Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields

2001/04/25 by Kristin Lauter, Jean-Pierre Serre
Mathematics · #math.AG #math.NT #msc:14G45 #msc:11G10 #msc:14K15 #msc:11D45

paper · pdf

published as Journal of Algebraic Geometry 10 (2001), 19-36 · 16 pages including the appendix. Main paper by Kristin Lauter, appendix by Jean-Pierre Serre

arxiv created 2001/04/25 · arxiv updated 2009/11/30

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=2³, 2⁵, 2¹³, 3³, 3⁵, 5³, 5⁷, and when q=2^2s, s>1. For large genera, isolated improvements are obtained for q=3,8,9.

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