Fix a valuation v of F and let p be the residue characteristic at v . For any prime number 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=4eb649e9d75c30832f6a1218fe3d6cd2" title="Click to view the MathML source">ℓ≠p, the representation 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=528275032338b4090101d91be212c478" title="Click to view the MathML source">ρℓ gives rise to a representation 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6af2bdab1a04d72afc7f43e9fa17ff61"> 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 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6edbf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">GA.
The p -adic representation 11" class="mathmlsrc">11.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=91de8287935340162f6c2b792b100e57" title="Click to view the MathML source">ρp defines a representation of the Weil–Deligne group 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=e407aa7cf8b8f8686307036dff8581a0">115" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1630213X-si12.gif">, where 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=18d98ff358b9963e75e34d83096f0fd1" title="Click to view the MathML source">Fv,0 is the maximal unramified extension of 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=45d4a71d0aee96cd054afc23d98f3c38" title="Click to view the MathML source">Qp contained in 15" class="mathmlsrc">15.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=d0017b6d89f51badcddd5b5566fe2dc5" title="Click to view the MathML source">Fv and 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=e377f7f92b5f52dc572d82c0a7162f80">
is an inner form of 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6edbf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">GA over 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=18d98ff358b9963e75e34d83096f0fd1" 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 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6edbf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">GA, this representation is Q-rational and that it is compatible with the above system of representations 111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6af2bdab1a04d72afc7f43e9fa17ff61">
.