The maximum or minimum number of rational points on curves of genus three over finite fields

2001/04/07 by Kristin Lauter, Jean-Pierre Serre
Mathematics · #math.AG #math.NT #msc:14G45 #msc:11G10 #msc:14K15 #msc:11D45

paper · pdf

published as Compositio Mathematica 134 (p. 87-111) 2002 · 28 pages including the appendix. Main paper by Kristin Lauter, appendix by Jean-Pierre Serre

arxiv created 2001/04/07 · arxiv updated 2009/11/30

Abstract

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.

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