论文标题
在特征p中具有许多自动形态的代数曲线上
On algebraic curves with many automorphisms in characteristic p
论文作者
论文摘要
令$ \ mathcal {x} $为在奇数特征$ p $的字段上定义的不可约，非单一的代数曲线。令$ g $和$γ$分别为$ \ Mathcal {x} $的属和$ p $ -Lank。 $ g $和$γ$对$ \ MATHCAL {x} $的自动形态组$ aut（\ Mathcal {x}）$的影响是文献中众所周知的。如果$ g \ geq 2 $，则$ aut（\ Mathcal {x}）$是一个有限的组，除非$ \ Mathcal {x} $是所谓的Hermitian曲线，否则其订单在$ g $ fortim of fout的$ g $四（Stichtenoth）中。 1978年，亨恩（Henn）提议将Stichtenoth的Cube Order Bound of Cube Order（$ g $）提高到少数例外，所有这些都有$ p $ -rank零。在本文中，提出了进一步的亨恩结果。首先，我们证明，如果属$ g \ geq 2 $的代数曲线具有超过$ 336G^2 $自动形态的$ 336G^2 $，那么其自动形态组恰好有两个短轨道，一个驯服和一个非驯服。然后，我们表明，如果$ | aut（\ Mathcal {x}）| \ geq 900g^2 $，商曲线$ \ MATHCAL {x}/aut（\ Mathcal {x}）_ p^{（1）} $，其中$ p $包含在非tame短轨道中是理性的$ \ MATHCAL {X} $等于零。
Let $\mathcal{X}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic $p$. Let $g$ and $γ$ be the genus and $p$-rank of $\mathcal{X}$, respectively. The influence of $g$ and $γ$ on the automorphism group $Aut(\mathcal{X})$ of $\mathcal{X}$ is well-known in the literature. If $g \geq 2$ then $Aut(\mathcal{X})$ is a finite group, and unless $\mathcal{X}$ is the so-called Hermitian curve, its order is upper bounded by a polynomial in $g$ of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth's bound of cube order in $g$ up to few exceptions, all having $p$-rank zero. In this paper a further refinement of Henn's result is proposed. First, we prove that if an algebraic curve of genus $g \geq 2$ has more than $336g^2$ automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame. Then we show that if $|Aut(\mathcal{X})| \geq 900g^2$, the quotient curve $\mathcal{X}/Aut(\mathcal{X})_P^{(1)}$ where $P$ is contained in the non-tame short orbit is rational, and the stabilizer of 2 points is either a $p$-group or a prime-to-$p$ group, then the $p$-rank of $\mathcal{X}$ is equal to zero.
