Summary: We consider families of cyclic covers of $\Bbb P^1$, 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.

11G15, 14H40, 14G35

special subvarieties, Jacobians, complex multiplication