The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields
Kristin Lauter, Jean-Pierre Serre · Compositio Mathematica · 2002
We show that for all finite fields F q , there exists a curve C over F q 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 F q .