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

Abstract

We extend the computations from our previous work 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, an optimal curve of gonality γ and large genus over ${\mathbb{F}}_q$ has $\gamma (q+1)$ rational points.

Correction Statement

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Declaration of Interest

No potential conflict of interest was reported by the authors.