Duke Mathematical Journal

Curves of every genus with many points, II: Asymptotically good families

Noam D. Elkies, Everett W. Howe, Andrew Kresch, Bjorn Poonen, Joseph L. Wetherell, and Michael E. Zieve

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Abstract

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant cq with the following property: for every integer g≥0, there is a genus-g curve over Fq with at least cqg rational points over Fq. Moreover, we show that there exists a positive constant d such that for every q we can choose cq=d log q. We show also that there is a constant c>0 such that for every q and every n>0, and for every sufficiently large g there is a genus-g curve over Fq that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (/n)r for some r>cg/n.

Article information

Source
Duke Math. J., Volume 122, Number 2 (2004), 399-422.

Dates
First available in Project Euclid: 14 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1081971771

Digital Object Identifier
doi:10.1215/S0012-7094-04-12224-9

Mathematical Reviews number (MathSciNet)
MR2053756

Zentralblatt MATH identifier
1072.11041

Subjects
Primary: 11G20: Curves over finite and local fields [See also 14H25]
Secondary: 14G05, 14G15: Finite ground fields

Citation

Elkies, Noam D.; Howe, Everett W.; Kresch, Andrew; Poonen, Bjorn; Wetherell, Joseph L.; Zieve, Michael E. Curves of every genus with many points, II: Asymptotically good families. Duke Math. J. 122 (2004), no. 2, 399--422. doi:10.1215/S0012-7094-04-12224-9. https://projecteuclid.org/euclid.dmj/1081971771


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References

  • J. A. Csirik, J. L. Wetherell, and M. E. Zieve, On the genera of $X_0(N)$, preprint., to appear in J. Number Theory.
  • N. D. Elkies, “Explicit modular towers” in Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing (Urbana, Ill., 1997), ed. T. Başar and A. Vardy, Univ. of Illinois at Urbana-Champaign, 1998, 23–32..
  • –. –. –. –., “Explicit towers of Drinfeld modular curves” in European Congress of Mathematics, Vol. II (Barcelona, 2000), ed. C. Casacuberta, R. M. Miró-Roig, J. Verdera, and S. Xambó-Descamps, Progr. Math. 202, Birkhauser, Basel, 2001, 189–198.
  • A. Garcia and H. Stichtenoth, A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vlăduţ bound, Invent. Math. 121 (1995), 211–222.
  • –. –. –. –., On the asymptotic behaviour of some towers of function fields over finite fields, J. Number Theory 61 (1996), 248–273.
  • A. Garcia, H. Stichtenoth, and M. Thomas, On towers and composita of towers of function fields over finite fields, Finite Fields Appl. 3 (1997), 257–274.
  • H. Hasse, Zur Theorie der abstrakten elliptischen Funktionkörper, I, II, III, J. Reine Angew. Math. 175 (1936), 55–62., 69–88., 193–208.
  • Y. Ihara, “Algebraic curves mod $\mathfrakp$ and arithmetic groups” in Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965), ed. A. Borel and G. D. Mostow, Amer. Math. Soc., Providence, 1966, 265–271.
  • ––––, On Congruence Monodromy Problems, Vol. 2, Lecture Notes 2, Department of Mathematics, Univ. Tokyo, 1969.
  • –. –. –. –., “On modular curves over finite fields” in Discrete Subgroups of Lie Groups and Applications to Moduli (Bombay, 1973), Oxford Univ. Press, Bombay, 1975, 161–202.
  • –. –. –. –., “Congruence relations and Shimūra curves” in Automorphic Forms, Representations, and $L$-Functions, Part 2 (Corvallis, Ore., 1977), ed. A. Borel and W. Casselman, Proc. Sympos. Pure Math 33, Amer. Math. Soc., Providence, 1979, 291–311.
  • –. –. –. –., Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 721–724.
  • –. –. –. –., “Shimura curves over finite fields and their rational points” in Applications of Curves over Finite Fields (Seattle, Wash., 1997), ed. M. D. Fried, Contemp. Math. 245, Amer. Math. Soc., Providence, 1999, 15–23.
  • A. Kresch, J. Wetherell, and M. E. Zieve, Curves of every genus with many points, I: Abelian and toric families, J. Algebra 250 (2002), 353–370.
  • W.-C. W. Li and H. Maharaj, Coverings of curves with asymptotically many rational points, J. Number Theory 96 (2002), 232–256.
  • Yu. I. Manin, What is the maximum number of points on a curve over $\mathbbF_2$?, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 715–720.
  • Y. Morita, Reduction modulo $\mathfrak P$ of Shimura curves, Hokkaido Math. J. 10 (1981), 209–238.
  • B. Poonen, Bertini theorems over finite fields, to appear in Ann. of Math. (2), preprint.
  • R. Schoof, Algebraic curves over $\mathbbF_2$ with many rational points, J. Number Theory 41 (1992), 6–14.
  • J.-P. Serre,“Nombres de points des courbes algébriques sur $\mathbbF_q$” in Seminar on Number Theory (Talence, France, 1982/1983), Univ. Bordeaux I, Talence, 1983, exp. 22.
  • –. –. –. –., Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Sér I Math. 296 (1983), 397–402.
  • ––––, Rational points on curves over finite fields, unpublished lecture notes by F. Q. Gouvêa, Harvard University, Cambridge, 1985.
  • G. Shimura, Arithmetic of unitary groups, Ann. of Math. (2) 79 (1964), 369–409.
  • –. –. –. –., On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 91 (1970), 144–222.
  • H. Stichtenoth, Algebraic Function Fields and Codes, Universitext, Springer, Berlin, 1993.
  • S. G. Vlăduţ [Vlèduts] and V. G. Drinfel'd, The number of points of an algebraic curve (in Russian), Funktsional. Anal. i Prilozhen. 17, no. 1 (1983), 68–69.; English translation in Funct. Anal. Appl. 17 (1983), 53–54.
  • A. Weil, Sur les fonctions algébriques à corps de constantes fini, C. R. Acad. Sci. Paris 210 (1940), 592–594.
  • –. –. –. –., On the Riemann hypothesis in functionfields, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 345–347.
  • ––––, Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Sci. Indust. 1041, Hermann, Paris, 1948.
  • ––––, Variétés abéliennes et courbes algébriques, Actualités Sci. Indust. 1064, Hermann, Paris, 1948.
  • T. Zink, “Degeneration of Shimura surfaces and a problem in coding theory” in Fundamentals of Computation Theory (Cottbus, Germany, 1985), ed. L. Budach, Lecture Notes in Comput. Sci. 199, Springer, Berlin, 1985, 503–511.

See also

  • First in series: A. Kresch, J. Wetherell, M.E. Zieve. Curves of every genus with many points, 1: Abelian and toric families. J Algebra 250 (2002), 353-370.