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

Howe, Everett W.; Lauter, Kristin E.

doi:10.5802/aif.1990

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.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)

Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1677-1737.

corrigé par Corrigendum to: Improved upper bounds for the number of points on curves over finite fields
[Corrigendum : Améliorations des majorations pour le nombre de points des courbes sur un corps fini]

Résumé
Abstract

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 $g / 2$ .

MR Zbl

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

Affiliations des auteurs :

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)

@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},
     zbl = {1065.11043},
     mrnumber = {2038778},
     language = {en},
     url = {https://aif.centre-mersenne.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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Tome 53 (2003) no. 6, pp. 1677-1737. doi : 10.5802/aif.1990. https://aif.centre-mersenne.org/articles/10.5802/aif.1990/

Bibliographie
Cité par

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

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

[3] P. Deligne Variétés abéliennes ordinaires sur un corps fini, Invent. Math., Volume 8 (1969), pp. 238-243 | DOI | EuDML | MR | Zbl

[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, Volume 73 (1998), pp. 426-450 | DOI | MR | Zbl

[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, Volume 83 (2000) no. 1, pp. 182 | Zbl

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

[6] G. van der Geer; M. van der Vlugt Tables of curves with many points, Math. Comp., Volume 69 (2000), pp. 797-810 | DOI | MR | Zbl

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

[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, Volume 92 (2002), pp. 139-163 | DOI | MR | Zbl

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

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

[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., Volume 328 (1999), pp. 1181-1185 | DOI | MR | Zbl

[12] K. Lauter Non-existence of a curve over $𝔽_{3}$ of genus 5 with 14 rational points, Proc. Amer. Math. Soc, Volume 128 (2000), pp. 369-374 | DOI | MR | Zbl

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

[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., Volume 10 (2001), pp. 19-36 | MR | Zbl

[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., Volume 134 (2002), pp. 87-111 | DOI | MR | Zbl

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

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

[18] D. Savitt with an Appendix by K. Lauter The maximum number of rational points on a curve of genus 4 over $𝔽_{8}$ is 25, Canad. J. Math., Volume 55 (2003), pp. 331-352 | DOI | MR | Zbl

[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., Volume 296 (1983), pp. 397-402 | MR | Zbl

[20] J.-P. Serre Nombres de points des courbes algébriques sur $𝔽_{q}$ , Sém. Théor. Nombres Bordeaux 1982/83, Volume Exp. No. 22 | Zbl

[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), Volume 46 (1945), pp. 302-312 | DOI | MR | Zbl

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

[25] H. M. Stark; Harold G. Diamond, ed. On the Riemann hypothesis in hyperelliptic function fields, Analytic number theory (Proc. Sympos. Pure Math), Volume 24 (1973), pp. 285-302 | Zbl

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

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

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

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

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

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT