Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points

TL;DR

This paper extends computations to determine the maximum rational points on small genus curves over finite fields with fixed gonality, proposing conjectures relating gonality, genus, and rational points.

Contribution

It provides new data on rational points for curves over finite fields with fixed gonality and small genus, and introduces two conjectures linking gonality, genus, and rational points.

Findings

No curve of genus 5 and gonality 6 over finite fields.

Proposed conjecture: optimal curve of genus g has gonality at most (g+3)/2.

Proposed conjecture: large genus curve with gonality γ over F_q has γ(q+1) rational points.

Abstract

We extend the computations from our previous paper arXiv:2005.07054 to determine the maximum number of rational points on a curve over $\mathbb{F}_3$ and $\mathbb{F}_4$ 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, a curve of gonality $\gamma$ and large genus over $\mathbb{F}_q$ has $\gamma(q+1)$ rational points.