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Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$ -covers in characteristic $2$

This paper investigates the Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$ -Galois covers in characteristic two, demonstrating that these types are fully determined by the base curve's genus and ramification when the base is ordinary, establishing general bounds otherwise, and introducing a theory of enhanced differentials of the second kind to analyze algebraic de Rham cohomology directly.

Original authors: Jeremy Booher, Steven R. Groen, Joe Kramer-Miller

Published 2026-02-13

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CC BY 4.0

Original authors: Jeremy Booher, Steven R. Groen, Joe Kramer-Miller

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 have a piece of fabric (a mathematical object called a curve) and you want to fold it over itself in a specific way to create a new, more complex piece of fabric. In the world of mathematics, this is called a cover.

This paper is about what happens to the "shape" and "texture" of that new fabric when you fold it, specifically in a strange, twisted universe where the rules of arithmetic are different (characteristic 2, where $1+1=0$ ).

Here is the breakdown of the paper's journey, using everyday analogies.

1. The Setup: Folding a Blanket

Think of your base curve $X$ as a flat, ordinary blanket. You are going to fold it to make a new blanket $Y$ .

The Fold: This is a "Z/2Z-cover." Imagine taking a blanket and folding it perfectly in half. Some parts of the blanket might get crinkled or torn (these are the ramification points).
The Goal: Mathematicians want to know: If I know the texture of the original blanket ( $X$ ) and exactly how I folded it (the ramification breaks), can I predict the texture of the new blanket ( $Y$ )?

2. The Texture: The "Ekedahl-Oort Type"

In this mathematical universe, the "texture" of a blanket is described by something called the Ekedahl-Oort type.

Think of this as a fingerprint or a DNA sequence for the blanket. It tells you how the fabric behaves when you try to stretch it, twist it, or fold it in specific ways (using mathematical operators called Frobenius and Verschiebung).
If you know the fingerprint, you know everything about the blanket's hidden structure.

3. The Big Discovery: The "Ordinary" Case

The authors first looked at a very specific type of blanket: an Ordinary curve.

The Analogy: Imagine an "Ordinary" blanket is one that is perfectly smooth and has no weird knots.
The Result: They found a perfect recipe! If your starting blanket is smooth (Ordinary), then the fingerprint of the new folded blanket ( $Y$ $Y$ ) is completely determined by two things:
1. The fingerprint of the original blanket.
2. The number and severity of the crinkles (ramification) where you folded it.
Why it's cool: It's like saying, "If I start with a smooth sheet of paper and I fold it 3 times with specific creases, I can tell you exactly what the final crumpled ball looks like, no matter how I hold it." There is no guessing.

4. The Complication: The "Non-Ordinary" Case

What if the starting blanket isn't smooth? What if it's already knotted or tangled (a Supersingular curve)?

The Problem: In this case, the recipe breaks. Knowing the original texture and the fold isn't enough to predict the exact final fingerprint. Two different folds with the same creases can result in slightly different textures.
The Solution: Instead of a perfect recipe, the authors built a safety net. They created a set of bounds.
- Think of it like a weather forecast: "We can't tell you the exact temperature tomorrow, but we can guarantee it will be between 60°F and 70°F."
- They proved that the final texture will always fall within a specific range. The "error margin" of this prediction depends on how wild the fold was (the ramification break). Interestingly, even if the fold is huge, the error doesn't grow linearly; it grows very slowly (logarithmically).

5. The Secret Weapon: "Enhanced Differentials"

How did they solve this? They invented a new tool called Enhanced Differentials of the Second Kind.

The Analogy: Imagine trying to measure the water level in a leaky bucket.
- Old Method: You try to measure the water directly, but the leaks (poles in the math) make it messy.
- New Method (The Paper's Tool): Instead of just measuring the water, you attach a "sponge" to the bucket that soaks up the leaks and records exactly how much water was lost at each hole.
- By keeping track of these "leaks" and their specific "sponges" (antiderivatives), they could reconstruct the entire water level (the cohomology) perfectly, even in this weird characteristic 2 world where normal math rules don't apply.

