Intersection theory on Shimura surfaces

The theory of complex multiplication (CM) describes abelian extensions of certain degree 4 number fields K: they are given by adjoining Igusa invariants of suitably chosen 2-dimensional abelian varieties to K. In this talk we focus on a new method for computing the minimal polynomial of these Igusa invariants. Our method uses the Galois action on the invariants coming from class field theory. We give an explicit description of this action and explain how to compute it. Many examples will be given.