Year: 2026
Type: preprint
Abstract: For every prime power $q$ and integer $n\ge1$, 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 $n\ge5$. For $q>2$ and even $n\ge4$, 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 $n\ge5$, we remove the Galois hypothesis from the known non-covering results for the Beelen--Montanucci and Garcia--Güneri--Stichtenoth curves. For even $n\ge4$, 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.
Source: arXiv (Cornell University)
Authors: Ma, Liming, Wang, Yipeng
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213598202