×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
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 Unpublished 1972 · Zbl 0345.14007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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.