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.

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 $\left(q,g\right)$. Given a pair $\left(q,g\right)$ 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, 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 $g/2$.

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 $\left(q,g\right)$. 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, 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 $g/2$.

DOI: 10.5802/aif.1990
Classification: 11G20,  14G05,  14G10,  14G15
Keywords: curve, rational point, zeta function, Weil bound, Serre bound, Oesterlé bound
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, Volume 53 (2003) no. 6, pp. 1677-1737. doi : 10.5802/aif.1990. http://archive.numdam.org/articles/10.5802/aif.1990/

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