Double coverings of Klein surfaces by a given Riemann surface (Q1602643)

In this paper the authors investigate, for a given Riemann surface \(X\) of genus \(g\) the number of double covers \(X\to Y\) where \(Y\) is either a Riemann surface or a Klein surface. In both cases they study the maximum number of ramified covers, or, if there are none, the number of unramified ones. The results depend on the power of 2 dividing \(g-1\). The results can also be interpreted as giving estimates for the maximal number of real forms of a given complex curve. The methods rely on the reduction of the questions to problems in group theory, in particular, of groups whose order is a power of 2. Some of the results depend on an unproved statement about such groups (Conjecture 3.3).