On a Hasse principle for Mordell–Weil groups

Grzegorz Banaszak

doi:10.1016/j.crma.2009.03.014

Number Theory

On a Hasse principle for Mordell–Weil groups
[Un principe de Hasse pour les groupes de Mordell–Weil]

Présenté par : Christophe Soulé

Grzegorz Banaszak ¹

¹ Department of Mathematics, Adam Mickiewicz University, 61614 Poznań, Poland

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 709-714.

Résumé
Abstract

Dans cette Note, on démontre un principe de Hasse concernant la dépendance linéaire sur $Z$ des points d'ordre infini dans le groupe de Mordell–Weil d'une variété abélienne définie sur un corps de nombres.

In this Note we establish a Hasse principle concerning the linear dependence over $Z$ of nontorsion points in the Mordell–Weil group of an abelian variety over a number field.

Reçu le : 2008-04-07
Accepté le : 2009-03-17
Publié le : 2009-05-07

DOI : 10.1016/j.crma.2009.03.014

Affiliations des auteurs :

Grzegorz Banaszak ¹

¹ Department of Mathematics, Adam Mickiewicz University, 61614 Poznań, Poland

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Grzegorz Banaszak. On a Hasse principle for Mordell–Weil groups. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 709-714. doi : 10.1016/j.crma.2009.03.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.014/

Bibliographie
Cité par

[1] G. Banaszak; W. Gajda; P. Krasoń Support problem for the intermediate Jacobians of l-adic representations, J. Number Theory, Volume 100 (2003) no. 1, pp. 133-168

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

[3] G. Banaszak; W. Gajda; P. Krasoń On reduction map for étale K-theory of curves, Homology Homotopy Appl., Volume 7 (2005) no. 3, pp. 1-10

[4] S. Barańczuk On reduction maps and support problem in K-theory and abelian varieties, J. Number Theory, Volume 119 (2006), pp. 1-17

[5] F.A. Bogomolov Sur l'algébricité des représentations l-adiques, C. R. Acad. Sci. Paris Sér. A-B, Volume 290 (1980), p. A701-A703

[6] C. Corralez-Rodrigáñez; R. Schoof Support problem and its elliptic analogue, J. Number Theory, Volume 64 (1997), pp. 276-290

[7] G. Faltings Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Volume 73 (1983), pp. 349-366

[8] W. Gajda, K. Górnisiewicz, Linear dependence in Mordell–Weil groups, J. Reine Angew. Math., in press

[9] M. Hindry; J.H. Silverman Diophantine Geometry an Introduction, Graduate Texts in Math., vol. 201, Springer, 2000

[10] N.M. Katz Galois properties of torsion points on abelian varieties, Invent. Math., Volume 62 (1981), pp. 481-502

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

[12] M. Larsen, R. Schoof, Whitehead's lemmas and Galois cohomology of abelian varieties, preprint

[13] A. Perucca, The l-adic support problem for abelian varieties, preprint, 2007

[14] R. Pink On the order of the reduction of a point on an abelian variety, Math. Ann., Volume 330 (2004), pp. 275-291

[15] K.A. Ribet Kummer theory on extensions of abelian varieties by tori, Duke Math. J., Volume 46 (1979) no. 4, pp. 745-761

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

[17] J.-P. Serre; J. Tate Good reduction of abelian varieties, Ann. of Math., Volume 68 (1968), pp. 492-517

[18] A. Weil Variétés Abélienne et Courbes Algébriques, Hermann, Paris, 1948

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

[20] J.G. Zarhin A finiteness theorem for unpolarized abelian varieties over number fields with prescribed places of bad reduction, Invent. Math., Volume 79 (1985), pp. 309-321

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