Existence of covers with fixed ramification in positive characteristic

We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at 4 points and obtain information on the structure of the Hurwitz curve parametrizing these covers.

Deformation data, Belyi maps, and the local lifting problem

We prove existence and nonexistence results for certain differential forms in positive characteristic, called {em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow $p$-subgroup of order $p$.