For an abelian variety over a number field with semistable reduction at v, the Mumford-Tate group valued Weil-Deligne representation at v is defined over Q and is independent of the auxiliary prime l.
Emerton, Formal algebraic stacks , (unpublished), available at https://www.math.uchicago.edu/ emerton/pdffiles/formal-stacks.pdf
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Strongly compatible systems associated to semistable abelian varieties
For an abelian variety over a number field with semistable reduction at v, the Mumford-Tate group valued Weil-Deligne representation at v is defined over Q and is independent of the auxiliary prime l.