Double coverings of Klein surfaces by a given Riemann surface

Let X be a Riemann surface. Two coverings p1 : X→Y1 and p2 : X→Y2 are said to be equivalent if p2=phip1 for some conformal homeomorphism phi : Y1→Y2. In this paper we determine, for each integer ggreater-or-equal, slanted2, the maximum number ρR(g) of inequivalent ramified coverings between compact Riemann surfaces X→Y of degree 2, where X has genus g. Moreover, for infinitely many values of g, we compute the maximum number ρU(g) of inequivalent unramified coverings X→Y of degree 2 where X has genus g and admits no ramified covering. For the remaining values of g, the computation of ρU(g) relies on a likely conjecture on the number of conjugacy classes of 2-groups. We also extend these results to double coverings X→Y, where Y is now a proper Klein surface. In the language of algebraic geometry, this means we calculate the number of real forms admitted by the complex algebraic curve X.