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Intersections of the automorphism and the Ekedahl-Oort strata in $M_2$

This paper investigates the intersections between automorphism strata and Ekedahl-Oort strata within the moduli space of genus two curves by explicitly classifying automorphism groups in positive characteristic, parametrizing their families, and developing an algorithm to compute the dimension and irreducible components of these intersections.

Original authors: Alvaro Gonzalez-Hernandez

Published 2026-04-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alvaro Gonzalez-Hernandez

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 cartographer trying to map a very strange, high-dimensional landscape called Moduli Space. This isn't a map of mountains or rivers, but a map of shapes—specifically, a special family of curved surfaces known as genus two curves (think of them as two-holed donuts).

The author of this paper, Álvaro González-Hernández, is trying to answer a very specific question: "If I pick a specific type of symmetry for my two-holed donut, and I also pick a specific type of 'arithmetic weather' (based on a prime number $p$ ), does a shape exist that fits both descriptions? And if it does, how many such shapes are there?"

Here is a breakdown of the paper's journey, using simple analogies:

1. The Map and the Coordinates (The Igusa Invariants)

To navigate this landscape, you need coordinates. In the world of these curves, mathematicians use a set of five numbers called Igusa invariants (like $J_2, J_4, J_6, J_8, J_{10}$ ).

The Analogy: Think of these invariants as the DNA of the curve. Just as your DNA determines your height, eye color, and other traits, these five numbers determine the shape of the curve. If two curves have the same DNA, they are essentially the same shape.
The paper uses these "DNA markers" to build a coordinate system (a moduli space) where every point represents a unique curve.

2. The Two Types of Filters

The author applies two different "filters" to this map to see which shapes survive.

Filter A: The Symmetry Strata (Automorphism Groups)

Some curves are very symmetrical. You can rotate or flip them, and they look exactly the same.

The Analogy: Imagine a snowflake. It has high symmetry; you can rotate it by 60 degrees, and it looks identical. A generic rock has low symmetry.
The paper categorizes curves by their "symmetry group" (e.g., $C_2$ , $D_4$ , $GL_2(\mathbb{F}_3)$ ). These groups are like the symmetry score of the curve.
The author maps out exactly where these highly symmetrical curves live on the map. For example, there is a specific "neighborhood" for curves with a symmetry group of order 48, and another for those with order 10.

Filter B: The Arithmetic Weather (Ekedahl-Oort Strata)

This is the trickier part. The curves exist in a world with "positive characteristic" (a mathematical universe based on a prime number $p$ , like 2, 3, 5, etc.). In this world, curves have an "arithmetic personality" defined by two numbers:

p-rank ( $f$ ): How "connected" the curve is to the prime number $p$ .
a-number ( $a$ ): A measure of how "rigid" or "flexible" the curve is in this arithmetic world.

The Analogy: Imagine the curve is a boat. The p-rank is how many engines it has (0, 1, or 2). The a-number is how much it rocks in the waves.
The Ekedahl-Oort strata are regions on the map where the boats have specific engine counts and rocking patterns.

3. The Main Event: The Intersections

The core of the paper is finding the intersection of these two filters.

The Question: "Is there a curve that has Symmetry Group X AND Arithmetic Weather (f, a)?"
The Result: The author creates a giant table (like a weather report for symmetrical shapes).
- For some combinations, the answer is "No such shape exists" (the intersection is empty).
- For others, the answer is "Yes, and here is exactly how many dimensions of space they occupy."
- For example, if you are in a world where $p=2$ (characteristic 2), and you want a curve with a specific high symmetry, the paper tells you exactly which "arithmetic weather" it can have.

4. The "Donut" and the "Elliptic Curves"

One of the most beautiful parts of the paper (Section 3.3) deals with curves that have a specific symmetry ( $C_2^2$ ).

The Analogy: The author discovers that these specific symmetrical curves are actually made by gluing two simpler curves (elliptic curves, or one-holed donuts) together.
It's like taking two separate circles, cutting them, and sewing them together to make a figure-eight (a two-holed donut).
The paper proves that the map of these symmetrical curves is essentially the same as the map of pairs of one-holed donuts. This allows the author to use known information about the simpler donuts to understand the complex ones.

