Papers › Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points
Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points
Xander Faber, Jon Grantham
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We extend the computations from our previous paper arXiv:2005.07054 to determine the maximum number of rational points on a curve over 𝔽₃ and 𝔽₄ 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 ⌊(g+3)/2 ⌋. Second, a curve of gonality γ and large genus over 𝔽_q has γ(q+1) rational points.
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