A counterexample to the local–global principle of linear dependence for Abelian varieties

Peter Jossen; Antonella Perucca

doi:10.1016/j.crma.2009.11.019

Number Theory

A counterexample to the local–global principle of linear dependence for Abelian varieties
[Un contre-exemple au principe de la dépendance linéaire des variétés abéliennes]

Présenté par : Jean-Pierre Serre

Peter Jossen ¹ ; Antonella Perucca ²

¹ NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany
² Section des mathématiques, École polytechnique fédérale de Lausanne, EPFL station 8, Ch-1015 Lausanne, Switzerland

Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 9-10.

Résumé
Abstract

Soient k un corps de nombres, A une variété abélienne sur k, P un point de $A (k)$ et X un sous-groupe de $A (k)$ . En 2002 Gajda et Kowalski ont demandé s'il est vrai que le point P appartient à X si et seulement si le point $(P \mod p)$ appartient à $(X \mod p)$ pour presque toute place finie $p$ 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 $A (k)$ and let X be a subgroup of $A (k)$ . Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point $(P \mod p)$ belongs to $(X \mod p)$ for all but finitely many primes $p$ of k. We provide a counterexample.

Reçu le : 2009-07-04
Accepté le : 2009-11-23
Publié le : 2009-12-23

DOI : 10.1016/j.crma.2009.11.019

Affiliations des auteurs :

Peter Jossen ¹ ; Antonella Perucca ²

¹ NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany
² Section des mathématiques, École polytechnique fédérale de Lausanne, EPFL station 8, Ch-1015 Lausanne, Switzerland

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     title = {A counterexample to the local{\textendash}global principle of linear dependence for {Abelian} varieties},
     journal = {Comptes Rendus. Math\'ematique},
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Peter Jossen; Antonella Perucca. A counterexample to the local–global principle of linear dependence for Abelian varieties. Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 9-10. doi : 10.1016/j.crma.2009.11.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.11.019/

Bibliographie
Cité par

[1] S. Barańczuk On a generalization of the support problem of Erdös and its analogues for abelian varieties and K-theory, Journal Pure Appl. Algebra, Volume 214 (2010), pp. 380-384

[2] G. Banaszak On a Hasse principle for Mordell–Weil groups, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009), pp. 709-714

[3] G. Banaszak; P. Krasoń On arithmetic in Mordell–Weil groups, 2009 | arXiv

[4] G. Banaszak; W. Gajda; P. Krasoń Detecting linear dependence by reduction maps, J. Number Theory, Volume 115 (2005) no. 2, pp. 322-342

[5] J. Cremona Elliptic curve data, 2009 http://www.warwick.ac.uk/staff/J.E.Cremona/

[6] W. Gajda; K. Górnisiewicz Linear dependence in Mordell–Weil groups, J. Reine Angew. Math., Volume 630 (2009), pp. 219-233

[7] P. Jossen Detecting linear dependence on a semiabelian variety, 2009 | arXiv

[8] P. Jossen, On the arithmetic of 1-motives, Ph.D. thesis, Central European University Budapest, July 2009

[9] C. Khare Compatible systems of mod p Galois representations and Hecke characters, Math. Res. Lett., Volume 10 (2003), pp. 71-83

[10] E. Kowalski Some local–global applications of Kummer theory, Manuscripta Math., Volume 111 (2003) no. 1, pp. 105-139

[11] A. Perucca, On the problem of detecting linear dependence for products of abelian varieties and tori, Acta Arith., in press

[12] A. Schinzel On power residues and exponential congruences, Acta Arith., Volume 27 (1975), pp. 397-420

[13] T. Weston Kummer theory of abelian varieties and reduction of Mordell–Weil groups, Acta Arith., Volume 110 (2003), pp. 77-88

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