Fix a valuation v of F and let p be the residue characteristic at v . For any prime number a1218fe3d6cd2" title="Click to view the MathML source">ℓ≠p, the representation 90101d91be212c478" title="Click to view the MathML source">ρℓ gives rise to a representation a17ff61"> 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 90234c6ccdb5848" title="Click to view the MathML source">GA.
The p -adic representation ρp defines a representation of the Weil–Deligne group 8686307036dff8581a0">, where e75e34d83096f0fd1" title="Click to view the MathML source">Fv,0 is the maximal unramified extension of aee96cd054afc23d98f3c38" title="Click to view the MathML source">Qp contained in Fv and is an inner form of 90234c6ccdb5848" title="Click to view the MathML source">GA over e75e34d83096f0fd1" title="Click to view the MathML source">Fv,0. It is proved, under the same conditions as in the previous theorem, that, as a representation with values in 90234c6ccdb5848" title="Click to view the MathML source">GA, this representation is Q-rational and that it is compatible with the above system of representations a17ff61">.
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