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The Maximum Number of Points on a Curve of Genus 4 over ${{\mathbb{F}}_{8}}$ is 25

Published online by Cambridge University Press:  20 November 2018

David Savitt*
Affiliation:
Microsoft Research, e-mail: klauter@microsoft.com
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Abstract

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We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Lauter, K., Non-existence of a curve over F3 of genus 5 with 14 rational points. Proc. Amer.Math. Soc. 128 (2000), 369374.Google Scholar
[2] Lauter, K., with an Appendix by Serre, J.-P., Geometric Methods for Improving the Upper Bounds on the Number of Rational Points on Algebraic Curves over Finite Fields. J. Algebraic Geom. (1) 10 (2001), 1936.Google Scholar
[3] Serre, J.-P., Rational Points on Curves over Finite Fields. Notes by F. Gouvea of lectures at Harvard University, 1985.Google Scholar
[4] Smyth, C., Totally Positive Algebraic Integers of Small Trace. Ann. Inst. Fourier (Grenoble) (3) 33 (1984), 128.Google Scholar
[5] Waterhouse, W. C., Abelian Varieties over Finite Fields. Ann. Sci. École Norm. Sup. (4) 2 (1969), 521560.Google Scholar
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