Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T14:17:55.386Z Has data issue: false hasContentIssue false
  • English
  • Français

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
David Savitt*
Affiliation:
Microsoft Research, e-mail: klauter@microsoft.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

11G2014H25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 2 , 01 April 2003 , pp. 331 - 352
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), 369–374.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), 19–36.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), 1–28.Google Scholar
[5] Waterhouse, W. C., Abelian Varieties over Finite Fields. Ann. Sci. École Norm. Sup. (4) 2 (1969), 521–560.Google Scholar