  An abelian variety over a number field is called L-abelian variety if, for
any element of the absolute Galois group of a number field L, the conjugated
abelian variety is isogenous to the given one by means of an isogeny that
preserves the Galois action on the endomorphism rings. We can think of them as
generalizations of abelian varieties defined over L with endomorphisms also
defined over L. In the one dimensional case, an elliptic curve defined over L
gives rise to a Galois representation provided by the Galois action on its Tate
module. This classical Galois representation has been a central object of study
in Number Theory over the last decades. Besides, given an elliptic L-curve one
can construct a projective analogue of the previous Galois representation. In
this work we construct similar projective representations in the
two-dimensional case, namely, attached to abelian L-surfaces with quaternionic
multiplication or fake elliptic curves. Moreover, we prove that such projective
representations also describe the Galois action on the isogeny class of the
elliptic L-curve, in the classical case, or the abelian L-surface with
quaternionic multiplication, in the general case.
