Curves of fixed gonality with many rational points

Floris Vermeulen

Published 2021-02-01Version 1

Given a gonality $\gamma$ and a prime power $q\geq \gamma -1$ we show that for every large genus $g$ there exists a 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 allowed. This answers a recent conjecture by Faber--Grantham. Our methods are based on Poonen's work on squarefree values of polynomials together with a Newton polygon argument.