Endomorphism rings of abelian surfaces and projective models of their moduli spaces

Abstract

We construct projective models for Humbert surfaces and QCM-curves, i.e., Shimura curves together with their natural embedding into the coarse moduli space for principally polarized abelian surfaces. The points of a QCM-curve correspond to an abelian surface, such that its algebra of complex multiplications is an order in an indefinite rational quaternian algebra. Moreover, we determine the structure of such orders.