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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

##### MSC:
 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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