Improved upper bounds for the number of points on curves over finite fields
Annales de l'Institut Fourier, Volume 53 (2003) no. 6, p. 1677-1737

We give new arguments that improve the known upper bounds on the maximal number N q (g) of rational points of a curve of genus g over a finite field 𝔽 q , for a number of pairs (q,g). Given a pair (q,g) and an integer N, we determine the possible zeta functions of genus-g curves over 𝔽 q with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-g curve over 𝔽 q with N points must have a low-degree map to another curve over 𝔽 q , and often this is enough to give us a contradiction. In particular, we are able to provide eight previously unknown values of N q (g), namely: N 4 (5)=17, N 4 (10)=27, N 8 (9)=45, N 16 (4)=45, N 128 (4)=215, N 3 (6)=14, N 9 (10)=54, and N 27 (4)=64. Our arguments also allow us to give a non-computer-intensive proof of the recent result of Savitt that there are no genus-4 curves over 𝔽 8 having exactly 27 rational points. Furthermore, we show that there is an infinite sequence of q’s such that for every g with 0<g<log 2 q, the difference between the Weil-Serre bound on N q (g) and the actual value of N q (g) is at least g/2.

Grâce à de nouveaux arguments, nous améliorons les majorations connues du nombre maximal N q (g) de points rationnels sur une courbe de genre g définie sur un corps fini 𝔽 q , pour certains couples (q,g). En particulier, nous donnons huit valeurs de N q (g) qui étaient jusqu’à présent inconnues : N 4 (5)=17, N 4 (10)=27, N 8 (9)=45, N 16 (4)=45, N 128 (4)=215, N 3 (6)=14, N 9 (10)=54, et N 27 (4)=64. Nous redémontrons aussi, avec une utilisation minimale de l’ordinateur, un résultat de Savitt : il n’y a pas de courbe de genre 4 sur 𝔽 8 ayant exactement 27 points rationnels. Enfin, nous démontrons qu’il y a une infinité de q tels que pour tout g satisfaisant 0<g<log 2 q, la différence entre la borne de Weil-Serre de N q (g) et la valeur exacte de N q (g) est au moins égale à g/2.

DOI : https://doi.org/10.5802/aif.1990
Classification:  11G20,  14G05,  14G10,  14G15
Keywords: curve, rational point, zeta function, Weil bound, Serre bound, Oesterlé bound
@article{AIF_2003__53_6_1677_0,
     author = {Howe, Everett W. and Lauter, Kristin E.},
     title = {Improved upper bounds for the number of points on curves over finite fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {6},
     year = {2003},
     pages = {1677-1737},
     doi = {10.5802/aif.1990},
     zbl = {1065.11043},
     mrnumber = {2038778},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_6_1677_0}
}
Howe, Everett W.; Lauter, Kristin E. Improved upper bounds for the number of points on curves over finite fields. Annales de l'Institut Fourier, Volume 53 (2003) no. 6, pp. 1677-1737. doi : 10.5802/aif.1990. http://www.numdam.org/item/AIF_2003__53_6_1677_0/

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