On ramified double covering maps of Riemann surfaces

Let X,Y1 and Y2 be compact Riemann surfaces. Two coverings p1 :hsp sp=0.25>X→Y1 and p2 : X→Y2, say of degree n, are said to be equivalent if p2= p1 for some conformal homeomorphism : Y1→Y2. We find an upper bound for the number of nonequivalent coverings of degree 2 by a compact Riemann surface of genus g≥2 and we show that our bound, which depends on g, is sharp for infinitely many odd values of g which we determine. We also consider the case of surfaces of even genus, in which the situation turns out to be dramatically different, since then X admits 1 or 3 coverings of degree 2. Moreover, if X is hyperelliptic, we determine the ramification data of such coverings.