Zeta Functions of Curves over Finite Fields with Many Rational Points

  • Kristin Lauter
Conference paper

Abstract

Currently, the best known bounds on the number of rational points on an absolutely irreducible, smooth, projective curve defined over a finite field in most cases come from the optimization of the explicit formulae method, which is detailed in [7] and [9]. In [4], we gave an example of a case where the best known bound could not be met. The purpose of this paper is to give another new example of this phenomenon, and to explain the method used to obtain these results. This method gives rise to lists of all possible zeta functions, even in cases where we cannot conclude that no curve exists to meet the bound.

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References

  1. 1.
    N. Bourbaki, Algèbre, chap. IV, Hermann, Paris, 1958.Google Scholar
  2. 2.
    G. van der Geer, M. van der Vlugt, Tables of Curves with Many Points. Regularly updated tables at:http://www.wins.uva.nl
  3. 3.
    Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokyo 28 (1981), p. 721–724.MathSciNetMATHGoogle Scholar
  4. 4.
    K. LauterNon-existence of a Curve over IF3 of Genus 5 with 14 Rational Points Max Planck Institut Preprint 98–51, submitted for publication March, 1998.Google Scholar
  5. 5.
    J. Milne, Etale Cohomology. Princeton University Press: Princeton, NJ, 1980.MATHGoogle Scholar
  6. 6.
    H. Niederreiter, C.P. Xing, Cyclotomic function fields, Hilbert class fields and global function fields with many rational places. Acta Arithm. 79 (1997), p.59–76.MathSciNetMATHGoogle Scholar
  7. 7.
    R. SchoofAlgebraic curves and coding theory UTM 336, Univ. of Trento, 1990.Google Scholar
  8. 8.
    J.-P. SerreSur le nombre des points rationnels d’une courbe algébrique sur un corps fini C.R. Acad. Sc. Paris Sér. I Math. 296 (1983), p.397–402.MATHGoogle Scholar
  9. J.-P. Serre, Rational Points on curves over finite fields. Notes by F. Gouvea of lectures at Harvard University, 1985.Google Scholar
  10. J.-P. Serre, Letter to K. Lauter, December 3, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Kristin Lauter
    • 1
  1. 1.University of MichiganAnn ArborUSA

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