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arXiv · 2102.00900

Curves of fixed gonality with many rational points

Floris Vermeulen
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Abstract

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$ $\mathbb{F}_q$-rational points. This is the maximal number of rational points possible. This answers a recent conjecture by Faber--Grantham. Our methods are based on curves on toric surfaces and Poonen's work on squarefree values of polynomials.

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Floris Vermeulen. 2021-02-01. Curves of fixed gonality with many rational points. https://arxiv.org/abs/2102.00900

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