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Diophantine approximation on abelian varieties with complex multiplication. (English) Zbl 0342.10018
Summary: Let \(A\) be an abelian variety of dimension \(d\), defined over an algebraic number field \(K\), with many complex multiplications in the sense of Shimura. There is a theta map \(\Theta\) from \(\mathbb C^d\) to \(A\), with algebraic derivative at the origin, by means of which the endomorphism ring \(\mathrm{End}\) of \(A\) is represented by a ring \(E\) of diagonal matrices with algebraic entries. Let \(u_1,\ldots, u_n\) be points of \(\mathbb C^d\), linearly independent over \(E\), such that \(\Theta(u_1),\ldots, \Theta(u_n)\) are points on \(A\) defined over \(K\). It

11J95 Results involving abelian varieties
14K22 Complex multiplication and abelian varieties
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