A new family of maximal curves not covered by the Hermitian curve

Abstract

For every prime power $q$ and integer $n\ge 1$, we give a single Kummer equation defining an ${\mathbb{F}}_{q^{2n}}$-maximal curve ${\mathcal{X}}_{q,n}$. The family recovers the Hermitian curves for $n=1,2$, the Giulietti--Korchmáros curve for $n=3$, and the Beelen--Montanucci curves for odd $n\ge 5$. For $q>2$ and even $n\ge 4$, it gives new maximal curves of genus $(q^2-1)q^n/2$ that are not covered by the Hermitian curve over ${\mathbb{F}}_{q^{2n}}$. We compute the genus for both parities of $n$ and give a uniform proof of maximality by counting rational places. For $q>2$ and odd $n\ge 5$, we remove the Galois hypothesis from the known non-covering results for the Beelen--Montanucci and Garcia--Güneri--Stichtenoth curves. For even $n\ge 4$, we determine the full automorphism group,of order $(q^n+1)q(q^2-1)$, and give explicit equations and genera for all Galois quotients, with closed-form equations for several classes of subgroups.