Representations of Weil-Deligne groups and Frobenius conjugacy classes

Let X be a smooth projective algebraic variety over a number field F, with an embedding τ:F↪C. The action of Gal(F¯/F) on ℓ-adic cohomology groups Heti(X/F¯,Qℓ), induces Galois representations ρℓi:Gal(F¯/F)→GL(Heti(X/F¯,Qℓ)). Fix a non-archimedean valuation v on F, of residual characteristic p. Let Fv be the completion of F at v and ′Wv be the Weil-Deligne group of Fv. We establish new cases, for which the linear representations ρℓi_ of ′Wv, associated to ρℓi, form a compatible system of representations of ′Wv defined over Q. Under suitable hypotheses, we show that in some cases, these representations actually form a compatible system of representations of ′Wv, with values in the Mumford-Tate group of HBi(τX(C),Q). When X has good reduction at v, we establish a motivic relationship between the compatibility of the system {ρℓi}ℓ≠p and the conjugacy class of the crystalline Frobenius of the reduction of X at v.