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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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