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Everett W. Howe; Kristin E. Lauter
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)
Annales de l'institut Fourier, 53 no. 6 (2003), p. 1677-1737
Article PDF | Analyses MR 2038778 | Zbl 1065.11043 | 2 citations dans Cedram
Voir aussi un erratum à cet article
Class. Math.: 11G20, 14G05, 14G10, 14G15
Mots clés: courbe, point rationnel, fonction zêta, borne de Weil, borne de Serre, borne d'Oesterlé
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
${\Bbb F}_q$, 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 ${\Bbb F}_8$ 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 $g/2$.
[1] W. Bosma, J. Cannon & C. Playoust, “The Magma algebra system I: The user language”, J. Symbolic Comput. 24 (1997), p. 235-265 MR 1484478 | Zbl 0898.68039
[2] I. I. Bouw, The p-rank of curves and covers of curves, Progr. Math., Birkhäuser, 2000, p. 267-277 Zbl 0979.14015
[3] P. Deligne, “Variétés abéliennes ordinaires sur un corps fini”, Invent. Math. 8 (1969), p. 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 73 (1998), p. 426-450 MR 1657992 | Zbl 0931.11023
[6] G. van der Geer & M. van der Vlugt, “Tables of curves with many points”, Math. Comp. 69 (2000), p. 797-810 MR 1654002 | Zbl 0965.11028
[7] E. W. Howe, “Principally polarized ordinary abelian varieties over finite fields”, Trans. Amer. Math. Soc. 347 (1995), p. 2361-2401 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 92 (2002), p. 139-163 MR 1880590 | Zbl 0998.11031
[9] G. Korchmáros & F. Torres, “On the genus of a maximal curve”, Math. Ann. 323 (2002), p. 589-608 MR 1923698 | Zbl 1018.11029
[10] R. B. Lakein, “Euclid's algorithm in complex quartic fields”, Acta Arith. 20 (1972), p. 393-400
Article | 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. 328 (1999), p. 1181-1185 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 128 (2000), p. 369-374 MR 1664414 | Zbl 0983.11036
[13] K. Lauter, Zeta functions of curves over finite fields with many rational points, Springer-Verlag, 2000, p. 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. 10 (2001), p. 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. 134 (2002), p. 87-111 MR 1931964 | Zbl 1031.11038
[16] D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Oxford University Press, 1985 Zbl 0583.14015
[17] F. Oort, Commutative group schemes, Lecture Notes in Math 15, Springer-Verlag, 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. 55 (2003), p. 331-352 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. 296 (1983), p. 397-402 MR 703906 | Zbl 0538.14015
[20] J.-P. Serre, Nombres de points des courbes algébriques sur $\smallF_q$
Article | Zbl 0538.14016
[21] J.-P. Serre, “Résumé des cours de 1983--1984”, Ann. Collège France (1984), p. 79-83
[22] J.-P. Serre, “Rational points on curves over finite fields”, unpublished notes by Fernando Q. Gouvéa of lectures at Harvard University, 1985
[23] C. L. Siegel, “The trace of totally positive and real algebraic integers”, Ann. of Math (2) 46 (1945), p. 302-312 MR 12092 | Zbl 0063.07009
[24] C. Smyth, “Totally positive algebraic integers of small trace”, Ann. Inst. Fourier (Grenoble) 33 (1984), p. 1-28
Cedram | MR 762691 | Zbl 0534.12002
[25] H. M. Stark, On the Riemann hypothesis in hyperelliptic function fields, Proc. Sympos. Pure Math, American Mathematical Society, 1973, p. 285-302 Zbl 0271.14012
[26] H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, 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) 52 (1986), p. 1-19 MR 812443 | Zbl 0593.14020
[28] D. Subrao, “The p-rank of Artin-Schreier curves”, Manuscripta Math. 16 (1975), p. 169-193
Article | MR 376693 | Zbl 0321.14017
[29] J. Tate, Classes d'isogénie des variétés abéliennes sur un corps fini, Lecture Notes in Math, Springer-Verlag, 1971, p. 95-110
Numdam | Zbl 0212.25702
[30] M. E. Zieve, “Improving the Oesterlé bound”, preprint
[5] R. Fuhrmann & F. Torres, “The genus of curves over finite fields with many rational points”, Manuscripta Math 89 (1996), p. 103-106