6. The "Unlikely Intersections"

The paper also found some very rare blankets.

The Analogy: Imagine a room full of people (all possible shapes of blankets). Most people have a common height. But the authors found a way to create a whole family of blankets that are all exactly the same height, even though they were made in different ways.
Statistically, finding such a group is like finding a needle in a haystack, or finding a group of people who all have the exact same birthday, height, and shoe size by pure chance. The authors showed these "unlikely" groups actually exist and are quite large.

Summary

The Problem: Predicting the hidden structure of a folded mathematical curve.
The Success: If the starting curve is "normal" (Ordinary), the prediction is 100% accurate.
The Fallback: If the starting curve is "weird" (Supersingular), the prediction is a tight range of possibilities.
The Tool: They used a new way of looking at "leaky" mathematical functions (Enhanced Differentials) to make these calculations possible.

In short, the authors mapped out the rules of folding in a strange mathematical universe, giving us a perfect map for smooth blankets and a reliable safety net for the knotty ones.

1. Problem Statement

The paper addresses the problem of determining the Ekedahl-Oort (EO) type of a curve $Y$ that is a $\mathbb{Z}/2\mathbb{Z}$ -Galois cover ( $\pi: Y \to X$ ) of a base curve $X$ in characteristic $p=2$ .

Context: The EO type classifies the $p$ -torsion subgroup scheme of the Jacobian, $\text{Jac}(Y)[p]$ , or equivalently, the structure of the algebraic de Rham cohomology $H^1_{dR}(Y)$ as a Dieudonné module.
Known Results:
- The Riemann-Hurwitz formula relates the genus of $Y$ to $X$ and ramification.
- The Deuring-Shafarevich theorem relates the $p$ -rank of $Y$ to that of $X$ and the number of ramification points.
- However, finer invariants like the $a$ -number or the full EO type are not generally determined solely by the base curve's invariants and ramification data.
Specific Challenge: In odd characteristics, even for ordinary base curves, the EO type of a cover is not determined by ramification. The authors investigate whether this changes in characteristic 2, specifically asking: Does the EO type of $Y$ depend only on the EO type of $X$ and the ramification breaks when $X$ is ordinary?

2. Methodology

The authors develop a novel framework to analyze de Rham cohomology directly via differential forms, avoiding standard Čech resolutions which are less natural for studying the interaction between Frobenius ( $F$ ) and Verschiebung ( $V$ ).

A. Enhanced Differentials of the Second Kind

The core technical innovation is the theory of enhanced differentials of the second kind.

Classical Theory: In characteristic 0, a differential of the second kind is a meromorphic differential with zero residues. It is determined up to exact differentials.
Characteristic $p$ Issue: In characteristic $p$ , $d(f^p) = 0$ , so the antiderivative of a meromorphic part is not unique.
Definition: An enhanced differential is a pair $(\omega, (f_Q)_{Q \in S})$ , where $\omega$ is a differential of the second kind and $f_Q$ is a choice of local antiderivative (a "tail") at each pole $Q$ .
Result: The authors prove that $H^1_{dR}(X)$ is isomorphic to the space of enhanced differentials modulo globally exact ones. This allows for a concrete representation of cohomology classes and the explicit computation of the $F$ and $V$ operators.

B. Decomposition of the Dieudonné Module

For the cover $Y$ , the authors decompose the cohomology $H^1_{dR}(Y)$ into three components based on the action of $F$ and $V$ :
$H^1_{dR}(Y) = U \oplus Z \oplus L$

$U$ : $V$ is bijective, $F=0$ (corresponds to the $p$ -rank).
$Z$ : $F$ is bijective, $V=0$ .
$L$ : Both $F$ and $V$ are nilpotent (the "local" part containing the interesting EO structure).
The authors construct a specific basis for a subspace $W \subset L$ using pullbacks of differentials from the base curve $X$ .

