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1999 Density measure of rational points on abelian varieties
Michel Waldschmidt
Nagoya Math. J. 155: 27-53 (1999).


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 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 $\ell \geq g^{2}-g+1$. 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.


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Michel Waldschmidt. "Density measure of rational points on abelian varieties." Nagoya Math. J. 155 27 - 53, 1999.


Published: 1999
First available in Project Euclid: 27 April 2005

zbMATH: 0931.11020
MathSciNet: MR1711387

Primary: 11J95
Secondary: 11J82, 14G05

Rights: Copyright © 1999 Editorial Board, Nagoya Mathematical Journal


Vol.155 • 1999
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