Density measure of rational points on abelian varieties

Abstract

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.