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

Dušan Dragutinović

TL;DR

The paper constructs a 3-dimensional family of smooth supersingular genus $5$ curves in characteristic $2$, each possessing a nontrivial automorphism group and admitting a double cover to an elliptic curve and to a genus $2$ curve. By restricting to curves canonically embedded on the quadric $X^2+YZ=0$ in $\mathbb{P}^4$ and applying a systematic parameter reduction, the authors obtain an explicit semimodel and a 4-parameter subfamily with $2$-rank $0$, from which they deduce supersingularity and compute the $a$-number. They show that the generic member has $a(C)=2$ and that the Jacobian is isogenous to the sum of the Jacobians of the quotient curves, all of which are supersingular, yielding $\mathrm{Aut}(C)$ containing a $\mathbb{Z}/2\mathbb{Z}$ subgroup (and sometimes more). The resulting 3-dimensional family $\mathcal{S}$ matches the expected dimension of the supersingular locus in $\mathcal{M}_5$ and provides a potential counterexample to an Oort-type conjecture in genus $5$ and characteristic $2$, while also informing the interplay between double covers and supersingularity. Overall, the work combines explicit projective models, Hasse–Witt computations, and isogeny considerations to illuminate the landscape of genus $5$ supersingular curves in characteristic $2$.

Abstract

We construct a family of smooth supersingular curves of genus $5$ in characteristic $2$ with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus $5$, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-$2$ curve. We also provide an explicit parametrization of this family.

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

TL;DR

The paper constructs a 3-dimensional family of smooth supersingular genus curves in characteristic , each possessing a nontrivial automorphism group and admitting a double cover to an elliptic curve and to a genus curve. By restricting to curves canonically embedded on the quadric in and applying a systematic parameter reduction, the authors obtain an explicit semimodel and a 4-parameter subfamily with -rank , from which they deduce supersingularity and compute the -number. They show that the generic member has and that the Jacobian is isogenous to the sum of the Jacobians of the quotient curves, all of which are supersingular, yielding containing a subgroup (and sometimes more). The resulting 3-dimensional family matches the expected dimension of the supersingular locus in and provides a potential counterexample to an Oort-type conjecture in genus and characteristic , while also informing the interplay between double covers and supersingularity. Overall, the work combines explicit projective models, Hasse–Witt computations, and isogeny considerations to illuminate the landscape of genus supersingular curves in characteristic .

Abstract

We construct a family of smooth supersingular curves of genus in characteristic with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus , its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus- curve. We also provide an explicit parametrization of this family.

Paper Structure

This paper contains 11 sections, 9 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariants of curves in positive characteristic
  4. The $p$-rank and the $a$-number of a smooth curve
  5. The Newton polygon of a smooth curve and supersingular curves
  6. The purity of de Jong and Oort
  7. Computing Hasse-Witt matrices and automorphism groups of curves
  8. Curves of genus $5$ in characteristic $2$ lying on $X^2 + YZ = 0$ in $\mathbb{P}^4$
  9. Reducing the number of parameters
  10. Properties of curves $\{q_1 = 0, q_2 = 0, q_3 = 0\} \subseteq \mathbb{P}^4$ with $q_3$ as in \ref{['eqn:genus5_2rank0_4dim_family']}
  11. Proof of the main theorem and some remarks

Key Result

Theorem 1.1

Let $b_1, b_2, b_3 \in \overline{\mathbb{F}}_2$, $b_3 \neq 0$ and let $C = V(q_1, q_2, q_3) \subseteq \mathbb{P}^4$, where Then, $C$ is a smooth canonical curve of genus $5$ over $\overline{\mathbb{F}}_2$ which is supersingular and satisfies the following properties: Furthermore, let $\mathcal{S}$ denote the family which consists of the isomorphism classes of all curves $C$ as in eqn:supersingul

Theorems & Definitions (20)

  • Theorem 1.1
  • Example 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 10 more