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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

##### MSC:
 11J95 Results involving abelian varieties 14K22 Complex multiplication and abelian varieties
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##### References:
 [1] Baker, A.: Transcendental numbers. Cambridge: Cambridge University Press 1975 · Zbl 0297.10013 [2] Coates, J.: An application of the division theory of elliptic functions to diophantine approximation. Inventiones math.11, 167-182 (1970) · Zbl 0216.04403 · doi:10.1007/BF01404611 [3] Coates, J.: On the analogue of Baker’s theorem for elliptic integrals. Unpublished 1972 · Zbl 0345.14007 [4] Lang, S.: Diophantine Geometry. New York: Interscience 1962 · Zbl 0115.38701 [5] Lang, S.: Diophantine approximation on abelian varieties with complex multiplication. Advances in Math. (1975) · Zbl 0306.14019 [6] Lang, S.: Diophantine approximations on toruses. Am. J. Math.86, 521-533 (1964) · Zbl 0142.29601 · doi:10.2307/2373022 [7] Masser, D.: Elliptic functions and transcendence. Lecture Notes in Math.437. Berlin-Heidelberg-New York: Springer 1975 · Zbl 0312.10023 [8] Masser, D.: Linear forms in algebraic points of abelian functions 1. Math. Proc. Camb. Phil. Soc.77, 499-513 (1975) · Zbl 0306.14018 · doi:10.1017/S030500410005132X [9] Ribet, K.: Division points on abelian varieties. To appear, Compositions Math. · Zbl 0348.14022 [10] Shimura, G., Taniyama, Y.: Complex multiplication of abelian varieties. Pub. Math. Soc. Japan, 1961 · Zbl 0112.03502 [11] Bashmakov, M.: Un théoreme de finitude sur la cohomologie des courbes élliptiques. C. R. Acad. Sci. Paris270, Série A, 999-1001 (1970) · Zbl 0194.52303
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