On maximal curves which are not Galois subcovers of the Hermitian curve

TL;DR

This paper proves that the generalized Giulietti-Korchmárós curve cannot be obtained as a Galois subcover of the Hermitian curve over certain finite fields, addressing a previously open question.

Contribution

It establishes a non-Galois subcover property for the generalized Giulietti-Korchmárós curve, expanding understanding of maximal curves and their relation to Hermitian curves.

Findings

The generalized Giulietti-Korchmárós curve is not a Galois subcover of the Hermitian curve.

Answers an open question by Garcia, G"uneri, and Stichtenoth.

Provides insights into the structure of maximal curves over finite fields.

Abstract

We show that the generalized Giulietti-Korchm\'aros curve is not a Galois subcover of the Hermitian curve over $\mathbb{F}_{q^{2n}}$. This answers a question raised by Garcia, G\"uneri and Stichtenoth.