
Contents
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V.A Oriented Number Fields V.A Oriented Number Fields
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V.B Hodge Structures With Special Endomorphisms V.B Hodge Structures With Special Endomorphisms
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V.C A Categorical Equivalence V.C A Categorical Equivalence
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V.D Polarization and Mumford-Tate Groups V.D Polarization and Mumford-Tate Groups
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V.E An Extended Example V.E An Extended Example
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V.F Proofs of Propositions V.D.4 And V.D.5 In The Galois Case V.F Proofs of Propositions V.D.4 And V.D.5 In The Galois Case
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Cite
Abstract
This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining an n-orientation of a totally imaginary number field and constructs a precise correspondence between these and certain important kinds of CM Hodge structures. In the classical case of weight n = 1, the abelian variety associated to a CM type is recovered. The notion of the Kubota rank and reflex field associated to a CM Hodge structure is then generalized to the totally imaginary number field setting. When the Kubota rank is maximal, the CM Hodge structure is non-degenerate. The discussion also covers oriented imaginary number fields, Hodge structures with special endomorphisms, polarization and Mumford-Tate groups, and the Mumford-Tate group in the Galois case.
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