Vol. 7, No. 2, 2013

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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

Fix a number field F , an abelian variety AF and let GA be the Mumford–Tate group of A. After replacing F by finite extension one can assume that, for every prime number , the action of the absolute Galois group ΓF = Gal(F̄F) on the étale cohomology group Hét1(AF̄, ) factors through a morphism ρ: ΓF GA(). Let v be a valuation of F and write ΓFv for the absolute Galois group of the completion Fv. For every with v() = 0, the restriction of ρ to ΓFv defines a representation WFv GAl of the Weil–Deligne group.

It is conjectured that, for every , this representation of WFv is defined over as a representation with values in GA and that the system above, for variable , forms a compatible system of representations of WFv with values in GA. A somewhat weaker version of this conjecture is proved for the valuations of F, where A has semistable reduction and for which ρ(Frv) is neat.

abelian variety, compatible system of Galois representations, Weil–Deligne group
Mathematical Subject Classification 2000
Primary: 11G10
Secondary: 14K15, 14F20
Received: 10 August 2009
Revised: 17 January 2012
Accepted: 25 March 2012
Published: 25 April 2013
Rutger Noot
Institut de Recherche Mathématique Avancée
UMR 7501, Université de Strasbourg and CNRS
7 rue René Descartes
67084 Strasbourg