The maximum or minimum number of rational points on curves of genus three over finite fields
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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.
28 pages including the appendix. Main paper by Kristin Lauter, appendix by Jean-Pierre Serre
28 pages including the appendix. Main paper by Kristin Lauter, appendix by Jean-Pierre Serre