Lifting according to an isogeny of \(\ell\)-adic Galois representations associated to motives (Q1895677)

Let \(E/\mathbb{Q}\) be a number field and write \(G_E= \text{Gal} (\overline{\mathbb{Q}}/E)\). Let \(H/\mathbb{Q}\) be an algebraic group and let \((\rho_\ell)\) be a system of \(\ell\)-adic Galois representations of \(G_E\) with values in \(H\). Suppose that \((\rho_\ell)\) is associated to a motive defined over \(E\). Let \(H'\to H\) be a central isogeny such that the complex Hodge structure lifts to \(H'\). The main result of the paper then states (under some mild hypothesis) that after restriction to a finite extension \(E'/E\) the system \((\rho_\ell)\) lifts to \(H'\).