In this Note we establish a Hasse principle concerning the linear dependence over of nontorsion points in the Mordell–Weil group of an abelian variety over a number field.
Dans cette Note, on démontre un principe de Hasse concernant la dépendance linéaire sur des points d'ordre infini dans le groupe de Mordell–Weil d'une variété abélienne définie sur un corps de nombres.
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@article{CRMATH_2009__347_13-14_709_0, author = {Banaszak, Grzegorz}, title = {On a {Hasse} principle for {Mordell{\textendash}Weil} groups}, journal = {Comptes Rendus. Math\'ematique}, pages = {709--714}, publisher = {Elsevier}, volume = {347}, number = {13-14}, year = {2009}, doi = {10.1016/j.crma.2009.03.014}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.03.014/} }
TY - JOUR AU - Banaszak, Grzegorz TI - On a Hasse principle for Mordell–Weil groups JO - Comptes Rendus. Mathématique PY - 2009 SP - 709 EP - 714 VL - 347 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.03.014/ DO - 10.1016/j.crma.2009.03.014 LA - en ID - CRMATH_2009__347_13-14_709_0 ER -
Banaszak, Grzegorz. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.03.014/
[1] Support problem for the intermediate Jacobians of l-adic representations, J. Number Theory, Volume 100 (2003) no. 1, pp. 133-168
[2] Detecting linear dependence by reduction maps, J. Number Theory, Volume 115 (2005) no. 2, pp. 322-342
[3] On reduction map for étale K-theory of curves, Homology Homotopy Appl., Volume 7 (2005) no. 3, pp. 1-10
[4] On reduction maps and support problem in K-theory and abelian varieties, J. Number Theory, Volume 119 (2006), pp. 1-17
[5] Sur l'algébricité des représentations l-adiques, C. R. Acad. Sci. Paris Sér. A-B, Volume 290 (1980), p. A701-A703
[6] Support problem and its elliptic analogue, J. Number Theory, Volume 64 (1997), pp. 276-290
[7] 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] Diophantine Geometry an Introduction, Graduate Texts in Math., vol. 201, Springer, 2000
[10] Galois properties of torsion points on abelian varieties, Invent. Math., Volume 62 (1981), pp. 481-502
[11] 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] On the order of the reduction of a point on an abelian variety, Math. Ann., Volume 330 (2004), pp. 275-291
[15] Kummer theory on extensions of abelian varieties by tori, Duke Math. J., Volume 46 (1979) no. 4, pp. 745-761
[16] On power residues and exponential congruences, Acta Arith., Volume 27 (1975), pp. 397-420
[17] Good reduction of abelian varieties, Ann. of Math., Volume 68 (1968), pp. 492-517
[18] Variétés Abélienne et Courbes Algébriques, Hermann, Paris, 1948
[19] Kummer theory of abelian varieties and reductions of Mordell–Weil groups, Acta Arith., Volume 110 (2003), pp. 77-88
[20] 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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