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