Geometric Methods for Improving the Upper Bounds on the Number of Rational Points on Algebraic Curves over Finite Fields
Geometric Methods for Improving the Upper Bounds on the Number of Rational Points on Algebraic Curves over Finite Fields
复制标题
改进有限域代数曲线上有理点数上界的几何方法
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
Jean
中科院分区：
文献类型：
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作者：
K. Lauter;Jean
目前，最好的上界的数量合理的点上的绝对不可约的，光滑的，射影代数曲线的genus g定义在有限域Fq要么来自塞尔的细化的Weil界相比，如果属小q，或从Oesterle的优化显式公式的方法，如果属大。本文提出了三种方法来改善这些界限。使用的参数是曲线的θ因子的不可分解性，伽罗瓦下降，和本田泰特理论。改进的例子包括当q = 2 3，2 5，2 13，3 3，3 5，5 3，5 7时，以及当q = 2 2 s，s > 1时，降低了大范围的小亏格的边界。对于大的亏格，q = 3，8，9时得到了孤立的改进.
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 Fq 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.