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The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

Published online by Cambridge University Press:  04 December 2007

Kristin Lauter and
Jean-Pierre Serre
Kristin Lauter
Affiliation:
Microsoft Corporation, One Microsoft Way, Redmond, WA 98052, U.S.A. klauter@microsoft.comwith an Appendix by Jean-Pierre Serre
Jean-Pierre Serre
Affiliation:
College de France, Paris, France
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Abstract

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We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.

Keywords

curverational pointszeta functionSerre bound

Information

Type
Research Article
Information
Compositio Mathematica , Volume 134 , Issue 1 , October 2002, pp. 87 - 111
Copyright
© 2002 Kluwer Academic Publishers