Genus-five hyperelliptic or trigonal curves with many rational points in characteristic three

TL;DR

This paper investigates genus five hyperelliptic and trigonal curves over finite fields of characteristic three, focusing on their rational points over 9, and suggests directions for finding curves with more rational points.

Contribution

It enumerates hyperelliptic and trigonal genus five curves over 3 with many 9-rational points, and highlights the need to explore sextic models with singularities for more points.

Findings

Maximal 9-rational points for these curves is 30.

Existing nonhyperelliptic, nontrigonal curves reach 32 points.

Further search in sextic models with singularities may yield more points.

Abstract

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.