Abstract

We extend the computations from our previous work to determine the maximum number of rational points on a curve over F3${\mathbb{F}}_3$${\mathbb{F}}_3$ and F4F4${\mathbb{F}}_4$ with fixed gonality and small genus. We find, for example, that there is no curve of genus 5 and gonality 6 over a finite field. We propose two conjectures based on our data. First, an optimal curve of genus g has gonality at most $\lfloor \frac{g+3}{2}\rfloor$. Second, an optimal curve of gonality γ and large genus over ${\mathbb{F}}_q$ has $\gamma (q+1)$ rational points.

中文翻译：

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具有多有理点的小固定亏格和性状的三元和四元曲线

摘要

我们扩展了之前工作的计算，以确定曲线上有理点的最大数量F3${\mathbb{F}}_3$${\mathbb{F}}_3$和F 4F4${\mathbb{F}}_4$具有固定的性和小的属。例如，我们发现在有限域上不存在属 5 和性 6 的曲线。我们根据我们的数据提出两个猜想。首先，亏格g的最优曲线最多具有无性$\lfloor \frac{G+3}{2}\rfloor$。其次，最优性γ和大亏格曲线${\mathbb{F}}_q$有$\gamma （q+1）$理性点。