Ben Moonen (Radboud University Nijmegen)

An old conjecture by Coleman predicts that if we fix a sufficiently large integer g, there exist only finitely many (smooth projective) complex curves of genus g whose Jacobians are abelian variety of CM type. Coleman originally conjectured this to be true for g>3; there are, however, counterexamples for g up to 7. I will explain how, through a programme initiated by Oort, this problem relates to the question of whether in the moduli space of abelian varieties there exists positive dimensional special subvarieties (Shimura varieties) that are contained in the Torelli locus. The goal of my talk is to explain what we currently know about this problem, and what not.