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 subset of C and let G(A) 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 Gamma(F) = Gal((F) over bar /F) on H-et(1)(A/(F) over bar, Q(l)) factors through a map Gamma(F)-> G(A) (Q(l)). Fix a valuation nu of F and let p be the residue characteristic at nu. For any prime number l not equal p, the representation pe gives rise to a representation 'WF nu -> G(A/Ql), of the Weil Deligne group. In the case where A has semistable reduction at nu it was shown in a previous paper that, with some restrictions, these representations form a compatible system of Q -rational representations with values in G(A). The p-adic representation p(p) defines a representation of the Weil Deligne group 'WF nu -> G(A)(l)/F-nu,F-0, where F-nu,F-0 is the maximal unramified extension of Q(p) contained in F-nu, and G(A)(l) is an inner form of G(A) over F-nu,F-0. It is proved, under the same conditions as in the previous theorem, that, as a representation with values in G(A), this representation is Q -rational and that it is compatible with the above system of representations 'WF nu -> G(A/Ql). (C) 2016 Elsevier Inc. All rights reserved.