We give new arguments that improve the known upper bounds on the maximal number of rational points of a curve of genus over a finite field , for a number of pairs . Given a pair and an integer , we determine the possible zeta functions of genus- curves over with points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus- curve over with points must have a low-degree map to another curve over , and often this is enough to give us a contradiction. In particular, we are able to provide eight previously unknown values of , namely: , , , , , , , and . Our arguments also allow us to give a non-computer-intensive proof of the recent result of Savitt that there are no genus- curves over having exactly rational points. Furthermore, we show that there is an infinite sequence of ’s such that for every with , the difference between the Weil-Serre bound on and the actual value of is at least .
Grâce à de nouveaux arguments, nous améliorons les majorations connues du nombre maximal de points rationnels sur une courbe de genre définie sur un corps fini , pour certains couples . En particulier, nous donnons huit valeurs de qui étaient jusqu’à présent inconnues : , , , , , , , et . Nous redémontrons aussi, avec une utilisation minimale de l’ordinateur, un résultat de Savitt : il n’y a pas de courbe de genre sur ayant exactement points rationnels. Enfin, nous démontrons qu’il y a une infinité de tels que pour tout satisfaisant , la différence entre la borne de Weil-Serre de et la valeur exacte de est au moins égale à .
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
@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}, pages = {1677--1737}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {6}, year = {2003}, doi = {10.5802/aif.1990}, mrnumber = {2038778}, zbl = {1065.11043}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1990/} }
TY - JOUR AU - Howe, Everett W. AU - Lauter, Kristin E. TI - Improved upper bounds for the number of points on curves over finite fields JO - Annales de l'Institut Fourier PY - 2003 SP - 1677 EP - 1737 VL - 53 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1990/ DO - 10.5802/aif.1990 LA - en ID - AIF_2003__53_6_1677_0 ER -
%0 Journal Article %A Howe, Everett W. %A Lauter, Kristin E. %T Improved upper bounds for the number of points on curves over finite fields %J Annales de l'Institut Fourier %D 2003 %P 1677-1737 %V 53 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1990/ %R 10.5802/aif.1990 %G en %F 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/articles/10.5802/aif.1990/
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