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Density measure of rational points on Abelian varieties. (English) Zbl 0931.11020
Let $${\mathcal A}$$ be a simple abelian variety of dimension $$g$$ over $${\mathbb Q}$$, and let $$\ell$$ be the rank of the Mordell-Weil group $${\mathcal A}({\mathbb Q})$$. Assume $$\ell\geq 1$$. A conjecture of Mazur [B. Mazur, Exp. Math. 1, No. 1, 35-45 (1992; Zbl 0784.14012)] asserts that the closure of $${\mathcal A}({\mathbb Q})$$ into $${\mathcal A}({\mathbb R})$$ for the real topology contains the neutral component $${\mathcal A}({\mathbb R})^0$$ of the origin. This is known only under the extra hypothesis $$\ell\geq g^2-g+1$$ [M. Waldschmidt, Exp. Math. 3, 329-352 (1994; Zbl 0837.11040) and 4, 255 (1995; Zbl 0853.11057)].
We investigate here a quantitative refinement of this question: for each given positive $$h$$, the set of points in $${\mathcal A}({\mathbb Q})$$ of Néron-Tate height $$\leq h$$ is finite, and we study how these points are distributed into the connected component $${\mathcal A}({\mathbb R})^0$$. More generally we consider an abelian variety $${\mathcal A}$$ over a number field $$K$$ embedded in $${\mathbb R}$$, and a subgroup $$\Gamma$$ of $${\mathcal A}(K)$$ of sufficiently large rank. The effective result of density we obtain relies on an estimate of diophantine approximation, namely a lower bound for linear combinations of determinants involving abelian logarithms.

##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 14G05 Rational points 11J89 Transcendence theory of elliptic and abelian functions 14K15 Arithmetic ground fields for abelian varieties 11G50 Heights
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