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On the classification of indecomposable Ekedahl-Oort strata in unitary Shimura varieties, and related Newton polygons

This paper provides a complete classification of indecomposable Ekedahl-Oort strata in unitary Shimura varieties over odd inert primes into four distinct types, offers an algorithm to translate their combinatorial descriptions into Weyl group representatives, and constructs tautological points to compute their Newton polygons, demonstrating that specific strata types always intersect the supersingular locus.

Original authors: Emerald Andrews, Deewang Bhamidipati, Maria Fox, Heidi Goodson, Steven R. Groen, Sandra Nair

Published 2026-06-16

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CC BY 4.0

Original authors: Emerald Andrews, Deewang Bhamidipati, Maria Fox, Heidi Goodson, Steven R. Groen, Sandra Nair

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect trying to understand the blueprints of a very special, mysterious city called the Unitary Shimura Variety. This city isn't made of bricks and mortar; it's built from abstract mathematical objects called abelian varieties (think of them as complex, multi-dimensional shapes that behave like doughnuts).

The authors of this paper, Emerald Andrews and her team, are trying to map out the different "neighborhoods" within this city. They are particularly interested in two things:

The Ekedahl-Oort Strata: These are neighborhoods defined by the specific "fingerprint" of the shapes' torsion (a type of internal structural weakness or pattern).
The Newton Strata: These are neighborhoods defined by the "slope" or overall shape of the city's underlying geometry.

The paper's main goal is to figure out exactly what the "indecomposable" neighborhoods look like. In plain English, an "indecomposable" neighborhood is one that cannot be broken down into smaller, simpler neighborhoods stuck together. It's a fundamental building block.

Here is the breakdown of their discovery, using everyday analogies:

1. The Four Fundamental Building Blocks

The authors discovered that every indecomposable neighborhood in this city falls into exactly one of four categories. They named them after the shapes of their internal diagrams (called Kraft diagrams), which look like wheels or loops:

Unitary Unicycle: A single loop with one wheel.
Unitary Bicycle: A structure with two wheels linked together.
Serre Unicycle: A single wheel, but constructed in a specific way using a "Serre tensor" (think of this as a special recipe for doubling a shape).
Serre Bicycle: Two wheels linked together, also using that special doubling recipe.

The Analogy: Imagine you are sorting a pile of LEGO structures. You might think there are infinite ways to build them, but the authors prove that if you strip away any structure that is just two smaller structures glued together, you are left with only these four specific types of "atomic" LEGO creations.

2. The Secret Code: Words of "f" and "v"

How do you describe these complex shapes without drawing them? The authors use a simple code: a string of letters, f and v.

Think of f as a "forward" step.
Think of v as a "backward" or "vertical" step.

By arranging these letters in a circle (a word), they can generate the blueprint for any of the four building blocks.

The Challenge: Mathematicians usually describe these neighborhoods using complex group theory (Weyl groups), which is like speaking a high-level dialect.
The Solution: The authors created a "dictionary" (an algorithm) that translates the simple f/v word code directly into the complex group theory language. This allows anyone to look at a simple word like fvfv and instantly know exactly which neighborhood it represents in the city.

3. The "Tautological Lift": Building a Bridge

One of the paper's clever tricks is constructing a "tautological lift."

The Problem: The "f/v" words describe the shape of the city in a simplified, "mod-p" world (a world where numbers wrap around, like a clock). But to understand the full geometry, you need to see the city in its full, 3D glory (the "p-adic" world).
The Solution: The authors built a specific, standard bridge (a "lift") that takes their simple word-based blueprint and constructs a full, 3D version of the shape.
The Result: Once they built this bridge, they could calculate the Newton Polygon (the slope map) of the shape. This tells them exactly which "Newton neighborhood" the shape belongs to.

4. The Big Discovery: Where the Neighborhoods Meet

The most exciting finding is about where these different neighborhoods overlap.

There is a special, closed-off area in the city called the Supersingular Locus. This is the "most singular" or "most extreme" part of the city.
The authors proved that if a neighborhood is built from a Unitary Unicycle or a Serre Unicycle, it always touches the Supersingular Locus.
The Metaphor: Imagine the Supersingular Locus as the "center of the universe" for these shapes. The authors found that if your shape is built like a "Unicycle" (one of their four types), it is guaranteed to have a path leading directly to the center. You can't build a Unicycle without it touching the core.

Summary

In short, this paper is a comprehensive catalog and translation guide for the fundamental building blocks of a specific type of mathematical city.

They identified the four atomic types of structures (Unicycles and Bicycles).
They wrote a dictionary to translate simple letter codes into complex mathematical maps.
They built a bridge to calculate the slopes of these structures.
They proved that the Unicycle structures are special because they always connect to the most central, extreme part of the city.

This work doesn't just list these shapes; it gives mathematicians the tools to instantly recognize them, translate them into different languages, and understand exactly where they fit in the grand geometry of the universe.

