The automorphism groups of a family of maximal curves
Let \(q=p^h\) be a prime power, and let \(n\geq 3\) be an odd integer. The generalized GK curve \(\mathcal C_n\) is a maximal curve defined over \(\mathbb F_q^{2n}\) by the equations \[ \mathcal H_q:~x^q+x-y^{q+1}=0, ~~ \mathcal X_n:~y^{q^2}-y-z^m. \] In this paper, the authors determine the automorphism group of this curve \(\mathcal C_n\). As it turns out, the automorphism group is a semidirect product \(\Gamma\) of the form \(Q\times\Sigma\), where \(\Sigma\) is a cyclic group of order \((q^n+1)(q-1)\) and \(Q\) is a non-abelian group of order \(q^3\). Their proof uses ideas from ramification theory and a new structural result about automorphism groups of curves in characteristic \(p\). It is worth noting that \textit{C. Güneri} et al. [Adv. Geom. 13, No. 2, 369--380 (2013; Zbl 1285.11093)] determined the automorphism group of the same curve using a different approach based on Riemann-Roch theory.
scientific article