Zeta Functions of Curves over Finite Fields with Many Rational Points
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  and . In , 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.
Unable to display preview. Download preview PDF.
- 1.N. Bourbaki, Algèbre, chap. IV, Hermann, Paris, 1958.Google Scholar
- 2.G. van der Geer, M. van der Vlugt, Tables of Curves with Many Points. Regularly updated tables at:http://www.wins.uva.nl
- 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
- 7.R. SchoofAlgebraic curves and coding theory UTM 336, Univ. of Trento, 1990.Google Scholar
- J.-P. Serre, Rational Points on curves over finite fields. Notes by F. Gouvea of lectures at Harvard University, 1985.Google Scholar
- J.-P. Serre, Letter to K. Lauter, December 3, 1997.Google Scholar