Article Dans Une Revue Duke Mathematical Journal Année : 2012

A uniform open image theorem for -adic representations i

Résumé

Let k be a field finitely generated over Q and let X be a smooth, separated and geometrically connected curve over k. Fix a prime . A representation ρ : π 1 (X) → GLm(Z ) is said to be geometrically Lie perfect if the Lie algebra of ρ(π 1 (X k )) is perfect. Typical examples of such representations are those arising from the action of π 1 (X) on the generic -adic Tate module T (Aη) of an abelian scheme A over X or, more generally, from the action of π 1 (X) on the -adic etale cohomology groups H i et (Y η , Q ), i ≥ 0 of the geometric generic fiber of a smooth proper scheme Y over X. Let G denote the image of ρ. Any k-rational point x on X induces a splitting x :

The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation ρ : π 1 (X) → GLm(Z ), the set Xρ of all x ∈ X(k) such that Gx is not open in G is finite and there exists an integer Bρ ≥ 1 such that [G : Gx] ≤ Bρ for every x ∈ X(k) Xρ.

Fichier principal
Vignette du fichier
UOI1.pdf (423.25 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Licence

Dates et versions

hal-05056368 , version 1 (05-05-2025)

Licence

Identifiants

  • HAL Id : hal-05056368 , version 1

Citer

Anna Cadoret, Akio Tamagawa. A uniform open image theorem for -adic representations i. Duke Mathematical Journal, 2012. ⟨hal-05056368⟩
296 Consultations
57 Téléchargements

Partager

  • More