POINTLESS CURVES OF GENUS THREE AND FOUR by
نویسندگان
چکیده
— A curve over a field k is pointless if it has no k-rational points. We show that there exist pointless genus-3 hyperelliptic curves over a finite field Fq if and only if q 6 25, that there exist pointless smooth plane quartics over Fq if and only if either q 6 23 or q = 29 or q = 32, and that there exist pointless genus-4 curves over Fq if and only if q 6 49. Résumé (Courbes de genre 3 et4 sans point). — Une courbe sur un corps k est appelée une courbe sans point si elle n’a aucun point k-rationnel. Nous prouvons qu’il existe des courbes hyperelliptiques de genre trois sans point sur un corps fini Fq si et seulement si q 6 25, qu’il existe des quartiques planes sans point sur un corps fini Fq si et seulement si q 6 23, q = 29 ou q = 32, et qu’il existe des courbes de genre quatre sans point sur un corps fini Fq si et seulement si q 6 49.
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