Related papers: Maximal curves from subcovers of the GK-curve

For every $q=l^3$ with $l$ a prime power greater than 2, the GK curve $X$ is an $F_{q^2}$-maximal curve that is not $F_{q^2}$-covered by any $F_{q^2}$-maximal Deligne-Lusztig curve. Interestingly, $X$ has a very large $F_{q^2}$-automorphism…

A (projective, geometrically irreducible, non-singular) curve $\mathcal{X}$ defined over a finite field $\mathbb{F}_{q^2}$ is maximal if the number $N_{q^2}$ of its $\mathbb{F}_{q^2}$-rational points attains the Hasse-Weil upper bound, that…

The Deligne-Lusztig curves associated to the algebraic groups of type $^2A_2$, $^2B_2$, and $^2G_2$ are classical examples of maximal curves over finite fields. The Hermitian curve $\mathcal H_q$ is maximal over $\mathbb F_{q^2}$, for any…

For each prime power $\ell$ the plane curve $\mathcal X_\ell$ with equation $Y^{\ell^2-\ell+1}=X^{\ell^2}-X$ is maximal over $\mathbb{F}_{\ell^6}$. Garcia and Stichtenoth in 2006 proved that $\mathcal X_3$ is not Galois covered by the…

A problem of current interest, also motivated by applications to Coding theory, is to find explicit equations for \textit{maximal} curves, that are projective, geometrically irreducible, non-singular curves defined over a finite field…

In this article we construct for any prime power $q$ and odd $n \ge 5$, a new $\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\" uneri--Stichtenoth maximal curves, our curves generalize the Giulietti--Korchm\'aros…

We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of \mathcal H by several automorphisms groups.…

Let $\mathbb{F}$ be the finite field of order $q^2$, $q=p^h$ with $p$ prime. It is commonly atribute to J.P. Serre the fact that any curve $\mathbb{F}$-covered by the Hermitian curve $\mathcal{H}_{q+1}:\, y^{q+1}=x^q+x$ is also…

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.

In the study of algebraic curves with many points over a finite field, a well known general problem is to understanding better the properties of $\mathbb{F}_{q^2}$-maximal curves whose genera fall in the higher part of the spectrum of the…

In this paper we compute the automorphism group of the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ introduced in Tafazolian et al. in 2016 as new examples of maximal curves which cannot be covered by the Hermitian curve. They…

In this paper we characterize the genera of those quotient curves $\mathcal{H}_q/G$ of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_q$ for which $G$ is contained in the maximal subgroup $\mathcal{M}_1$ of ${\rm…

We investigate the genera of quotient curves $\mathcal H_q/G$ of the $\mathbb F_{q^2}$-maximal Hermitian curve $\mathcal H_q$, where $G$ is contained in the maximal subgroup $\mathcal M_q\leq{\rm Aut}(\mathcal H_q)$ fixing a pole-polar pair…

We determine the full automorphism group of two recently constructed families $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ of maximal curves over finite fields. These curves are covers of the Suzuki and Ree curves, and are analogous…

The second generalized GK function fields $K_n$ are a recently found family of maximal function fields over the finite field with $q^{2n}$ elements, where $q$ is a prime power and $n \ge 1$ an odd integer. In this paper we construct many…

Some new results on plane F_{q^2}-maximal curves are stated and proved. It is known that the degree d of such curves is upper bounded by q+1 and that d=q+1 if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that d\le…

Let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements. Most of the known $\mathbb{F}_{q^2}$-maximal curves arise as quotient curves of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_{q}$. After a seminal paper by…

In 2017 Skabelund constructed two new examples of maximal curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the…

We prove two "large images" results for the Galois representations attached to a degree $d$ Q-curve $E$ over a quadratic field $K$: if $K$ is arbitrary, we prove maximality of the image for every prime $p >13$ not dividing $d$, provided…

The second generalized GK maximal curves $\mathcal{GK}_{2,n}$ are maximal curves over finite fields with $q^{2n}$ elements, where $q$ is a prime power and $n \geq 3$ an odd integer, constructed by Beelen and Montanucci. In this paper we…