Transitive automorphism groups of maximal curves

Abstract

Let $q=p^h$ be an odd prime power and let $X/\F_{q^2}$ be a maximal curve of genus at least two. We classify the curves for which the full geometric automorphism group is transitive on the set of $\F_{q^2}$-rational points. We prove that, if $h>1$, then $X$ is the Hermitian curve. If $h=1$, the only additional possibility occurs for $q=5$: the unique $\F_{25}$-maximal genus-three curve, namely the maximal $S_4$-model of the Klein quartic. The proof separates tame and wild actions. In the tame case, genus bounds, signatures of quotient maps, specialization to characteristic zero, low-genus automorphism classifications, and a Hasse--Witt computation reduce the problem to the Klein quartic. In the wild case, the rational points are identified with the Sylow $p$-subgroups of the automorphism group. Noncyclic Sylow subgroups are treated using a finite-group classification theorem together with Henn's large-automorphism classification, while cyclic Sylow subgroups are excluded by local ramification and the Riemann--Hurwitz formula. Combined with the known characteristic-two result, this yields the classification for all prime powers.