Curves of fixed gonality with many rational points

Vermeulen, Floris

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 1, pp. 135-149

Given an integer $\gamma \ge 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over ${𝔽}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma (q+1)$ ${𝔽}_q$-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.

Étant donné un entier $\gamma \ge 2$ et une puissance $q$ d’un nombre premier impair, nous montrons que pour chaque genre $g$ suffisamment grand, il existe une courbe $C$ définie sur ${𝔽}_q$, non singulière, de genre $g$ et de gonalité $\gamma$, telle que son nombre de points rationnels est exactement $\gamma (q+1)$, c’est-à-dire le maximal possible, démontrant ainsi une conjecture récente de Faber-Grantham. Les méthodes que nous employons sont en lien avec l’étude des courbes sur les surfaces toriques et avec les travaux de Poonen sur les valeurs sans facteur carré de polynômes.

Classification : 11G20, 14G05, 14G15, 14M25
Keywords: Curves over finite fields, rational points, gonality, toric surfaces

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