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]

Howe, Everett W. ; Lauter, Kristin E.

Annales de l'Institut Fourier, Tome 53 (2003), p. 1677-1737 / Harvested from Numdam

Résumé

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 ${\mathbb {F}}_q$, 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 ${\mathbb {F}}_8$ 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 $g/2$.

Abstract

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 ${\mathbb {F}}_q$, 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 ${\mathbb {F}}_q$ 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 ${\mathbb {F}}_q$ with $N$ points must have a low-degree map to another curve over ${\mathbb {F}}_q$, 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 ${\mathbb {F}}_8$ 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 $g/2$.

Détail

Publié le : 2003-01-01

MR 2038778 | Zbl 1065.11043 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1990
Classification:  11G20,  14G05,  14G10,  14G15
Mots clés: courbe, point rationnel, fonction zêta, borne de Weil, borne de Serre, borne d'Oesterlé

Bibliographie

[1] W. Bosma; J. Cannon; C. Playoust The Magma algebra system I: The user language, J. Symbolic Comput., Tome 24 (1997), pp. 235-265 | Article | MR 1484478 | Zbl 0898.68039

[2] I. I. Bouw; Jean-Benoît Bost, François Loeser, And Michel Raynaud, Eds. The p-rank of curves and covers of curves, Courbes semi-stables et groupe fondamental en géométrie algébrique, Birkhäuser, Basel (Progr. Math.) Tome 187 (2000), pp. 267-277 | Zbl 0979.14015

[3] P. Deligne Variétés abéliennes ordinaires sur un corps fini, Invent. Math., Tome 8 (1969), pp. 238-243 | Article | MR 254059 | Zbl 0179.26201

[4] S. A. Dipippo; E. W. Howe Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory, Tome 73 (1998), pp. 426-450 | Article | MR 1657992 | Zbl 0931.11023

[4] S.A. Dilippo; E.W. Howe Corrigendum: Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory, Tome 83 (2000) no. 1, pp. 182 | Zbl 0931.11023

[5] R. Fuhrmann; F. Torres The genus of curves over finite fields with many rational points, Manuscripta Math, Tome 89 (1996), pp. 103-106 | Article | MR 1368539 | Zbl 0857.11032

[6] G. Van Der Geer; M. Van Der Vlugt Tables of curves with many points, Math. Comp., Tome 69 (2000), pp. 797-810 | Article | MR 1654002 | Zbl 0965.11028

[7] E. W. Howe Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc., Tome 347 (1995), pp. 2361-2401 | Article | MR 1297531 | Zbl 0859.14016

[8] E. W. Howe; H. J. Zhu On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field, J. Number Theory, Tome 92 (2002), pp. 139-163 | Article | MR 1880590 | Zbl 0998.11031

[9] G. Korchmáros; F. Torres On the genus of a maximal curve, Math. Ann., Tome 323 (2002), pp. 589-608 | Article | MR 1923698 | Zbl 1018.11029

[10] R. B. Lakein Euclid's algorithm in complex quartic fields, Acta Arith., Tome 20 (1972), pp. 393-400 | MR 304350 | Zbl 0224.12001

[11] K. Lauter Improved upper bounds for the number of rational points on algebraic curves over finite fields, C. R. Acad. Sci. Paris, Sér. I Math., Tome 328 (1999), pp. 1181-1185 | Article | MR 1701382 | Zbl 0948.11024

[12] K. Lauter Non-existence of a curve over $\smallF_3$ of genus 5 with 14 rational points, Proc. Amer. Math. Soc, Tome 128 (2000), pp. 369-374 | Article | MR 1664414 | Zbl 0983.11036

[13] K. Lauter; Johannes Buchmann, Tom Høholdt, Henning Stichtenoth, Horacio Ta Zeta functions of curves over finite fields with many rational points, Coding Theory, Cryptography and Related Areas, Springer-Verlag, Berlin (2000), pp. 167-174 | Zbl 1009.11049

[14] K. Lauter With An Appendix By J-P. Serre Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, J. Algebraic Geom., Tome 10 (2001), pp. 19-36 | MR 1795548 | Zbl 0982.14015

[15] K. Lauter With An Appendix By J-P. Serre The maximum or minimum number of rational points on genus three curves over finite fields, Compositio Math., Tome 134 (2002), pp. 87-111 | Article | MR 1931964 | Zbl 1031.11038

[16] D. Mumford Abelian Varieties, Oxford University Press, Oxford, Tata Institute of Fundamental Research Studies in Mathematics, Tome 5 (1985) | Zbl 0583.14015

[17] F. Oort Commutative group schemes, Springer-Verlag, Berlin, Lecture Notes in Math, Tome 15 (1966) | MR 213365 | Zbl 0216.05603

[18] D. Savitt With An Appendix By K. Lauter The maximum number of rational points on a curve of genus 4 over $\smallF_8$ is 25, Canad. J. Math., Tome 55 (2003), pp. 331-352 | Article | MR 1969795 | Zbl 02005249

[19] J.-P. Serre Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris, Sér. I Math., Tome 296 (1983), pp. 397-402 | MR 703906 | Zbl 0538.14015

[20] J.-P. Serre Nombres de points des courbes algébriques sur $\smallF_q$, Sém. Théor. Nombres Bordeaux 1982/83, Tome Exp. No. 22 | Zbl 0538.14016

[21] J.-P. Serre Résumé des cours de 1983--1984, Ann. Collège France (1984), pp. 79-83

[22] J.-P. Serre Rational points on curves over finite fields (1985) (unpublished notes by Fernando Q. Gouvéa of lectures at Harvard University)

[23] C. L. Siegel The trace of totally positive and real algebraic integers, Ann. of Math (2), Tome 46 (1945), pp. 302-312 | Article | MR 12092 | Zbl 0063.07009

[24] C. Smyth Totally positive algebraic integers of small trace, Ann. Inst. Fourier (Grenoble), Tome 33 (1984) no. 3, pp. 1-28 | Article | Numdam | MR 762691 | Zbl 0534.12002

[25] H. M. Stark; Harold G. Diamond, Ed. On the Riemann hypothesis in hyperelliptic function fields, Analytic number theory, American Mathematical Society, Providence, R.I. (Proc. Sympos. Pure Math) Tome 24 (1973), pp. 285-302 | Zbl 0271.14012

[26] H. Stichtenoth Algebraic Function Fields and Codes, Springer-Verlag, Berlin (1993) | MR 1251961 | Zbl 0816.14011

[27] K.-O. Stöhr; J. F. Voloch Weierstrass points and curves over finite fields, Proc. London Math. Soc (3), Tome 52 (1986), pp. 1-19 | Article | MR 812443 | Zbl 0593.14020

[28] D. Subrao The p-rank of Artin-Schreier curves, Manuscripta Math., Tome 16 (1975), pp. 169-193 | Article | MR 376693 | Zbl 0321.14017

[29] J. Tate Classes d'isogénie des variétés abéliennes sur un corps fini, Séminaire Bourbaki 1968/69, Springer-Verlag, Berlin (Lecture Notes in Math) Tome 179 (1971), pp. 95-110 | Numdam | Zbl 0212.25702

[30] M. E. Zieve Improving the Oesterlé bound (preprint)