论文标题
特征三的属五五曲线或三角曲线具有许多合理点
Genus-five hyperelliptic or trigonal curves with many rational points in characteristic three
论文作者
论文摘要
数字$ n_9（5）$，$ \ mathbb {f} _9 $ - 合理点的最大数量超过$ \ mathbb {f} _9 $ 5 $ $ 5 $是未知的，但众所周知，$ 32 \ le N_9（5）（5）\ le 35 $。在本文中，我们列举了超过$ \ mathbb {f} _3 $超过$ \ mathbb {f} _9 $ -rational-rational Points（和$ \ Mathbb {f} _3 $ rational-rational-rational-rational-points），尤其是这些$ \ mathbb的最大数量的$ clive curve cultect of $ clive culte clive of $ f imate $ clive cultiant， Kudo-Harashita研究了非遗传性和非三角形案例，在那里他们找到了一个新的曲线示例（超过$ \ m athbb {f} _3 $）属的五属属于$ 32 $，并证明在Sextic平面曲线中，没有示例，并证明没有示例可获得$ 32 $。我们从本文的主要结果中得出结论，我们需要搜索具有不良奇异性的六型模型（即非遗传性和非三角形），以便在$ \ mathbb {f} _3 $上找到具有至少$ 33 $ $ 33 $ $ \ thybb {f} f} f} f} $ _9 $ _9 $ - $ _9 $ - 有限的点。
The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational points (and $\mathbb{F}_3$-rational points), especially the maximal number of $\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over $\mathbb{F}_3$) of genus five which attains $32$ and proved that there is no example attaining more than $32$, among sextic plane curves with mild singularities. We conclude from the main results in this paper that we need to search sextic models (i.e., nonhyperelliptic and nontrigonal) with bad singularities, in order to find a genus-five curve over $\mathbb{F}_3$ with at least $33$ $\mathbb{F}_9$-rational points.
