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Density measure of rational points on Abelian varieties

Published online by Cambridge University Press:  22 January 2016

Michel Waldschmidt
Michel Waldschmidt*
Affiliation:
Université P. et M. Curie (Paris VI), Institut Mathématique de Jussieu, Problèmes Diophantiens, Case 247, 4, Place Jussieu, F – 75252 PARIS CEDEX 05, miw@math.jussieu.fr
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Abstract

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Let be a simple Abelian variety of dimension g over ℚ, and let ℓ be the rank of the Mordell-Weil group (ℚ). Assume ℓ ≥ 1. A conjecture of Mazur asserts that the closure of (ℚ) into (ℝ) for the real topology contains the neutral component (ℝ)0 of the origin. This is known only under the extra hypothesis ℓ ≥ g2 - g + 1. We investigate here a quantitative refinement of this question: for each given positive h, the set of points in (ℚ) of Néron-Tate height ≤ h is finite, and we study how these points are distributed into the connected component (ℝ)0. More generally we consider an Abelian variety A over a number field K embedded in ℝ, and a subgroup Γ of (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.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 155 , 1999 , pp. 27 - 53
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

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