Families of Complex Tori
The endomorphism algebra Endℚ(X) of a simple complex torus X is a skew field of finite dimension over ℚ. According to the Theorem of Oort-Zarhin (see Section 1.9) every skew field of finite dimension over ℚ occurs as the endomorphism algebra of a complex torus. For nondegenerate complex tori the situation is completely different: The existence of a polarization H of index k on X gives strong restrictions for Endℚ(X): The hermitian form H induces an anti-involution ’ on Endℚ(X). The skew fields F of finite type over ℚ with anti-involution ′ were classified by Albert. In this chapter we work out which of these algebras can be realized as endomorphism algebras of nondegenerate complex tori.
KeywordsAbelian Variety Hermitian Form Quaternion Algebra Complex Torus Algebraic Number Field
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