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

Chun Yin Hui

doi:10.1016/j.crma.2011.12.012

Number Theory

Specialization of monodromy group and ℓ-independence

Presented by: Jean-Pierre Serre

Chun Yin Hui ¹

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

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

Abstract
Résumé

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].

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].

Received: 2011-11-13
Accepted: 2011-12-21
Published online: 2012-01-01

DOI: 10.1016/j.crma.2011.12.012

Author's affiliations:

Chun Yin Hui ¹

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

@article{CRMATH_2012__350_1-2_5_0,
     author = {Chun Yin Hui},
     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},
}

TY  - JOUR
AU  - Chun Yin Hui
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
DO  - 10.1016/j.crma.2011.12.012
LA  - en
ID  - CRMATH_2012__350_1-2_5_0
ER  -

%0 Journal Article
%A Chun Yin Hui
%T Specialization of monodromy group and ℓ-independence
%J Comptes Rendus. Mathématique
%D 2012
%P 5-7
%V 350
%N 1-2
%I Elsevier
%R 10.1016/j.crma.2011.12.012
%G en
%F CRMATH_2012__350_1-2_5_0

Chun Yin Hui. Specialization of monodromy group and ℓ-independence. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 5-7. doi : 10.1016/j.crma.2011.12.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.12.012/

References
Cited by

[1] F.A. Bogomolov Sur lʼalgébricité des représentations ℓ-adiques, C. R. Acad. Sci. Paris, Ser. I, Volume 290 (1980), pp. 701-703

[2] A. Borel 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] G. Faltings Complements to Mordell (G. Faltings; G. Wüstholz, eds.), Rational Points, Seminar Bonn/Wuppertal, 1983–1984, Vieweg, 1984 (Chapter 6)

[5] J.S. Milne 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] Yu.G. Zarhin Abelian varieties, ℓ-adic representations and Lie algebras. Rank independence on ℓ, Invent. Math., Volume 55 (1979), pp. 165-176

[8] Yu.G. Zarhin Abelian varieties having a reduction of K3 type, Duke Math. J., Volume 65 (1992), pp. 511-527

Cited by Sources:

Comments - Policy