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.

中文翻译：

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$\mathrm{GL}_2$-type 的虚拟阿贝尔变体

本文研究了一类具有 $\mathrm{GL}_2$-type 并且在数域 $k$（即 $k$-virtual）上定义同构类的阿贝尔变体。当它们的自同态代数是（1）一个完全实数域 $K$ 或（2）一个完全不定域 $K$ 上的完全不定四元数代数时，我们处理这两种情况。在这种阿贝尔变体的同质类中，我们确定了一个其伽罗瓦共轭可以用 Atkin-Lehner 算子和 $K$ 类群的某些作用来描述的类。我们推断出这样的阿贝尔变体是由某些 PEL Shimura 变体的有限商参数化的。这些新的模空间族在维度为 $2$ 时被进一步分析。当它们是一般类型的表面时，我们提供了明确的数值界限。此外，对于两个特定的例子，我们精确计算不等价椭圆点的坐标，研究某些 Hirzebruch 循环与特殊除数的交集。我们能够证明它们都是有理曲面。