Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields
Abstract
Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus is small compared to q, or from Oesterle's optimization of the explicit formulae method if the genus is large. This paper presents three methods for improving these bounds. The arguments used are the indecomposability of the theta divisor of a curve, Galois descent, and Honda-Tate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q=2^3, 2^5, 2^{13}, 3^3, 3^5, 5^3, 5^7, and when q=2^{2s}, s>1. For large genera, isolated improvements are obtained for q=3,8,9.
Keywords
Cite
@article{arxiv.math/0104247,
title = {Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields},
author = {Kristin Lauter and Jean-Pierre Serre},
journal= {arXiv preprint arXiv:math/0104247},
year = {2007}
}
Comments
16 pages including the appendix. Main paper by Kristin Lauter, appendix by Jean-Pierre Serre