AI 中文总结
本文在特征二代数闭域上分类了亏格 g≥2 的曲线的大循环自同构子群及具有指数二循环子群的群，给出其阶的尖锐界与显式方程，并讨论了 Hermitian 曲线的极大性条件。
AI 中文摘要
设 $\mathcal{X}$ 是特征二代数闭域上亏格 $g\ge2$ 的射影、几何不可约、非奇异代数曲线。我们分类 $\Aut(\mathcal{X})$ 中阶 $N\ge2g+1$ 的循环子群。除在三个点分歧的奇次数 Kummer 扩张外，恰好出现两种情况：$y^2+y=x^m$，其中 $N=2m=4g+2$，以及 $y^2+y=c/(x^m+1)$，其中 $m$ 为奇数，$c\ne0$，且 $N=2m=2g+2$。特别地，$4\nmid N$。然后我们分类具有指数二循环子群且 $|H|>4g+4$ 的群 $H$。它们恰好是曲线 $y^k=x^M+x^{-M}$ 上的群 $C_k\times D_{2M}$，其中 $k,M\ge3$ 为奇数且互素，且 $2g=M(k-1)$。它们的可能阶为 $4g+2M$，其中 $M$ 遍历 $g$ 的某些奇数因子，且 $|H|\le6g$。此族中没有曲线是普通的。Hermitian 曲线出现在该族中，我们给出在有限域上极大性的两个充分条件。对于二面体群，尖锐界在偶亏格时为 $4g+4$，在奇亏格时为 $4g$。等号情形由显式 Artin--Schreier 方程给出，且在每种情形下二面体群都是完全自同构群。
英文摘要
Let $\mathcal{X}$ be a projective, geometrically irreducible, nonsingular algebraic curve of genus $g\ge2$ over an algebraically closed field of characteristic two. We classify cyclic subgroups of $\Aut(\mathcal{X})$ of order $N\ge2g+1$. Besides Kummer extensions of odd degree ramified over three points, precisely two cases occur: $y^2+y=x^m$, with $N=2m=4g+2$, and $y^2+y=c/(x^m+1)$, with $m$ odd, $c\ne0$, and $N=2m=2g+2$. In particular, $4\nmid N$. We then classify groups $H$ with a cyclic subgroup of index two and $|H|>4g+4$. They are precisely the groups $C_k\times D_{2M}$ on the curves $y^k=x^M+x^{-M}$, where $k,M\ge3$ are odd and coprime, and $2g=M(k-1)$. Their possible orders are $4g+2M$, where $M$ runs over certain odd divisors of $g$, and $|H|\le6g$. No curve in this family is ordinary. Hermitian curves occur in the family, and we give two sufficient conditions for maximality over finite fields. For dihedral groups the sharp bound is $4g+4$ in even genus and $4g$ in odd genus. The equality cases are given by explicit Artin--Schreier equations, and in each case the dihedral group is the full automorphism group.