5. The "Supersingular" Mystery

The paper also looks at the most extreme arithmetic weather: Supersingular curves (where the p-rank is 0).

The Analogy: These are the "super-stiff" boats that don't move at all in the waves.
The author calculates exactly how many distinct "islands" (irreducible components) exist in the region where highly symmetrical curves meet this super-stiff weather.
The number of these islands depends on the prime number $p$ in a very specific, formulaic way involving things called Legendre symbols (which are like a mathematical way of checking if a number is a "square" in that specific arithmetic world).

Summary of Findings

The Map is Complete: The author has successfully mapped out where symmetrical curves live and what their arithmetic properties are.
The Rules are Strict: Not every combination of symmetry and arithmetic weather is possible. The paper provides a definitive "Yes/No" list for every prime number.
The Connection: Highly symmetrical curves are deeply connected to pairs of simpler curves (elliptic curves).
The Tool: The author used a computer program (Magma) to do the heavy lifting of calculating these complex formulas, essentially letting the computer crunch the numbers to find the patterns that human intuition might miss.

In short, this paper is a comprehensive catalog that tells us exactly which "symmetrical, two-holed donuts" can exist in different mathematical universes (defined by prime numbers), and it explains the geometry of how they fit together.

1. Problem Statement

The paper addresses a fundamental question in the arithmetic geometry of moduli spaces: Given a prime $p$ , a $p$ -rank $f$ , an $a$ -number $a$ , and a finite group $G$ , does there exist a genus two curve over a field of characteristic $p$ such that:

Its automorphism group is exactly $G$ (or contains $G$ ).
Its Jacobian variety has $p$ -rank $f$ and $a$ -number $a$ .

If such curves exist, the paper aims to characterize the locus of these curves within the coarse moduli space of genus two curves, $M_2$ . Specifically, it seeks to determine the dimension and the number of irreducible components of the intersection between the automorphism strata (curves with specific symmetries) and the Ekedahl-Oort (EO) strata (curves whose Jacobians have specific $p$ -torsion structures).

2. Methodology

The author employs a combination of explicit algebraic geometry, computational algebra, and theoretical classification:

Moduli Space Representation: The study is conducted within the coarse moduli space $M_2$ , represented via Igusa invariants ( $J_2, J_4, J_6, J_8, J_{10}$ ). The space is realized as a weighted projective space $\mathbb{P}(1,2,3,4,5)$ subject to a specific relation, with the discriminant locus removed.
Automorphism Strata Construction:
- The paper classifies all possible automorphism groups for genus two curves in various characteristics ( $p=2, 5$ and $p \geq 7$ ).
- It constructs explicit universal models for curves in each stratum (e.g., $W_{\geq G}$ ) by imposing symmetry conditions on the curve equation coefficients.
- For the stratum $W_{\geq C_2^2}$ , the author establishes a geometric connection to the moduli space of pairs of elliptic curves with level 2 structures, denoted as $M_{ell}$ .
Ekedahl-Oort Strata Computation:
- The EO strata are determined by the Hasse-Witt matrix ( $M_{HW}$ ) of the curve's Jacobian.
- The author utilizes the Cartier-Manin matrix to compute $M_{HW}$ explicitly from the curve coefficients.
- The $p$ -rank ( $f$ ) and $a$ -number ( $a$ ) are derived from the rank and nilpotency properties of $M_{HW}$ (e.g., $f=2$ iff $\text{rank}(M_{HW})=2$ ; $f=0, a=2$ iff $M_{HW}=0$ ).
Computational Tools: The author uses Magma to compute the relations between Igusa invariants for specific families of curves, verify irreducibility, and calculate the number of components for specific primes.

