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

Abstract

Let A be an abelian variety defined over a number field F⊂C and let GA be the Mumford–Tate group of A/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).Fix a valuation v of F and let p be the residue characteristic at v. For any prime number ℓ≠p, the representation ρℓ 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 GA.The p-adic representation ρp defines a representation of the Weil–Deligne group WFv′→GA/Fv,0ι, where Fv,0 is the maximal unramified extension of Qp contained in Fv and GAι is an inner form of GA over Fv,0. It is proved, under the same conditions as in the previous theorem, that, as a representation with values in GA, this representation is Q-rational and that it is compatible with the above system of representations WFv′→GA/Qℓ.