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An abelian variety over a number field is called L-abelian variety if, forany element of the absolute Galois group of a number field L, the conjugatedabelian variety is isogenous to the given one by means of an isogeny thatpreserves the Galois action on the endomorphism rings. We can think of them asgeneralizations of abelian varieties defined over L with endomorphisms alsodefined over L. In the one dimensional case, an elliptic curve defined over Lgives rise to a Galois representation provided by the Galois action on its Tatemodule. This classical Galois representation has been a central object of studyin Number Theory over the last decades. Besides, given an elliptic L-curve onecan construct a projective analogue of the previous Galois representation. Inthis work we construct similar projective representations in thetwo-dimensional case, namely, attached to abelian L-surfaces with quaternionicmultiplication or fake elliptic curves. Moreover, we prove that such projectiverepresentations also describe the Galois action on the isogeny class of theelliptic L-curve, in the classical case, or the abelian L-surface withquaternionic multiplication, in the general case.

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