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

We consider families of cyclic covers of the projective line, 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.Comment: Minor improvements. To appear in Documenta Mat

A remark on the Tate conjecture

The Tate conjecture has two parts: an assertion (S) about semisimplicity of Galois representations, and an assertion (T) which says that every Tate class is algebraic. We show that in characteristic 0, (T) implies (S). In characteristic p an analogous result is true under stronger assumptions.Comment: 3 pages; updated version that includes a result in characteristic p. To appear in the J. of Alg. Geo