The p-adic representation of the Weil–Deligne group associated to an abelian variety

Let A   be an abelian variety defined over a number field F⊂CF⊂C and let GAGA be the Mumford–Tate group of A/CA/C. After replacing F by a finite extension, we can assume that, for every prime number ℓ  , the action of ΓF=Gal(F¯/F) on Hét1(A/F¯,Qℓ) factors through a map ρℓ:ΓF→GA(Qℓ)ρℓ:ΓF→GA(Qℓ).Fix a valuation v of F and let p be the residue characteristic at v  . For any prime number ℓ≠pℓ≠p, the representation ρℓρℓ gives rise to a representation WFv′→GA/Qℓ of the Weil–Deligne group. In the case where A has semistable reduction at v it was shown in a previous paper that, with some restrictions, these representations form a compatible system of Q-rational representations with values in GAGA.The p  -adic representation ρpρp defines a representation of the Weil–Deligne group WFv′→GA/Fv,0ι, where Fv,0Fv,0 is the maximal unramified extension of QpQp contained in FvFv and GAι is an inner form of GAGA over Fv,0Fv,0. It is proved, under the same conditions as in the previous theorem, that, as a representation with values in GAGA, this representation is Q-rational and that it is compatible with the above system of representations WFv′→GA/Qℓ.