Images of two-dimensional motivic Galois representations

Let M be a two-dimensional motive which is pure of weight w over a number field K and let (: G_K Aut(H(M) )) be the system of the -adic realizations. Choose G_K-invariant -lattices T of H(M) and let (:G_K GL (T))be the corresponding system of integral representations. Then either for almost all primes (G_K) consist of all the elements of GL(T) with determinant in ( ^*)^–w or the system () is associated to algebraic Hecke characters. We also can prove an adelic version of our results.Mathematics Subject Classification (2000): 11F80