Complex Tori pp 115-178 | Cite as

Families of Complex Tori

  • Christina Birkenhake
  • Herbert Lange
Part of the Progress in Mathematics book series (PM, volume 177)


The endomorphism algebra End(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 End(X): The hermitian form H induces an anti-involution ’ on End(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.


Abelian Variety Hermitian Form Quaternion Algebra Complex Torus Algebraic Number Field 
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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Christina Birkenhake
    • 1
  • Herbert Lange
    • 2
  1. 1.Fakultät für Mathematik und PhysikUniversität BayreuthBayreuthGermany
  2. 2.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany

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