Virtual Abelian varieties of GL2-type

Abstract

This paper studies a class of Abelian varieties that are of GL 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 nume..