The system of representations of the Weil–Deligne group associated to an abelian variety

Noot, Rutger

doi:10.2140/ant.2013.7.243

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The system of representations of the Weil–Deligne group associated to an abelian variety

Rutger Noot

Vol. 7 (2013), No. 2, 243–281

DOI: 10.2140/ant.2013.7.243

Abstract

Fix a number field $F \subset ℂ$ , an abelian variety $A ∕ F$ and let $G_{A}$ be the Mumford–Tate group of $A_{∕ ℂ}$ . After replacing $F$ by finite extension one can assume that, for every prime number $ℓ$ , the action of the absolute Galois group $Γ_{F} = Gal (\bar{F} ∕ F)$ on the étale cohomology group $H_{ét}^{1} (A_{\bar{F}}, ℚ_{ℓ})$ factors through a morphism $ρ_{ℓ} : Γ_{F} \to G_{A} (ℚ_{ℓ})$ . Let $v$ be a valuation of $F$ and write $Γ_{F_{v}}$ for the absolute Galois group of the completion $F_{v}$ . For every $ℓ$ with $v (ℓ) = 0$ , the restriction of $ρ_{ℓ}$ to $Γ_{F_{v}}$ defines a representation $^{'} W_{F_{v}} \to G_{A ∕ ℚ_{l}}$ of the Weil–Deligne group.

It is conjectured that, for every $ℓ$ , this representation of $^{'} W_{F_{v}}$ is defined over $ℚ$ as a representation with values in $G_{A}$ and that the system above, for variable $ℓ$ , forms a compatible system of representations of $^{'} W_{F_{v}}$ with values in $G_{A}$ . A somewhat weaker version of this conjecture is proved for the valuations of $F$ , where $A$ has semistable reduction and for which $ρ_{ℓ} ({Fr}_{v})$ is neat.

Keywords

abelian variety, compatible system of Galois representations, Weil–Deligne group

Mathematical Subject Classification 2000

Primary: 11G10

Secondary: 14K15, 14F20

Milestones

Received: 10 August 2009

Revised: 17 January 2012

Accepted: 25 March 2012

Published: 25 April 2013

Authors

Rutger Noot
Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg and CNRS 7 rue René Descartes 67084 Strasbourg France
http://www-irma.u-strasbg.fr/~noot/

Vol. 7, No. 2, 2013