Virtual Abelian varieties of $mathrm{GL}_2$-type

This paper studies a class of Abelian varieties that are of $mathrm{GL}_2$-type and with isogenous classes defined over a number field $k$ (i.e., $k$-virtual). We treat both cases when their endomorphism algebras are (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of Atkin-Lehner operators and certain action of the class group of $K$. We deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties. These new families of moduli spaces are further analysed when they are of dimension $2$. We provide explicit numerical bounds for when they are surfaces of general type. In addition, for two particular examples, we calculate precisely the coordinates of inequivalent elliptic points, study intersections of certain Hirzebruch cycles with exceptional divisors. We are able to show that they are both rational surfaces.