Some remarks on K2 of Fermat curve xN+yN=1

doi:10.7523/j.issn.2095-6134.2012.2.002

›› 2012, Vol. ›› Issue (2): 154-161.DOI: 10.7523/j.issn.2095-6134.2012.2.002

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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 Online:2012-03-15
Supported by:
Supported by National NSFC (11071247)

Abstract

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

CLC Number:

O154.3

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

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