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

Abstract

Fix a number field F⊂ℂ, an abelian variety A∕F and let GA 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 ΓF= Gal(F∕F) on the etale cohomology group Het1(AF,ℚl) factors through a morphism ρl:ΓF→GA(ℚl). Let v be a valuation of F and write ΓFv for the absolute Galois group of the completion Fv. For every l with v(l)=0, the restriction of ρl to ΓFv defines a representation ′WFv→GA∕ℚl of the Weil–Deligne group.It is conjectured that, for every l, this representation of ′WFv is defined over ℚ as a representation with values in GA and that the system above, for variable l, 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 ρl(Frv) is neat.