Abelian Surfaces with an Automorphism and Quaternionic Multiplication

Abstract

AbstractWe construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)- polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by .