Given an integer and an odd prime power we show that for every large genus there exists a non-singular curve defined over of genus and gonality and with exactly -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 et une puissance d’un nombre premier impair, nous montrons que pour chaque genre suffisamment grand, il existe une courbe définie sur , non singulière, de genre et de gonalité , telle que son nombre de points rationnels est exactement , 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.
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Keywords: Curves over finite fields, rational points, gonality, toric surfaces
Floris Vermeulen 1

@article{JTNB_2023__35_1_135_0, author = {Floris Vermeulen}, title = {Curves of fixed gonality with many rational points}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {135--149}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {35}, number = {1}, year = {2023}, doi = {10.5802/jtnb.1240}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1240/} }
TY - JOUR AU - Floris Vermeulen TI - Curves of fixed gonality with many rational points JO - Journal de théorie des nombres de Bordeaux PY - 2023 SP - 135 EP - 149 VL - 35 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1240/ DO - 10.5802/jtnb.1240 LA - en ID - JTNB_2023__35_1_135_0 ER -
%0 Journal Article %A Floris Vermeulen %T Curves of fixed gonality with many rational points %J Journal de théorie des nombres de Bordeaux %D 2023 %P 135-149 %V 35 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1240/ %R 10.5802/jtnb.1240 %G en %F JTNB_2023__35_1_135_0
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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