Curves of fixed gonality with many rational points
Floris Vermeulen1
1 KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium
Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 135-149.

Given an integer γ≥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 γ and with exactly γ(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 γ≥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 γ, telle que son nombre de points rationnels est exactement γ(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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1240
Classification: 11G20, 14G05, 14G15, 14M25
Keywords: Curves over finite fields, rational points, gonality, toric surfaces

Floris Vermeulen 1

1 KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Floris Vermeulen. Curves of fixed gonality with many rational points. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 135-149. doi : 10.5802/jtnb.1240. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1240/

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