Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many 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.

Final published version as it will appear in Experimental Mathematics; 19 pages; code available at https://github.com/RationalPoint/gonality [Note that an error was corrected since the previous version: up to isomorphism, there is exactly 1 curve of genus 4 and gonality 5 over GF(3).]

References in corpus (3)

• Binary Curves of small fixed genus and gonality with many rational points
• A census of all genus 4 curves over the field with 2 elements
• Existence d'une courbe de genre 5 sur F_3 avec 13 points rationnels

Cited by in corpus (2)

• On the Maximum Gonality of a Curve over a Finite Field
• On abelian covers of the projective line with fixed gonality and many rational points