关于费玛曲线xN+yN=1的K2群的一些注记
收稿日期: 2011-03-28
修回日期: 2011-05-09
网络出版日期: 2012-03-15
基金资助
Supported by National NSFC (11071247)
Some remarks on K2 of Fermat curve xN+yN=1
Received date: 2011-03-28
Revised date: 2011-05-09
Online published: 2012-03-15
Supported by
Supported by National NSFC (11071247)
首先简单介绍了对于有理数域上光滑射影曲线的Beilinson猜想,然后应用椭圆簇的知识指出了存在于费玛曲线K2群中的一个元素, 最后在费玛曲线X3这个特殊情形下,将其L-函数与Eisenstein-Kronecker-Lerch级数明确地联系起来,从而验证了其L-函数满足的函数方程,以及它能在复平面上解析延拓的事实.
关键词： Beilinson 猜想; K2群; 费玛曲线; 椭圆簇; CM 椭圆曲线; L-函数; Eisenstein-Kronecker-Lerch级数
田博 , 唐国平 , 陈虹 . 关于费玛曲线xN+yN=1的K2群的一些注记[J]. 中国科学院大学学报, 2012 , (2) : 154 -161 . DOI: 10.7523/j.issn.2095-6134.2012.2.002
We first review Beilinson’s conjecture for a smooth projective curve C over Q. Then we exhibit an element in K2-group of the Fermat curve XN:xN+yN=1 from a toric variety viewpoint. Finally, we focus on the special case of X3 and explicitly express its Hasse-Weil L-function L(X3,s) in terms of the Eisenstein-Kronecker-Lerch series, which allows us to verify that L(X3,s) satisfies a certain functional equation and has a meromorphic continuation in the entire complex plane.
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