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On complex tori with many endomorphisms. (English) Zbl 0581.14032
The author studies the structures of complex tori with many endomorphisms up to isogenies.
Let \({\mathbb{T}}\) and \({\mathbb{T}}'\) be two complex tori. Denote by Hom(\({\mathbb{T}},{\mathbb{T}}')\) the set of all homomorphisms of \({\mathbb{T}}\) to \({\mathbb{T}}'\) and put \(End({\mathbb{T}})=Hom({\mathbb{T}},{\mathbb{T}})\), \(End^{{\mathbb{Q}}}({\mathbb{T}})=End({\mathbb{T}})\otimes {\mathbb{Q}}\). - In § 1 the author proves the following theorem 1-2: Let \({\mathbb{T}}\) and \({\mathbb{T}}'\) be complex tori of dimension n and n’, respectively. If rank Hom(\({\mathbb{T}},{\mathbb{T}}')=2nn'\), then \({\mathbb{T}}\) and \({\mathbb{T}}'\) are respectively isogenous to the direct products of n and n’ copies of an elliptic curve C with complex multiplication. - In § 2 the author studies a period matrix of a complex torus \({\mathbb{T}}\) such that \(End^{{\mathbb{Q}}}({\mathbb{T}})\) contains a division algebra \(D\supsetneqq {\mathbb{Q}}\) (theorem 2-3). - In § 3 the author studies the condition that a complex torus \({\mathbb{T}}\) is isogenous to the direct product of some copies of a simple torus in terms of the structure of \(End^{{\mathbb{Q}}}({\mathbb{T}})\) (theorem 3-3). - In § 4 the author determines the structure of \(End^{{\mathbb{Q}}}({\mathbb{T}})\) for a 2- dimensional complex torus \({\mathbb{T}}\). If \(End^{{\mathbb{Q}}}({\mathbb{T}})\neq {\mathbb{Q}}\), there are ten types of \(End^{{\mathbb{Q}}}({\mathbb{T}})\) and for each type there exists a complex torus \({\mathbb{T}}\) such that \(End^{{\mathbb{Q}}}({\mathbb{T}})\) has the given type. Moreover, the author shows that a 2-dimensional complex torus \({\mathbb{T}}\) is isogenous to one of the above ten complex tori, if and only if \(End^{{\mathbb{Q}}}({\mathbb{T}})\) is isomorphic to one of the above ten types as \({\mathbb{Q}}\)-algebra. - Using the results in § 4, in § 5 the author proves the following theorem 4-5: Let \({\mathbb{T}}\) be a 2-dimensional simple complex torus with non- trivial endomorphisms. Then \({\mathbb{T}}\) is an abelian variety if and only if \(End^{{\mathbb{Q}}}({\mathbb{T}})\) contains a real quadratic field over \({\mathbb{Q}}\) as a \({\mathbb{Q}}\)-subalgebra.
Reviewer: K.Ueno

14K20 Analytic theory of abelian varieties; abelian integrals and differentials
14E99 Birational geometry
32J99 Compact analytic spaces
14K22 Complex multiplication and abelian varieties
14J25 Special surfaces
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