An unusual family of supersingular curves of genus five in characteristic two

Technical Summary: An Unusual Family of Supersingular Curves of Genus Five in Characteristic Two

Problem Statement
The paper investigates the structure of the supersingular locus $M^{ss}_g$ within the moduli space $M_g$ of smooth curves of genus $g$ over a field $k$ of characteristic $p > 0$. Specifically, the author focuses on the case $g=5$ and $p=2$. While the dimensions of supersingular loci for $g \le 4$ are established in the literature, the case $g=5$ presents a gap in understanding. The expected dimension of any component of $M^{ss}_5$ is 3. However, existing constructions of supersingular families, such as the family $\mathcal{F}_{GV,5}$ by van der Geer and van der Vlugt, have a dimension of only 2. Furthermore, a conjecture by Oort suggests that the generic point of a supersingular component should have a trivial automorphism group (assuming the curve is non-hyperelliptic). The paper seeks to determine if a larger family exists in genus 5 that might challenge these expectations or provide new structural insights.

Methodology
The study proceeds through a combination of computational data analysis and explicit algebraic construction:

1. Data-Driven Motivation: The author utilizes existing datasets of smooth genus 5 curves over $\mathbb{F}_2$ (from [Dra24a]) to identify a specific subset of interest. The data reveals that all non-hyperelliptic, non-trigonal supersingular curves with non-trivial $\mathbb{F}_2$-automorphism groups lie on a specific quadric surface in $\mathbb{P}^4$, namely $X^2 + YZ = 0$.
2. Explicit Parametrization: Motivated by this observation, the author constructs a family of curves $C$ defined as the complete intersection of three quadrics in $\mathbb{P}^4$:
   • $q_1 = X^2 + YZ$
   • $q_2 = XY + ZT + T^2$
   • $q_3 = b_2XZ + b_3XT + b_3^2YT + b_8ZT + ZU + U^2$
     where $b_2, b_3, b_8 \in \mathbb{F}_2$ and $b_3 \neq 0$.
3. Invariant Computation: The paper employs the Kudo-Harashita method to compute the Hasse-Witt matrix $H$ for these curves. The $p$-rank (specifically the 2-rank) and the $a$-number are derived from the rank of $H$ and its iterated Frobenius products.
4. Automorphism and Cover Analysis: The author analyzes the projective transformations in $\text{PGL}_5(\mathbb{F}_2)$ that fix the defining quadrics to determine the automorphism group $\text{Aut}(C)$. Additionally, the function fields of quotient curves $C/\langle \iota \rangle$ and $C/\langle \eta \rangle$ are computed to establish double cover structures.
5. Supersingularity Verification: Supersingularity is proven by demonstrating that the Jacobian $J_C$ is isogenous to a product of Jacobians of lower-genus curves ($g=1$ and $g=2$) which are themselves supersingular.

Key Contributions and Results
The main result is the construction of a 3-dimensional family $\mathcal{S}$ of smooth supersingular curves of genus 5 over $\mathbb{F}_2$, satisfying the following properties (Theorem 1.1 / Theorem 4.1):

• Dimension: The dimension of the family $\mathcal{S}$ is 3, which matches the expected dimension of the supersingular locus in genus 5. This is strictly larger than the dimension of the previously known family $\mathcal{F}_{GV,5}$ (which has dimension 2).
• Automorphisms: The generic curve in $\mathcal{S}$ possesses a non-trivial automorphism group containing $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. This contrasts with the expectation that generic supersingular curves should have trivial automorphism groups.
• Covering Structures: Every curve in the family admits double covers over:
  • An elliptic curve $E$ (specifically the unique supersingular elliptic curve $y^2+y=x^3$).
  • A smooth curve $D$ of genus 2.
• Invariants: The curves have a 2-rank of 0 and an $a$-number of 2. The $a$-number of 2 is notable because the generic point of the supersingular locus in the moduli space of abelian varieties $A_g$ typically has an $a$-number of 1.
• Explicit Equations: The paper provides explicit equations for the curves and their automorphisms, allowing for concrete verification of these properties.

Significance and Claims
The paper claims that the family $\mathcal{S}$ exhibits "rather unusual properties" for a supersingular locus:

1. Dimensional Match: It is the first known family in genus 5 that achieves the expected dimension of the supersingular locus (3), whereas previous constructions were lower-dimensional.
2. Counterexample Potential: If the closure of $\mathcal{S}$ constitutes a component of $M^{ss}_5$, it would serve as a counterexample to the analog of Oort's conjecture for curves in genus 5 and characteristic 2, as the generic point has a non-trivial automorphism group. The author notes that while they cannot definitively prove $\mathcal{S}$ is a full component, the dimensionality and properties make this a strong possibility.
3. Structural Uniqueness: The family lies in the intersection of loci of double covers of elliptic curves and genus 2 curves, and possesses an $a$-number of 2, distinguishing it from the generic behavior of supersingular abelian varieties.

The author concludes that while the question of whether $\mathcal{S}$ is a full component of $M^{ss}_5$ remains open, the construction provides a significant new object of study in the arithmetic geometry of curves in characteristic 2.