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The $p$ -rank stratification of the moduli space of double covers of a fixed elliptic curve

This paper investigates the $p$ -rank stratification of the moduli space of genus $g$ curves admitting a double cover to a fixed elliptic curve in characteristic $p>2$ , proving that its closed strata are equidimensional of the expected dimension and that smooth double covers exist for all possible $p$ -rank values.

Original authors: Kevin Chang, Dušan Dragutinović, Steven R. Groen, Yuxin Lin, Natalia Pacheco-Tallaj, Deepesh Singhal

Published 2026-03-03

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CC BY 4.0

Original authors: Kevin Chang, Dušan Dragutinović, Steven R. Groen, Yuxin Lin, Natalia Pacheco-Tallaj, Deepesh Singhal

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 an architect designing a vast, infinite city called Moduli City. This city isn't made of brick and mortar, but of mathematical shapes called curves. In this city, every building has a specific number of "rooms" (genus) and a unique "fingerprint" called the p-rank.

The p-rank is like a measure of how "connected" or "complex" a building is in a specific mathematical universe (characteristic $p$ ). Some buildings are very simple (low p-rank), while others are incredibly intricate (high p-rank).

The Mission: Mapping the Neighborhoods

The authors of this paper are cartographers. Their goal is to map out a specific neighborhood in Moduli City: the Double Cover District.

In this district, every building (a curve of genus $g$ ) is constructed by taking a specific, fixed "parent" building (an elliptic curve, which is a fancy donut shape) and wrapping a new structure around it twice. Think of it like taking a single loop of string (the elliptic curve) and wrapping a second, larger loop around it, creating a double-layered structure.

The question the authors ask is: "If we build these double-layered structures, what are all the possible p-ranks we can get? And how many different ways can we build them for each rank?"

The Challenge: Smooth vs. Cracked

In mathematics, it's often easy to build "cracked" or "singular" structures (buildings with corners or joints where pieces are glued together). It's much harder to build "smooth" structures (perfectly round, seamless buildings).

Usually, when mathematicians try to find a smooth building with a specific p-rank, they hit a wall. They can't find one, or they don't know if it exists.

The Authors' Secret Weapon: The Boundary
The authors realized that while smooth buildings are hard to find, the "cracked" ones at the edge of the city (the boundary) are easy to construct. You can just glue two smaller, well-understood buildings together at a single point.

They used a clever trick:

They studied the cracked buildings at the edge of the district.
They calculated the dimensions (the "size" or "volume") of the space these cracked buildings occupy.
They proved that the space of smooth buildings is just as "big" as the space of cracked ones.
Therefore, if you can find a cracked building with a certain p-rank, you can be almost certain that a smooth building with that same p-rank exists nearby, even if you can't see it immediately.

The Main Discoveries

1. The Map is Complete and Neat (Theorem 1.1)
The authors proved that the "p-rank neighborhoods" in this Double Cover District are perfectly organized.

Pure Dimensions: If you look at all the buildings with a p-rank of, say, 3, they form a neat, solid block of space. They aren't scattered randomly; they fill up a specific volume exactly as expected.
No Missing Pieces: There are no "holes" in the map. If a certain p-rank is theoretically possible, there is a whole neighborhood of buildings with that rank.

2. Smooth Buildings Exist (Corollary 1.2)
This is the big "Aha!" moment.

The Good News: For almost every possible p-rank, there exists a perfectly smooth double-layered building.
The One Exception: There is one tiny, weird case (in a specific mathematical universe where $p=3$ and the building has 2 rooms) where a smooth building cannot exist. It's like a rule in physics that says, "You can build a house out of anything, except for a house made of pure light in a vacuum."
The "Singular" Clusters: They also found that for lower p-ranks, there are entire neighborhoods made only of cracked buildings. These are special zones where you simply cannot find a smooth version.

Why Does This Matter? (The "So What?")

The Supersingular Mystery
In this mathematical world, there is a special type of building called a Supersingular building. These are the "superheroes" of the city—they have a p-rank of 0 (the lowest possible).

For a long time, mathematicians wondered: "Can we build a smooth Supersingular building that is a double cover of a parent building?"

