奇特征下零 $p$-秩曲线的自同构大群
Large automorphism groups of curves of zero $p$-rank in odd characteristic
- IMECC, Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学数学、计算和计算机科学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在奇特征下分类了零 $p$-秩曲线上无公共不动点且阶大于 $24g(g-1)$ 的自同构群，确定了相应曲线为显式循环覆盖、Hermitian 或 Ree 曲线，并推广了零 $2$-秩情形的分类，进而给出广义 Suzuki 曲线的新刻画。
AI中文摘要：
设 $\cX$ 是定义在奇特征 $p$ 的代数闭域上、亏格 $g\ge2$ 且零 $p$-秩的曲线。我们对满足 $G\le\Aut(\cX)$ 无公共不动点且 $|G|>24g(g-1)$ 的配对 $(\cX,G)$ 进行分类。对于 $g\ge4$，这些曲线是射影直线的显式循环覆盖、Hermitian 曲线或 Ree 曲线。我们确定了它们的完全自同构群以及可能的子群 $G$，包括中心扩张。特征 $5$ 中例外的 $A_7$ 作用决定了 6 次 Hermitian 曲线。亏格 $2$ 和 $3$ 的情形单独处理。这些结果给出了零 $2$-秩曲线大自同构群分类的奇特征对应物。作为应用，我们获得了广义 Suzuki 曲线的一个新刻画，其中不再需要不动点假设。
英文摘要：
Let $\cX$ be a curve of genus $g\ge2$ and zero $p$-rank over an algebraically closed field of odd characteristic $p$. We classify the pairs $(\cX,G)$ for which $G\le\Aut(\cX)$ has no common fixed point and $|G|>24g(g-1)$. For $g\ge4$, the curves are explicit cyclic covers of the projective line, the Hermitian curve, or the Ree curve. We determine their full automorphism groups and the possible subgroups $G$, including the central extensions. The exceptional $A_7$ action in characteristic $5$ determines the Hermitian curve of degree $6$. The cases of genus $2$ and $3$ are treated separately. These results give an odd-characteristic counterpart to the classification of large automorphism groups of zero $2$-rank curves. As an application, we obtain a new characterization of the generalized Suzuki curve in which the fixed-point hypothesis is no longer required.
