Soit E un schéma abélien sur une variété lisse et géométriquement connexe X, définie sur un corps k de type fini sur . Soit η le point générique de X et soit un point fermé. Si et sont les algèbres de Lie des représentations ℓ-adiques de Galois des variétés abéliennes et , alors est plongée dans par spécialisation. Nous démontrons que lʼensemble est indépendant de ℓ, ce qui confirme la Conjecture 5.5 de Cadoret et Tamagawa [3].
Let E be an abelian scheme over a geometrically connected, smooth variety X defined over k, a finitely generated field over . Let η be the generic point of X and a closed point. If and are the Lie algebras of the ℓ-adic Galois representations for abelian varieties and , then is embedded in by specialization. We prove that the set is independent of ℓ and confirm Conjecture 5.5 in Cadoret and Tamagawa [3].
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@article{CRMATH_2012__350_1-2_5_0, author = {Hui, Chun Yin}, title = {Specialization of monodromy group and \protect\emph{\ensuremath{\ell}}-independence}, journal = {Comptes Rendus. Math\'ematique}, pages = {5--7}, publisher = {Elsevier}, volume = {350}, number = {1-2}, year = {2012}, doi = {10.1016/j.crma.2011.12.012}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2011.12.012/} }
TY - JOUR AU - Hui, Chun Yin TI - Specialization of monodromy group and ℓ-independence JO - Comptes Rendus. Mathématique PY - 2012 SP - 5 EP - 7 VL - 350 IS - 1-2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2011.12.012/ DO - 10.1016/j.crma.2011.12.012 LA - en ID - CRMATH_2012__350_1-2_5_0 ER -
Hui, Chun Yin. Specialization of monodromy group and ℓ-independence. Comptes Rendus. Mathématique, Tome 350 (2012) no. 1-2, pp. 5-7. doi : 10.1016/j.crma.2011.12.012. http://archive.numdam.org/articles/10.1016/j.crma.2011.12.012/
[1] Sur lʼalgébricité des représentations ℓ-adiques, C. R. Acad. Sci. Paris, Ser. I, Volume 290 (1980), pp. 701-703
[2] Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126, Springer-Verlag, 1991
[3] A. Cadoret, A. Tamagawa, A uniform open image theorem for ℓ-adic representations I, preprint.
[4] Complements to Mordell (Faltings, G.; Wüstholz, G., eds.), Rational Points, Seminar Bonn/Wuppertal, 1983–1984, Vieweg, 1984 (Chapter 6)
[5] Jacobian varieties, Arithmetic Geometry, Springer-Verlag, New York, 1986
[6] J.-P. Serre, Letter to K.A. Ribet, Jan. 1, 1981, reproduced in Collected Papers, vol. IV, no. 133.
[7] Abelian varieties, ℓ-adic representations and Lie algebras. Rank independence on ℓ, Invent. Math., Volume 55 (1979), pp. 165-176
[8] Abelian varieties having a reduction of K3 type, Duke Math. J., Volume 65 (1992), pp. 511-527
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