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

TextThe 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 [10] follows from this.The third theorem states that some Pfister form is isomorphic to the orthogonal sum of TrK∗(Φ)/Q(a¯a) defined on the reflex fields ⊕Φ∈ΛK∗(Φ). This result suggests that the theory of complex multiplication on abelian varieties has a relationship with the multiplicative forms in higher dimension.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=IIwksVYV5YE.