Fields of definition of p-adic covers

This paper concerns fields of definition and fields of moduli of G-Galois covers of the line over p-adic fields, and more generally over henselian discrete valuation fields. We show that the field of moduli of a p-adic cover will be a field of definition provided that the residue characteristic p does not divide \G\ and that the branch points do not coalesce module p (or in the more general case, that the branch locus is smooth on the special fibre). Hence if p does not divide \G\, then a G-Galois cover of the (Q) over bar-line with field of moduli Q will be defined over a number field contained in Q(p), if the branch points do not coalesce module p. This provides an explicit global-to-local principle for p-adic covers.