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Maximal Subcovers of the Skabelund Curve: Uniqueness via Genus and Automorphism Groups

Gilberto B. Almeida Filho, Saeed Tafazolian, Stéfani C. Vieira

TL;DR

The paper proves a rigidity phenomenon for a family of maximal curves arising as intermediate covers of the Suzuki-based Skabelund curve over $\\mathbb{F}_{q^4}$. By explicitly constructing the family $\\mathcal{S}_s$, computing their genera, Weierstrass semigroups at rational points, and full automorphism groups $\\operatorname{Aut}(\\mathcal{S}_s)$, it shows that each curve is uniquely determined up to isomorphism by the pair $(g,\\mathrm{Aut})$, with the Skabelund curve itself singled out in this way. The results hinge on lifting properties of automorphisms from the Suzuki curve $S_q$, a precise analysis of ramification in the cyclic Kummer extensions, and the computation of a symmetric Weierstrass semigroup at a canonical rational point. This rigidity provides a precise classification within Suzuki-type covers and highlights the interplay between genus, automorphism structure, and ramification in maximal curves over finite fields.

Abstract

We establish a rigidity phenomenon for a family of intermediate covers of the Skabelund curve over $\mathbb{F}_{q^4}$. The Skabelund curve, introduced by D.~Skabelund as a cyclic cover of the Suzuki curve, is a maximal curve with a large automorphism group and plays a central role in the theory of maximal curves over finite fields. For the intermediate covers arising from this construction, we determine their full automorphism groups and compute the Weierstrass semigroups at all $\mathbb{F}_{q^4}$-rational points. Using these structural and arithmetic invariants, we prove that each curve in the family is uniquely determined, up to isomorphism over its field of definition, by the pair consisting of its genus and its full automorphism group. This provides a rigidity-type classification of intermediate Suzuki-type covers; in particular, the Skabelund curve itself is uniquely characterized within this family by its genus and automorphism group.

Maximal Subcovers of the Skabelund Curve: Uniqueness via Genus and Automorphism Groups

TL;DR

The paper proves a rigidity phenomenon for a family of maximal curves arising as intermediate covers of the Suzuki-based Skabelund curve over . By explicitly constructing the family , computing their genera, Weierstrass semigroups at rational points, and full automorphism groups , it shows that each curve is uniquely determined up to isomorphism by the pair , with the Skabelund curve itself singled out in this way. The results hinge on lifting properties of automorphisms from the Suzuki curve , a precise analysis of ramification in the cyclic Kummer extensions, and the computation of a symmetric Weierstrass semigroup at a canonical rational point. This rigidity provides a precise classification within Suzuki-type covers and highlights the interplay between genus, automorphism structure, and ramification in maximal curves over finite fields.

Abstract

We establish a rigidity phenomenon for a family of intermediate covers of the Skabelund curve over . The Skabelund curve, introduced by D.~Skabelund as a cyclic cover of the Suzuki curve, is a maximal curve with a large automorphism group and plays a central role in the theory of maximal curves over finite fields. For the intermediate covers arising from this construction, we determine their full automorphism groups and compute the Weierstrass semigroups at all -rational points. Using these structural and arithmetic invariants, we prove that each curve in the family is uniquely determined, up to isomorphism over its field of definition, by the pair consisting of its genus and its full automorphism group. This provides a rigidity-type classification of intermediate Suzuki-type covers; in particular, the Skabelund curve itself is uniquely characterized within this family by its genus and automorphism group.

Paper Structure

This paper contains 6 sections, 17 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. An explicit covering of the Suzuki curve
  4. The automorphism group of $\mathcal{S}_s$
  5. The Weierstrass Semigroup for rational points
  6. Uniqueness via Genus and Automorphism Groups

Key Result

Theorem 1

The curve $\mathcal{S}_s$ is maximal over $\mathbb{F}_{q^4}$.

Theorems & Definitions (34)

  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 24 more