关于CM-域的CM-型及其自反的若干代数性质
On some algebraic properties of CM-types of CM-fields and their reflexes
摘要：
本文的目的是表明，一个给定的CM-场的反射场配备了一定的组合结构，尚未开发。我们证明了三个定理，使用这种结构，第一个定理是由CM-型的阿贝尔簇的模和b-扭点生成的阿贝尔扩张，对任何自然数b。它推广了Wei关于由模和所有挠点得到的阿贝尔扩张的结果。第二个定理给出了CM-域K的Artin L-函数和K的反射域的特征恒等式。志村所指出的人物同一性由此而来。第三个定理指出某些Pfister形式同构于定义在反射场的直和上的Tr（\bar{a} a）的正交和（\bar{a}是a的复共轭）。这一结果表明，阿贝尔簇上的复乘理论与高维乘法形式有联系。
英文摘要：
The purpose of this paper is to show that the reflex fields of a given CM-field is equipped with a certain combinatorial structure that has not been exploited yet. We prove three theorems using this structure; 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 follows from this. The third theorem states that some Pfister form is isomorphic to the orthogonal sum of Tr(\bar{a}a) (\bar{a} is the complex conjugation of a) defined on a direct sum of reflex fields. This result suggests that the theory of complex multiplication on abelian varieties has a relationship with the multiplicative forms in higher dimension.
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