Local-global principles for algebraic covers

This paper is devoted to some local-global type questions about fields of definition of algebraic covers. Letf:X→B be a covera priori defined over . Assume that the coverf can be defined over each completion ?{p} of ?. Does it follow that the cover can be defined over ?? This is thelocal-to-global principle. It was shown to hold for G-covers DbDo], i.e., for Galois covers given with their automorphisms. Here we prove that, in the situation ofmere covers, the local-to-global principle holds under some additional assumptions on the groupG of the cover and the monodromy representationG→S _d (withd=deg(f)). This local-to-global problem is closely related to the obstruction to the field of moduli being a field of definition. This problem was studied in DbDo], which is the main tool of the present paper.