具有指数为2的循环子群的大自同构群的曲线
Curves with a large automorphism group admitting a cyclic subgroup of index $2$
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中文总结 AI 辅助
研究任意特征域上具有指数为2的循环子群的曲线，给出特征\(p\neq2\)的代数闭域上曲线自同构二面体群大小的上界，还得到大于\(4g(\mathcal{X}) + 4\)且具指数为2循环子群的（非二面体）群的分类结果。
中文摘要 AI 辅助
零特征域\(\mathbb{K}\)上亏格\(g(\mathcal{X})\geq2\)的代数曲线\(\mathcal{X}\)的\(\mathbb{K}\)-自同构群\({\rm{Aut}}({\mathcal{X}})\)的阶有Hurwitz界\(|{\rm{Aut}}({\mathcal{X}})|\leq84(g(\mathcal{X}) - 1)\)，某些子群有改进界。本文探索任意特征域上曲线及具有指数为2的循环子群的群\(H\)的更一般情形。证明了特征\(p\neq2\)的代数闭域上曲线的自同构二面体群大小有相同上界，还给出了关于大于\(4g(\mathcal{X}) + 4\)且具有指数为2的循环子群的（非二面体）群的一些分类结果。
英文摘要
The Hurwitz bound on the order of the $\mathbb K$-automorphism group ${\rm{Aut}}({\mathcal{X}})$ of an algebraic curve ${\mathcal{X}}$ of genus $g(\mathcal{X})\ge 2$ defined over a field $\mathbb K$ of zero characteristic states that $|{\rm{Aut}}({\mathcal{X}})|\le 84(g(\mathcal{X})-1)$. Improved bounds are available for the order of certain types of subgroups within automorphism groups. For instance, if a subgroup $H$ of ${\rm{Aut}}({\mathcal{X}})$ is dihedral, then in the complex case, $|H| \leq 4g(\mathcal{X}) + 4$. More recently it has been shown that a tighter bound holds for $H$ a generalized quasi-dihedral group. In this paper we explore the more general setting of a curve defined over a field of any characteristic, and $H$ a group admitting a cyclic subgroup of index two. We show that the same upper bound for the size of a dihedral group of automorphisms holds for curves defined over an algebraically closed field of characteristic $p\ne 2$. Then we provide some classification results about (non-dihedral) groups of size larger than $4g(\mathcal{X})+4$ admitting a cyclic subgroup of index $2$.
