Specialization of monodromy group and <i>ℓ</i>-independence

Hui, Chun Yin

doi:10.1016/j.crma.2011.12.012

Number Theory

Specialization of monodromy group and ℓ-independence
[Spécialisation du groupe de monodromie et ℓ-indépendance]

Présenté par : Jean-Pierre Serre

Hui, Chun Yin ¹

¹ Department of Mathematics, Indiana University, Rawles Hall, 831 E 3rd Street, Bloomington, IN 47405, USA

Comptes Rendus. Mathématique, Tome 350 (2012) no. 1-2, pp. 5-7.

Résumé
Abstract

Soit E un schéma abélien sur une variété lisse et géométriquement connexe X, définie sur un corps k de type fini sur $Q$ . Soit η le point générique de X et soit $x \in X$ un point fermé. Si $g_{ℓ}$ et ${(g_{ℓ})}_{x}$ sont les algèbres de Lie des représentations ℓ-adiques de Galois des variétés abéliennes $E_{η}$ et $E_{x}$ , alors ${(g_{ℓ})}_{x}$ est plongée dans $g_{ℓ}$ par spécialisation. Nous démontrons que lʼensemble ${x \in X point fermé | {(g_{ℓ})}_{x} ⊊ g_{ℓ}}$ 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 $Q$ . Let η be the generic point of X and $x \in X$ a closed point. If $g_{ℓ}$ and ${(g_{ℓ})}_{x}$ are the Lie algebras of the ℓ-adic Galois representations for abelian varieties $E_{η}$ and $E_{x}$ , then ${(g_{ℓ})}_{x}$ is embedded in $g_{ℓ}$ by specialization. We prove that the set ${x \in X closed point | {(g_{ℓ})}_{x} ⊊ g_{ℓ}}$ is independent of ℓ and confirm Conjecture 5.5 in Cadoret and Tamagawa [3].

Reçu le : 2011-11-13
Accepté le : 2011-12-21
Publié le : 2012-01-01

DOI : 10.1016/j.crma.2011.12.012

Affiliations des auteurs :

Hui, Chun Yin ¹

¹ Department of Mathematics, Indiana University, Rawles Hall, 831 E 3rd Street, Bloomington, IN 47405, USA

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     title = {Specialization of monodromy group and \protect\emph{\ensuremath{\ell}}-independence},
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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://www.numdam.org/articles/10.1016/j.crma.2011.12.012/

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