On Shimura subvarieties generated by families of abelian covers of P1

We investigate the occurrence of Shimura (special) subvarieties in the locus of Jacobians of abelian Galois covers of P1 in Ag and give classifications of families of such covers that give rise to Shimura subvarieties in the Torelli locus Tg inside Ag. Our methods are based on Moonen-Oort works as well as characteristic p techniques of Dwork and Ogus and Monodromy computations.