Improved upper bounds for the number of points on curves over finite fields
[Améliorations des majorations pour le nombre de points des courbes sur un corps fini]
Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1677-1737.

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.

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.

DOI : 10.5802/aif.1990
Classification : 11G20, 14G05, 14G10, 14G15
Keywords: curve, rational point, zeta function, Weil bound, Serre bound, Oesterlé bound
Mot clés : courbe, point rationnel, fonction zêta, borne de Weil, borne de Serre, borne d'Oesterlé
Howe, Everett W. 1 ; Lauter, Kristin E. 2

1 Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967 (USA)
2 Microsoft Research, One Microsoft Way, Redmond, WA 98052 (USA)
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Howe, Everett W.; Lauter, Kristin E. Improved upper bounds for the number of points on curves over finite fields. Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1677-1737. doi : 10.5802/aif.1990. http://www.numdam.org/articles/10.5802/aif.1990/

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