发表机构
Institute of Mathematics, Statistics and Scientific Computing, University of Campinas（坎皮纳斯大学数学、统计与科学计算研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分类了亏格至少二的极大曲线上完全几何自同构群在有理点集上传递的所有情形，证明除 Hermite 曲线外仅 $q=5$ 时 Klein 四次曲线例外，并给出完整分类。
AI 中文摘要
设 $q=p^h$ 为奇素数幂，$X/\F_{q^2}$ 为亏格至少为二的极大曲线。我们分类了其完全几何自同构群在 $\F_{q^2}$-有理点集合上传递的曲线。我们证明，若 $h>1$，则 $X$ 为 Hermite 曲线。若 $h=1$，唯一额外可能发生在 $q=5$：即唯一的 $\F_{25}$-极大亏格三曲线，即 Klein 四次曲线的极大 $S_4$-模型。证明区分了 tame 作用与 wild 作用。在 tame 情形，亏格界、商映射的签名、特征零特化、低亏格自同构分类以及 Hasse--Witt 计算将问题归结为 Klein 四次曲线。在 wild 情形，有理点与自同构群的 Sylow $p$-子群等同。非循环 Sylow 子群利用有限群分类定理以及 Henn 的大自同构分类处理，而循环 Sylow 子群则通过局部分歧和 Riemann--Hurwitz 公式排除。结合已知的特征二结果，这给出了所有素数幂情形的分类。
英文摘要
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.