3. Key Contributions

A. Classification and Parametrization of Automorphism Strata

The paper provides a comprehensive, characteristic-uniform description of the geometry of automorphism strata:

Explicit Models: It extends previous work (Cardona and Quer) to provide universal models for curves with automorphism groups $C_2^2$ , $D_4$ , $D_6$ , and others in all characteristics, including the exceptional cases $p=2$ and $p=5$ .
Geometric Connection: A significant contribution is Theorem 3.3, which proves an isomorphism between the moduli space of pairs $(C, \sigma)$ (where $C$ has $C_2^2$ symmetry and $\sigma$ is a non-hyperelliptic involution) and the moduli space of pairs of elliptic curves with level 2 structures. This allows the stratum $W_{\geq C_2^2}$ to be understood via the geometry of elliptic curves.

B. Algorithm for EO Strata

The paper details an algorithm to compute the equations defining the pullback of EO strata into $M_2$ . By expressing the Hasse-Witt matrix entries as polynomials in the curve coefficients (or Igusa invariants), the author defines the loci where specific $p$ -ranks and $a$ -numbers occur.

C. Intersection Analysis

The core result is the complete enumeration of the intersections $V_{(f,a)} \cap W_{=G}$ (curves with specific EO type and specific automorphism group).

Dimensions: The paper provides tables (Tables 3–7) listing the dimensions of these intersections for all primes $p$ $p$ .
- Key Finding: For curves with automorphism groups larger than $C_2^2$ (e.g., $D_4, D_6$ ), the $p$ -rank of the Jacobian can only be $0$ or $2$. It is impossible to have a $p$ -rank of $1$ for these highly symmetric curves (Corollary 5.3).
Irreducibility:
- For $p \leq 5$ , all non-empty intersections are irreducible.
- For $p \geq 7$ , the paper provides explicit formulas for the number of irreducible components, $n(f,a),G$ .
- Key Finding: The number of components for the intersection of the $C_2^2$ stratum with the $p$ -rank $\leq 1$ locus is exactly the number of supersingular elliptic curves in characteristic $p$ (Proposition 5.6).

4. Key Results Summary

Feature	Result
Automorphism Groups	Complete list for all $p$ , including $C_2^2, D_4, D_6, C_{10}, GL_2(\mathbb{F}_3)$ , and exceptional groups in $p=2,5$ .
Stratum $W_{\geq C_2^2}$	Isomorphic to a quotient of pairs of elliptic curves with level 2 structures. Rational in all characteristics.
$p$ -Rank Constraints	If $\text{Aut}(C) > C_2^2$ , then $p\text{-rank}(J(C)) \in \{0, 2\}$ . No genus 2 curve with high symmetry has $p$ -rank 1.
Supersingular Locus	The intersection of the $C_2^2$ stratum with the supersingular locus ( $f=0, a=2$ ) has $n$ components, where $n$ is the class number of supersingular elliptic curves.
Rationality	The locus $V_{\leq 1}$ (curves with $p$ -rank $\leq 1$ ) is rational for $p \leq 19$ .
Computational Data	Tables 3–9 provide the dimension and number of irreducible components for all combinations of $G, f, a$ across various primes.

5. Significance

Bridging Symmetry and Arithmetic: The paper successfully links the geometric symmetry of a curve (automorphism group) with the arithmetic properties of its Jacobian (Ekedahl-Oort type). This is a non-trivial connection, as high symmetry often forces constraints on the $p$ -torsion structure.
Explicit Computations: By providing explicit equations and models, the paper moves beyond existence proofs to allow for concrete computations and verification using software like Magma. This is crucial for testing conjectures in higher genus or different characteristics.
Foundational for Higher Genus: The methodology and results serve as a blueprint for studying similar intersections in $M_3$ or the hyperelliptic locus, as suggested in the "Future Directions" section.
Clarification of Supersingular Loci: The result connecting the components of the $C_2^2$ stratum intersection to the number of supersingular elliptic curves provides a deeper understanding of how the supersingular locus of $A_2$ (abelian surfaces) is distributed within the moduli space of curves.

In summary, this work provides a definitive, computationally verified map of how the stratification of genus two curves by symmetry intersects with the stratification by $p$ -torsion, resolving questions about existence, dimension, and irreducibility for all characteristics.