Article | MR 1368539 | Zbl 0857.11032
[<L>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 83 (2000) Zbl 0931.11023
Everett W. Howe; Kristin E. Lauter
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)
Annales de l'institut Fourier, 53 no. 6 (2003), p. 1677-1737
Article PDF | Analyses MR 2038778 | Zbl 1065.11043 | 2 citations dans Cedram
Voir aussi un erratum à cet article
Class. Math.: 11G20, 14G05, 14G10, 14G15
Mots clés: courbe, point rationnel, fonction zêta, borne de Weil, borne de Serre, borne d'Oesterlé
Résumé - Abstract
Bibliographie
[2] I. I. Bouw, The p-rank of curves and covers of curves, Progr. Math., Birkhäuser, 2000, p. 267-277 Zbl 0979.14015
[3] P. Deligne, “Variétés abéliennes ordinaires sur un corps fini”, Invent. Math. 8 (1969), p. 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 73 (1998), p. 426-450 MR 1657992 | Zbl 0931.11023
[6] G. van der Geer & M. van der Vlugt, “Tables of curves with many points”, Math. Comp. 69 (2000), p. 797-810 MR 1654002 | Zbl 0965.11028
[7] E. W. Howe, “Principally polarized ordinary abelian varieties over finite fields”, Trans. Amer. Math. Soc. 347 (1995), p. 2361-2401 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 92 (2002), p. 139-163 MR 1880590 | Zbl 0998.11031
[9] G. Korchmáros & F. Torres, “On the genus of a maximal curve”, Math. Ann. 323 (2002), p. 589-608 MR 1923698 | Zbl 1018.11029
[10] R. B. Lakein, “Euclid's algorithm in complex quartic fields”, Acta Arith. 20 (1972), p. 393-400
Article | 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. 328 (1999), p. 1181-1185 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 128 (2000), p. 369-374 MR 1664414 | Zbl 0983.11036
[13] K. Lauter, Zeta functions of curves over finite fields with many rational points, Springer-Verlag, 2000, p. 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. 10 (2001), p. 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. 134 (2002), p. 87-111 MR 1931964 | Zbl 1031.11038
[16] D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Oxford University Press, 1985 Zbl 0583.14015
[17] F. Oort, Commutative group schemes, Lecture Notes in Math 15, Springer-Verlag, 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. 55 (2003), p. 331-352 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. 296 (1983), p. 397-402 MR 703906 | Zbl 0538.14015
[20] J.-P. Serre, Nombres de points des courbes algébriques sur $\smallF_q$
Article | Zbl 0538.14016
[21] J.-P. Serre, “Résumé des cours de 1983--1984”, Ann. Collège France (1984), p. 79-83
[22] J.-P. Serre, “Rational points on curves over finite fields”, unpublished notes by Fernando Q. Gouvéa of lectures at Harvard University, 1985
[23] C. L. Siegel, “The trace of totally positive and real algebraic integers”, Ann. of Math (2) 46 (1945), p. 302-312 MR 12092 | Zbl 0063.07009
[24] C. Smyth, “Totally positive algebraic integers of small trace”, Ann. Inst. Fourier (Grenoble) 33 (1984), p. 1-28
Cedram | MR 762691 | Zbl 0534.12002
[25] H. M. Stark, On the Riemann hypothesis in hyperelliptic function fields, Proc. Sympos. Pure Math, American Mathematical Society, 1973, p. 285-302 Zbl 0271.14012
[26] H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, 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) 52 (1986), p. 1-19 MR 812443 | Zbl 0593.14020
[28] D. Subrao, “The p-rank of Artin-Schreier curves”, Manuscripta Math. 16 (1975), p. 169-193
Article | MR 376693 | Zbl 0321.14017
[29] J. Tate, Classes d'isogénie des variétés abéliennes sur un corps fini, Lecture Notes in Math, Springer-Verlag, 1971, p. 95-110
Numdam | Zbl 0212.25702
[30] M. E. Zieve, “Improving the Oesterlé bound”, preprint
[5] R. Fuhrmann & F. Torres, “The genus of curves over finite fields with many rational points”, Manuscripta Math 89 (1996), p. 103-106
Article | MR 1368539 | Zbl 0857.11032
[<L>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 83 (2000) Zbl 0931.11023
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