Counting real Galois covers of the projective line

Abstract

For Galois covers of ‫ސ‬ 1 of a given ramification type -essentially, a given monodromy group G and branch locus, assumed to be defined over ‫ޒ‬ -we ask: How many covers are defined over ‫ޒ‬ 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 ‫,ޒ‬ 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 ‫ޑ‬ tr of totally real algebraic numbers with ‫-ޑ‬rational branch locus.

Recovery of Plane Curves from Branch Points

“…We now turn to branched covers that are real. In the abstract setting of Hurwitz numbers H d , this has been studied in [3,15,20]. A cover f : C → P 1 is called real if the Riemann surface C has an involution which is compatible with complex conjugation on the Riemann sphere P 1 .…”

“…Solving this system means computing a fiber of the map (3) over B. Recovery is not unique because discr z (A) is invariant under the action of the subgroup G of PGL (3) given by g : x → g 0 x , y → g 0 y , z → g 1 x + g 2 y + g 3 z with g 0 g 3 = 0.…”

On the lower bounds for real double Hurwitz numbers

“…Note that the integer H R g (λ, µ; s) depends on the positions of points in x. The symmetric group can also be used to study real double Hurwitz number [2,10]. In the following, we give an equivalent description of real double Hurwitz number using symmetric group.…”

Hurwitz numbers for real polynomials

“…One approach uses the representation theory of the symmetric groups. We refer to Zagier's appendix to [8] for a review of the complex case and to A. Cadoret's work [4] for the real case (with h = 0). Another approach is based on a tropical correspondence theorem proved by B. Bertrand, E. Brugallé and G. Mikhalkin.…”

“…In the complex case, as well as in the setting of [10], the invariance of Hurwitz numbers (that is, independence of the number of coverings from the positions of the branch points, provided that the ramification profiles are fixed) is immediate. The papers [4] and [3] do not contain invariance statements: they enumerate all coverings sign '+' which, in general, gives rise to different Hurwitz numbers for different positions of branch points. So far, our attempts to generalize the signed count and the invariance theorem to this general situation have failed.…”