C. Symplectic Pairing Analysis

The EO type is determined by the dimensions of subspaces defined by words in $F, V$ , and the symplectic complement ( $\perp$ ). The authors analyze the symplectic pairing $\langle \cdot, \cdot \rangle$ on the constructed enhanced differentials to determine how the "local" contributions from ramification points interact with the base curve's structure.

3. Key Contributions and Results

A. The Ordinary Case (Main Theorem)

When the base curve $X$ is ordinary (i.e., $p$ -rank equals genus, $l_X = 0$ ), the authors prove a complete classification.

Theorem 1.2: If $X$ is ordinary, the EO type of $Y$ is completely determined by the genus of $X$ and the ramification breaks $\{d_i\}$ .
Structure: The Dieudonné module of $Y$ decomposes as:
$D(\text{Jac}(Y)[2]) \cong M_{2g_X - 1 + m}^{\text{ord}} \oplus \bigoplus_{i=1}^m M_{d_i}$
where $M_{d}$ is a specific polarized Dieudonné module determined by the ramification break $d$ .
Significance: This generalizes previous results by Elkin-Pries (for $X=\mathbb{P}^1$ ) and Voloch (for $a$ -numbers). It establishes a "local-to-global" principle for EO types in characteristic 2 when the base is ordinary.

B. The Non-Ordinary Case (Bounds)

When $X$ is not ordinary (e.g., supersingular), the EO type is not uniquely determined by ramification.

Counterexamples: The authors provide examples (e.g., covers of a supersingular elliptic curve) where two covers with identical ramification data have different EO types.
Theoretical Bounds: Despite the lack of uniqueness, they derive logarithmic bounds on the final type.
- Theorem 6.19 & 6.22: For a cover branched at one point with break $d$ , the final type $\nu_l$ satisfies:
  $\left| \nu_l - \left( f_Y + \left\lfloor \frac{l - f_Y}{2} \right\rfloor \right) \right| \leq C \cdot l_X \cdot \lceil \log_2(d-1) \rceil$
  where $l_X = g_X - f_X$ is the "local rank" of the base.
Implication: As the ramification break $d$ grows, the relative error in approximating the EO type vanishes, meaning the EO type becomes increasingly predictable for highly ramified covers.

C. Construction of Families with Constant EO Type

Using these results, the authors construct positive-dimensional families of curves with constant EO types (Corollary 1.5).

For any genus $g$ , there exists a $(g-1)$ -dimensional family of curves with final type $[0, 1, 1, 2, 2, \dots]$ .
These are described as "unlikely intersections" because the codimension of the EO stratum in the moduli space of abelian varieties often exceeds the dimension of the moduli space of curves.

4. Significance

Resolution of a Conjecture: The paper confirms a conjecture (Groen, 2023) that in characteristic 2, the EO type of a cover of an ordinary curve is determined by ramification data.
New Cohomological Tool: The theory of enhanced differentials of the second kind provides a robust method for computing de Rham cohomology and Dieudonné module structures in positive characteristic, offering an alternative to Čech cohomology that is better suited for $F$ and $V$ analysis.
Local-to-Global Principle: The work demonstrates that while EO types are generally global invariants, in characteristic 2 with an ordinary base, they exhibit a strong local-to-global behavior where ramification breaks contribute independently.
Quantitative Bounds: For non-ordinary bases, the paper moves beyond "no determination" to providing precise, logarithmic bounds on the variation of the EO type, quantifying how much the geometry of the base curve influences the cover.

In summary, this paper provides a complete solution for the EO types of $\mathbb{Z}/2\mathbb{Z}$ -covers over ordinary curves in characteristic 2 and establishes tight bounds for the general case, utilizing a new differential-geometric approach to de Rham cohomology.