Virtual Abelian Varieties of GL_2-type
Virtual Abelian Varieties of GL_2-type
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GL_2 型的虚拟阿贝尔簇
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通讯作者：
Chenyan Wu
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作者：
Chenyan Wu
本文研究了数域k上的一类$\GL_2 $-型的同构类阿贝尔簇。我们处理的情况下，当它们的自同态代数是（1）一个全真实的域$K$或（2）一个全不定四元数代数在一个全真实的域$K$。其中的同构类的阿贝尔品种，我们确定了一个伽罗瓦共轭可以描述的Atkin-Lehner运营商的行动和类组的$K$。因此，我们推断，这样的阿贝尔品种参数化的有限向量的某些PEL志村品种。这些新的家庭的模空间进一步分析时，他们的尺寸为2 $。当它们是一般类型的曲面时，我们提供了明确的数值界。另外，对于两个特殊的例子，通过计算不等价椭圆点的坐标和研究具有例外因子的Hirzebruch圈的交，证明了它们都是有理曲面.
This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (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 actions of Atkin-Lehner operators and the class group of $K$. Thus 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 show that they are both rational surfaces by computing the coordinates of inequivalent elliptic points and studying the intersections of Hirzebruch cycles with exceptional divisors.