Tate classes and poles of L-functions of twisted quaternionic Shimura surfaces

In this article we generalize a result obtained by Harder, Langlands and Rapoport in the case of Hilbert modular surfaces and we prove in particular the equality between the dimension of the space of Tate classes of twisted quaternionic Shimura surfaces defined over arbitrary solvable extensions of totally real fields and the order of the pole at s = 2 of the zeta functions of these surfaces.