On some algebraic properties of CM-types of CM-fields and their reflexes

JOURNAL OF NUMBER THEORY

卷:130

Article

Oishi-Tomiyasu, Ryoko

关键词: Complex multiplication;    CM-field;    Abelian extension;    Pfister form;   

DOI  :  10.1016/j.jnt.2010.03.021

来源: Elsevier

【 摘 要 】

Text. The purpose of this paper is to show that the reflex fields of a given CM-field K are equipped with a certain combinatorial structure that has not been exploited yet. The first theorem is on the abelian extension generated by the moduli and the b-torsion points of abelian varieties of CM-type, for any natural number b. It is a generalization of the result by Wei on the abelian extension obtained by the moduli and all the torsion points. The second theorem gives a character identity of the Artin L-function of a CM-field K and the reflex fields of K. The character identity pointed out by Shimura (1977) in 110] follows from this. The third theorem states that some Pfister form is isomorphic to the orthogonal sum of Trk*(phi)/Q((a) over bara) defined on the reflex fields circle plus(phi is an element of Lambda) K*(phi). This result suggests that the theory of complex multiplication on abelian varieties has a relationship with the multiplicative forms in higher dimension. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=IIwksVYV5YE. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free