Galois action on $\bar{\mathbb Q}$-isogeny classes of abelian $L$-surfaces with quaternionic multiplication

We construct a projective Galois representation attached to an abelian L-surface with quaternionic multiplication, describing the Galois action on its Tate module. We prove that such representation characterizes the Galois action on the isogeny class of the abelian L-surface, seen as a set of points of certain Shimura curves.Peer ReviewedPostprint (author's final draft