Fix a number field , an abelian variety and let be the Mumford–Tate group of . After replacing by finite extension one can assume that, for every prime number , the action of the absolute Galois group on the étale cohomology group factors through a morphism . Let be a valuation of and write for the absolute Galois group of the completion . For every with , the restriction of to defines a representation of the Weil–Deligne group.
It is conjectured that, for every , this representation of is defined over as a representation with values in and that the system above, for variable , forms a compatible system of representations of with values in . A somewhat weaker version of this conjecture is proved for the valuations of , where has semistable reduction and for which is neat.
Rutger Noot. "The system of representations of the Weil–Deligne group associated to an abelian variety." Algebra Number Theory 7 (2) 243 - 281, 2013. https://doi.org/10.2140/ant.2013.7.243