发表机构
University of Campinas (UNICAMP)（坎皮纳斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文将 Giulietti 和 Korchmáros 的二次亏格界从偶亏格推广到所有亏格，证明正 $p$-秩曲线自同构群阶小于 $900g^2$，通过分歧估计与最小亏格论证结合有限单群分类完成证明。
AI 中文摘要
设 $X$ 是特征为奇素数 $p$ 的代数闭域上的亏格 $g\ge2$ 的非奇异射影曲线。Giulietti 和 Korchmáros 建立了常数 $900$ 的二次亏格界，该界迫使偶亏格曲线具有零 $p$-秩。我们将此界推广到奇亏格情形：对每个 $g\ge2$，正 $p$-秩蕴含 $|\Aut(X)|<900g^2$。从 Montanucci 的二短轨道约化出发，我们结合分歧估计与最小亏格论证，将反例约化为一个几乎单群作用在具有两个分支点的射影直线覆盖上。主要工具是通过 Hasse--Arf 给出的第二分歧群的内在描述，以及与抛物子群相关的中间商曲线的亏格界。这些连同阶与局部结构估计，排除了所有可能的单基座。证明使用了有限单群分类定理。
英文摘要
Let $X$ be a nonsingular projective curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$. Giulietti and Korchmáros established a quadratic genus bound with constant $900$ forcing zero $p$-rank for curves of even genus. We extend this bound to odd genus: for every $g\ge2$, positive $p$-rank implies $|\Aut(X)|<900g^2$. Starting from Montanucci's two-short-orbit reduction, we combine ramification estimates and a minimum-genus argument to reduce a counterexample to an almost-simple group acting on a cover of the projective line with two branch points. The main ingredients are an intrinsic description of the second ramification group via Hasse--Arf and genus bounds for intermediate quotients associated with parabolic subgroups. These, together with order and local-structure estimates, exclude every possible simple socle. The proof uses the classification of finite simple groups.
