关于费玛曲线xN+yN=1的K2群的一些注记

doi:10.7523/j.issn.2095-6134.2012.2.002

中国科学院大学学报 ›› 2012, Vol. ›› Issue (2): 154-161.DOI: 10.7523/j.issn.2095-6134.2012.2.002

关于费玛曲线x^N+y^N=1的K₂群的一些注记

田博, 唐国平, 陈虹

中国科学院研究生院数学科学学院, 北京 100049

收稿日期:2011-03-28 修回日期:2011-05-09 发布日期:2012-03-15
通讯作者: 田博, 唐国平
基金资助:
Supported by National NSFC (11071247)

Some remarks on K₂ of Fermat curve x^N+y^N=1

TIAN Bo, TANG Guo-Ping, CHEN Hong

School of Mathematical Sciences, Graduate University, Chinese Academy of Sciences, Beijing 100049, China

Received:2011-03-28 Revised:2011-05-09 Published:2012-03-15
Supported by:
Supported by National NSFC (11071247)

摘要/Abstract

摘要：

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

Abstract:

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 X_N:x^N+y^N=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.

Key words: Beilinson’s conjecture, K₂-group, Fermat curve, toric variety, CM elliptic curve, Hasse-Weil L-function, Eisenstein-Kronecker-Lerch series

中图分类号:

O154.3

田博, 唐国平, 陈虹. 关于费玛曲线x^N+y^N=1的K₂群的一些注记[J]. 中国科学院大学学报, 2012, (2): 154-161.

TIAN Bo, TANG Guo-Ping, CHEN Hong. Some remarks on K₂ of Fermat curve x^N+y^N=1[J]. , 2012, (2): 154-161.

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关于费玛曲线x^N+y^N=1的K₂群的一些注记

Some remarks on K₂ of Fermat curve x^N+y^N=1

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