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

TL;DR

This paper proves that intermediate covers of the Skabelund curve are uniquely determined by their genus and automorphism group, establishing a rigidity phenomenon that classifies these maximal curves over finite fields.

Contribution

It introduces a rigidity classification for intermediate Skabelund curve covers, showing they are uniquely identified by genus and automorphism group.

Findings

Full automorphism groups determined for the covers

Weierstrass semigroups computed at rational points

Each curve uniquely characterized by genus and automorphism group

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.