On abelian surfaces with potential quaternionic multiplication

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An abelian surface A over a field K has potential quaternionic multiplication if the ring End K (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.