Counting real Galois covers of the projective line

Abstract

For Galois covers of P1 of a given rami?cation type ? essentially, a given monodromy group G and branch locus, assumed to be de?ned over R ?we ask: How many covers are de?ned over R and how many are not? J.-P. Serre showed that the number of all Galois covers with given rami?cation type can be computed from the character table of G. We adapt Serre?s method of calculation to the more re?ned situation of Galois covers de?ned over R, for which there is a group-theoretic characterization due to P. D?s and M. Fried. We obtain explicit answers to our problem. As an application, we exhibit new families of covers not de?ned over their ?eld of moduli, the monodromy group of which can be chosen arbitrarily large. We also give examples of Galois covers de?ned over the ?eld Qtr of totally real algebraic numbers with Q-rational branch locus.