standing Open Problem: This work completes the classification of generic automorphism groups on the supersingular locus, a central topic in the arithmetic geometry of Shimura varieties and moduli spaces.
• Methodological Advancement: The explicit coordinate description of the $a=1$ locus for arbitrary $g$ is a significant technical achievement. Previous proofs for $g=3, 4$ relied on case-specific geometric descriptions; Viehmann's approach is uniform and scalable to higher dimensions.
• Implications for Shimura Varieties: The results have direct applications to the study of the basic locus of unitary Shimura varieties and the structure of their reduction modulo $p$. Understanding the generic automorphism group is crucial for determining the dimension of the supersingular locus and the behavior of Hecke operators.
• Foundation for Future Work: The explicit equations derived for the self-dual lattices provide a framework for investigating further questions, such as the generic automorphism groups on specific Ekedahl-Oort strata or intersections with the supersingular locus.