Special subvarieties arising from families of cyclic covers of the projective line

Abstract

We consider families of cyclic covers of ${\mathbb{P}}^1$, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special subvariety in the moduli space of abelian varieties. Our proof uses techniques in mixed characteristics due to Dwork and Ogus.