Abstract
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.
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© 1999 Springer Science+Business Media New York
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Birkenhake, C., Lange, H. (1999). Families of Complex Tori. In: Complex Tori. Progress in Mathematics, vol 177. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1566-0_5
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DOI: https://doi.org/10.1007/978-1-4612-1566-0_5
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7195-6
Online ISBN: 978-1-4612-1566-0
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