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 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 , where Fv,0 is the maximal unramified extension of 45d4a71d0aee96cd054afc23d98f3c38" title="Click to view the MathML source">Qp contained in Fv and
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
.