GALOIS COVERS AND THE HILBERT-GRUNWALD PROPERTY

Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert's irreducibility theorem and a p-adic form a la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over Q. The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.