4 research outputs found
On some algebraic properties of CM-types of CM-fields and their reflexes
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