For small buildings (genus 2 and 3), the answer is yes (or no, depending on the specific rules).
For a building with 4 rooms, the authors proved that no smooth Supersingular double cover exists. This helps solve a long-standing puzzle about the structure of these mathematical cities.

The Fiber Product Analogy
To prove these things, the authors used a construction technique called a Fiber Product. Imagine you have two different maps of a city. You want to find the intersection where the streets match up perfectly.

They took their fixed "parent" donut (the elliptic curve).
They took a "hyperelliptic" curve (a more complex shape).
They "glued" them together along a common path.
By carefully choosing the shapes of these paths, they could force the resulting double-layered building to have a very low p-rank (like 0 or 1), effectively "engineering" the specific mathematical properties they needed.

Summary

Think of this paper as a guidebook for a very specific, high-tech construction zone.

The Problem: We wanted to know what kinds of double-layered structures could be built and if they could be perfectly smooth.
The Method: Instead of trying to build the smooth ones directly, the authors studied the "cracked" ones at the edge of the construction site to understand the rules of the space.
The Result: They drew a perfect map showing that smooth structures exist for almost every configuration, except for one tiny, forbidden corner. They also solved a mystery about "Supersingular" structures, proving that some of them simply cannot be smooth double covers.

It's a story of using the edges of a problem to understand the center, proving that in the world of abstract math, even the most complex shapes follow a beautiful, predictable order.

1. Problem Statement

The paper investigates the $p$ -rank stratification of the moduli space of smooth curves of genus $g$ ( $g \ge 2$ ) in characteristic $p > 2$ that admit a double cover to a fixed elliptic curve $E$ .

Context: The $p$ -rank of a curve $C$ is defined as the $p$ -rank of its Jacobian, $f_C = \dim_{\mathbb{F}_p} \text{Hom}(\mu_p, \text{Jac}(C))$ . It satisfies $0 \le f_C \le g$ .
The Object of Study: Let $B_{E,g}$ denote the locus of smooth genus- $g$ curves that are double covers of a fixed elliptic curve $E$ . The authors study the closed strata $V_f(B_{E,g})$ , consisting of curves in $B_{E,g}$ with $p$ -rank $\le f$ .
The Challenge: While the $p$ -rank stratification of the full moduli space of curves $\mathcal{M}_g$ and hyperelliptic curves $\mathcal{H}_g$ is well-understood (strata are pure of expected codimension), the behavior of these strata within specific subloci (like bielliptic curves or covers of a fixed curve) is subtler. Specifically, determining the dimension and non-emptiness of these strata, and whether they contain smooth curves, requires new geometric techniques.

2. Methodology

The authors employ a combination of moduli theory, deformation theory, and boundary analysis using Deligne-Mumford stacks.

A. Construction of Moduli Stacks

To analyze $B_{E,g}$ , the authors construct a moduli stack $\mathcal{M}_{E,g}$ (and its marked version $\mathcal{M}_{E,g;n}$ ) that parametrizes admissible double covers $D \to C$ where:

The target curve $C$ is a nodal curve of genus 1 with one component identified with the fixed elliptic curve $E$ and other components being $\mathbb{P}^1$ (bubbles).
The source curve $D$ is a nodal curve of genus $g$ .
The map is an admissible $\mathbb{Z}/2\mathbb{Z}$ -cover (allowing branching at nodes under specific conditions).

This stack $\mathcal{M}_{E,g}$ is a proper, smooth, irreducible Deligne-Mumford stack of dimension $2g-2$ . The locus of smooth curves $B_{E,g}$ is the image of the forgetful map $\mathcal{M}_{E,g} \to \mathcal{M}_g$ (extracting the source curve $D$ ).

B. Boundary Analysis

A core technique is the detailed description of the boundary $\overline{B}_{E,g} \setminus B_{E,g}$ . The authors classify boundary components using clutching morphisms:

Unramified Clutching ( $\Xi$ ): Gluing a double cover of $E$ and a double cover of $\mathbb{P}^1$ at unramified points.
Ramified Clutching ( $\Delta$ ): Gluing at ramification points.
They prove that the boundary has codimension 1 and that the smooth locus is dense. This allows them to use the boundary to infer properties of the interior (smooth locus).

