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