标题： 几何方法在提高有限域上代数曲线有理点数量的上界中的应用 显示英文标题

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

摘要： 目前，对于在有限域F_q上定义的绝对不可约、光滑、射影代数曲线，其有理点数量的最佳上界要么来自Serre对Weil界限的改进，当曲线的亏格相对于q较小时；要么来自Oesterle对显式公式法的优化，当曲线的亏格较大时。 本文提出了三种改进这些界限的方法。 所使用的论证包括曲线的theta除子的不可分解性、Galois下降和Honda-Tate理论。 改进的示例包括在q=2^3, 2^5, 2^{13}, 3^3, 3^5, 5^3, 5^7，以及当q=2^{2s}，s>1时，对一系列小亏格的界限进行降低。 对于大亏格，当q=3,8,9时获得了孤立的改进。 显示英文摘要

摘要： 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.