Improved upper bounds for the number of points on curves over finite fields

Howe, Everett W.; Lauter, Kristin E.

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 $N_q\left(g\right)$ 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 $N_q\left(g\right)$ qui étaient jusqu’à présent inconnues : $N_4\left(5\right)=17$, $N_4\left(10\right)=27$, $N_8\left(9\right)=45$, $N_{16}\left(4\right)=45$, $N_{128}\left(4\right)=215$, $N_3\left(6\right)=14$, $N_9\left(10\right)=54$, et $N_{27}\left(4\right)=64$. 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 $N_q\left(g\right)$ et la valeur exacte de $N_q\left(g\right)$ est au moins égale à $g/2$.

We give new arguments that improve the known upper bounds on the maximal number $N_q\left(g\right)$ 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 $N_q\left(g\right)$, namely: $N_4\left(5\right)=17$, $N_4\left(10\right)=27$, $N_8\left(9\right)=45$, $N_{16}\left(4\right)=45$, $N_{128}\left(4\right)=215$, $N_3\left(6\right)=14$, $N_9\left(10\right)=54$, and $N_{27}\left(4\right)=64$. 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 $N_q\left(g\right)$ and the actual value of $N_q\left(g\right)$ is at least $g/2$.

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

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