[Un contre-exemple au principe de la dépendance linéaire des variétés abéliennes]
Soient k un corps de nombres, A une variété abélienne sur k, P un point de et X un sous-groupe de . En 2002 Gajda et Kowalski ont demandé s'il est vrai que le point P appartient à X si et seulement si le point appartient à pour presque toute place finie de k. Nous donnons une réponse négative à cette question.
Let A be an Abelian variety defined over a number field k. Let P be a point in and let X be a subgroup of . Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point belongs to for all but finitely many primes of k. We provide a counterexample.
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@article{CRMATH_2010__348_1-2_9_0,
author = {Jossen, Peter and Perucca, Antonella},
title = {A counterexample to the local{\textendash}global principle of linear dependence for {Abelian} varieties},
journal = {Comptes Rendus. Math\'ematique},
pages = {9--10},
publisher = {Elsevier},
volume = {348},
number = {1-2},
year = {2010},
doi = {10.1016/j.crma.2009.11.019},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.11.019/}
}
TY - JOUR AU - Jossen, Peter AU - Perucca, Antonella TI - A counterexample to the local–global principle of linear dependence for Abelian varieties JO - Comptes Rendus. Mathématique PY - 2010 SP - 9 EP - 10 VL - 348 IS - 1-2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.11.019/ DO - 10.1016/j.crma.2009.11.019 LA - en ID - CRMATH_2010__348_1-2_9_0 ER -
%0 Journal Article %A Jossen, Peter %A Perucca, Antonella %T A counterexample to the local–global principle of linear dependence for Abelian varieties %J Comptes Rendus. Mathématique %D 2010 %P 9-10 %V 348 %N 1-2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.11.019/ %R 10.1016/j.crma.2009.11.019 %G en %F CRMATH_2010__348_1-2_9_0
Jossen, Peter; Perucca, Antonella. A counterexample to the local–global principle of linear dependence for Abelian varieties. Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 9-10. doi : 10.1016/j.crma.2009.11.019. http://www.numdam.org/articles/10.1016/j.crma.2009.11.019/
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