The Maximum Number of Points on a Curve of Genus

Our aim in this paper is to prove that a smooth geometrically irreducible curve C of genus 4 over the finite field F8 may have at most 25 F8-points. Our strategy is as follows: if C has more than 18 F8-points, then C may not be hyperelliptic, and so the canonical divisor of C yields an embedding of C into P3F8 . The image of C under this embedding is a degree 6 curve which is precisely the intersection of an irreducible cubic hypersurface with an irreducible quadric hypersurface, both defined over F8. (This is Example IV.5.2.2 in [Har]. Hartshorne works over an algebraically closed field, but his argument is equally valid over the smaller field. See, for example, Theorem III.5.1 in [Har] and Theorem A.4.2.1 in [HS] for the necessary tools.)