关于费玛曲线x^N＋y^N=1的K_2群的一些注记(英文)

Some remarks on K_2 of Fermat curve x^N+y^N=1

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.

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.