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 $g/2$.

[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)