C. Dimensional Arguments and Purity

The authors utilize the Purity Theorem (Faber-van der Geer, de Jong-Oort), which states that if a stratum of $p$ -rank $\le f$ is non-empty in an irreducible scheme, it is pure of codimension 1 relative to the stratum of $p$ -rank $\le f+1$ .

They compute the dimensions of $p$ -rank strata on the boundary components (which involve hyperelliptic curves and lower-genus bielliptic curves).
By induction on the genus $g$ , they establish that the dimension of the strata in the interior matches the expected dimension derived from the boundary.

D. Explicit Constructions

To prove the existence of smooth curves with specific $p$ -ranks, the authors construct explicit families using fiber products.

They consider the fiber product $D' = E \times_{\mathbb{P}^1} H$ , where $E$ is the fixed elliptic curve and $H$ is a hyperelliptic curve.
By analyzing the Hasse-Witt matrix of the resulting curve $D$ (the normalization of $D'$ ), they impose conditions on the branching locus and the characteristic $p$ to force the $p$ -rank to be low (e.g., $f_D \approx g/2$ or even $0$).

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

For any fixed elliptic curve $E$ with $p$ -rank $f_E \in \{0, 1\}$ , and for any $g \ge 2$ :
The closed $p$ -rank stratum $V_f(B_{E,g})$ (curves of $p$ -rank $\le f$ ) is pure of dimension:
$\dim V_f(B_{E,g}) = g - 2 + f - f_E$
for all $f_E \le f \le g - 1 + f_E$ .

Corollary for Bielliptic Locus: Consequently, the $p$ -rank strata of the full bielliptic locus $B_g$ (curves admitting some double cover to some elliptic curve) are pure of dimension $g - 2 + f$ .

Existence of Smooth Curves (Corollary 1.2)

The authors prove the existence of smooth curves with prescribed $p$ -ranks:

Ordinary Case ( $f_E=1$ ): For any $1 \le f \le g$ , there exists a smooth bielliptic curve of genus $g$ and $p$ -rank $f$ .
Supersingular Case ( $f_E=0$ ): For any $0 \le f \le g$ $0 \leq f \leq g$ , there exists a smooth bielliptic curve of genus $g$ $g$ and $p$ $p$ -rank $f$ $f$ , except for the specific case $(p, g, f) = (3, 2, 0)$ $(p, g, f) = (3, 2, 0)$ .
- Note: The exception at $p=3, g=2, f=0$ arises because there are no smooth superspecial curves of genus 2 in characteristic 3.

Supersingular Curves

The paper addresses the existence of supersingular bielliptic curves (curves with $p$ -rank 0 and Newton polygon $(1/2, \dots, 1/2)$ ):

Genus 2: Only finitely many exist (0-dimensional stratum).
Genus 3: The locus is 1-dimensional.
Genus 4: The generic point of the supersingular locus in $\mathcal{M}_4$ is not a double cover of an elliptic curve (dimension 2 vs. expected 3 or 4).
Genus 5: The authors construct explicit examples of supersingular double covers of elliptic curves for $p \equiv 11 \pmod{12}$ , showing the existence of a 1-dimensional family.

4. Significance

Refinement of Stratification: This work extends the understanding of $p$ -rank stratifications from the general moduli space $\mathcal{M}_g$ and hyperelliptic locus $\mathcal{H}_g$ to the bielliptic locus and covers of fixed curves. It confirms that these subloci behave "generically" regarding purity and dimension.
Resolution of Existence Questions: It settles the question of whether smooth bielliptic curves exist for all possible $p$ -ranks, providing a complete classification with a single known exception in characteristic 3.
Connection to Oort's Conjecture: The results are relevant to Oort's conjecture regarding the automorphism groups of supersingular abelian varieties. The analysis of the $p$ -rank 0 stratum in $B_4$ was instrumental in proving the conjecture for $g=4, p>2$ (as cited in the introduction).
Methodological Innovation: The paper provides a robust framework for studying covers of fixed curves by utilizing the boundary of the moduli space of admissible covers. This "boundary-to-interior" dimension argument is a powerful tool for similar problems in arithmetic geometry.
Explicit Constructions: The explicit construction of low $p$ -rank curves via fiber products and Hasse-Witt matrix analysis offers concrete examples for testing conjectures about supersingular curves in higher genera.