Technical Summary: On the Classification of Indecomposable Ekedahl-Oort Strata in Unitary Shimura Varieties, and Related Newton Polygons

Problem Statement
This paper addresses the classification of indecomposable Ekedahl-Oort (EO) strata within unitary Shimura varieties $M(a, b)$ defined over an algebraically closed field $k$ of characteristic $p > 2$ . These varieties parameterize abelian varieties of dimension $a+b$ equipped with an action of an order in an imaginary quadratic field where $p$ is inert, satisfying a signature $(a, b)$ condition. While the structure of EO strata for Siegel modular varieties and specific unitary signatures (such as $(q-1, 1)$ ) has been previously studied, a complete classification for arbitrary signatures $(a, b)$ in the unitary setting remained open. Furthermore, the paper seeks to elucidate the interaction between the EO stratification and the Newton stratification, specifically determining when an EO stratum intersects the supersingular locus.

Methodology
The authors employ a combination of Dieudonné theory, combinatorial group theory, and geometric lifting techniques. The core methodology proceeds as follows:

Combinatorial Classification via Words: Building on Kraft's structure theorem for mod- $p$ Dieudonné modules, the authors represent indecomposable modules using words in the alphabet $\{f, v\}$ (representing Frobenius and Verschiebung operators). They extend the classification of polarized BT1 modules (associated with Siegel varieties) to unitary BT1 modules (associated with unitary Shimura varieties).
Decomposition Theorems: The paper establishes that any unitary BT1 module decomposes uniquely into a direct sum of specific indecomposable types. This involves analyzing the "Kraft diagrams" associated with these words and identifying constraints imposed by the unitary structure (specifically the Rosati involution condition and the Lagrangian property of the subspaces $M_0$ and $M_1$ ).
Translation to Weyl Groups: Utilizing Moonen's correspondence between EO strata and cosets in the Weyl group $S_q$ (where $q=a+b$ ), the authors develop an algorithm to translate the combinatorial "word" description of a stratum into the corresponding Weyl group coset representative. This bridges the gap between the algebraic description of the $p$ -torsion and the geometric topology of the strata.
Tautological Lifts and Newton Polygons: To study the intersection with Newton strata, the authors construct a "tautological lift" of a unitary BT1 module to a unitary $p$ -adic Dieudonné module. By computing the Newton polygon of this lift, they determine the specific Newton stratum that the EO stratum intersects.

Key Contributions and Results

Classification of Indecomposable Types: The primary result (Theorem 4.21) is a complete classification of indecomposable unitary BT1 modules. The authors prove that every such module is isomorphic to exactly one of four types:
1. Unitary Unicycle: Arising from a primitive self-dual word $w = yy^\star$ where $y$ has odd length.
2. Unitary Bicycle: Arising from a primitive non-self-dual word $w$ of even length (specifically the module $M(w) \oplus M(w^\star)$ ).
3. Serre Unicycle: Arising from a primitive self-dual word $w$ of length divisible by 4. These correspond to the Serre tensor construction of a polarized unicycle.
4. Serre Bicycle: Arising from a word $w = zz$ where $z$ is a primitive non-self-dual word of odd length. These correspond to the Serre tensor construction of a polarized bicycle.
  The paper provides explicit formulas for the signatures $(a, b)$ associated with each type based on the distribution of $f$ and $v$ in the defining words.
Algorithmic Translation (Word to Weyl): Section 5 presents an algorithm (Theorem 5.3) that converts the multiset of words describing a unitary BT1 module into the minimal-length Weyl group coset representative $\gamma \in W(a, b)$ . This allows for the explicit identification of the EO stratum in terms of the Bruhat order, facilitating the study of closure relations.
Intersection with Newton Strata: Using the tautological lift construction, the authors compute the Newton polygon for any given EO stratum (Theorem 6.10). They prove that the EO stratum corresponding to a set of words $W$ intersects the Newton stratum defined by the slopes of those words.
- Supersingular Intersection: A significant application (Corollaries 6.12 and 6.13) demonstrates that EO strata corresponding to Unitary Unicycles and Serre Unicycles always intersect the supersingular locus (the unique closed Newton stratum). This is because the defining words for these types are necessarily "balanced" (containing an equal number of $f$ and $v$ ), resulting in a Newton polygon with constant slope $1/2$ .

Significance
The paper provides a foundational structural theorem for the $p$ -torsion of abelian varieties in the unitary setting, analogous to Oort's results for the Siegel case. By categorizing indecomposable strata into four distinct types and providing a computational bridge to Weyl group representatives, the work enables a more precise analysis of the geometry of unitary Shimura varieties in arbitrary signatures. The construction of tautological lifts and the subsequent determination of Newton polygon intersections offer a concrete method for verifying the non-emptiness of intersections between EO and Newton strata, extending previous results known only for specific signatures (such as Coxeter types or low dimensions) to the general case. The identification of the supersingular locus intersection for unicycle-type strata resolves a specific question regarding the topology of these varieties.

1. The Four Fundamental Building Blocks

2. The Secret Code: Words of "f" and "v"

3. The "Tautological Lift": Building a Bridge

4. The Big Discovery: Where the Neighborhoods Meet

Summary

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