Abstract
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.
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with an Appendix by Jean-Pierre Serre
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Lauter, K., Serre, JP. The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields. Compositio Mathematica 134, 87–111 (2002). https://doi.org/10.1023/A:1020246226326
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DOI: https://doi.org/10.1023/A:1020246226326