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2013 The system of representations of the Weil–Deligne group associated to an abelian variety
Rutger Noot
Algebra Number Theory 7(2): 243-281 (2013). DOI: 10.2140/ant.2013.7.243


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:ΓFGA(). 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 WFvGAl 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.


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Rutger Noot. "The system of representations of the Weil–Deligne group associated to an abelian variety." Algebra Number Theory 7 (2) 243 - 281, 2013.


Received: 10 August 2009; Revised: 17 January 2012; Accepted: 25 March 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1319.11038
MathSciNet: MR3123639
Digital Object Identifier: 10.2140/ant.2013.7.243

Primary: 11G10
Secondary: 14F20 , 14K15

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

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 2 • 2013
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