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

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Abstract

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 [10] follows from this.The third theorem states that some Pfister form is isomorphic to the orthogonal sum of TrK*(Φ)/Q(a{topbar}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. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=IIwksVYV5YE.

Original languageEnglish
Pages (from-to)2442-2466
Number of pages25
JournalJournal of Number Theory
Volume130
Issue number11
DOIs
Publication statusPublished - Nov 2010
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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