Non-existence of a curve over 𝔽₃ of genus 5 with 14 rational points

Non-existence of a curve over $\mathbb {F}_3$ of genus 5 with 14 rational points
HTML articles powered by AMS MathViewer

by Kristin Lauter PDF

Proc. Amer. Math. Soc. 128 (2000), 369-374 Request permission

We show that an absolutely irreducible, smooth, projective curve of genus $5$ over $\mathbb {F}_3$ with $14$ rational points cannot exist.

References

Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Rainer Fuhrmann and Fernando Torres, The genus of curves over finite fields with many rational points, Manuscripta Math. 89 (1996), no. 1, 103–106. MR 1368539, DOI 10.1007/BF02567508
Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 721–724 (1982). MR 656048
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
R. Schoof, Algebraic curves and coding theory, UTM 336, Univ. of Trento, 1990.
J.-P. Serre, Rational Points on curves over finite fields. Notes by F. Gouvea of lectures at Harvard University, 1985.
J.-P. Serre, Letter to K. Lauter, December 3, 1997.
H. M. Stark, On the Riemann hypothesis in hyperelliptic function fields, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 285–302. MR 0332793
Chao Ping Xing and Henning Stichtenoth, The genus of maximal function fields over finite fields, Manuscripta Math. 86 (1995), no. 2, 217–224. MR 1317746, DOI 10.1007/BF02567990