Double Coverings Of Klein Surfaces By A Given Riemann Surface

Abstract

Let X be a Riemann surface. Two coverings p1 : X → Y1 and p2 : X → Y2 are said to be equivalent if p2 =’p1 for some conformal homeomorphism ’: Y1 → Y2. In this paper we determine, for each integer g¿2, the maximum number R(g) of inequivalent rami>ed coverings between compact Riemann surfaces X → Y of degree 2; where X has genus g. Moreover, for in>nitely many values of g, we compute the maximum number U(g) of inequivalent unrami>ed coverings X → Y of degree 2 where X has genus g and admits no rami>ed covering. For the remaining values of g, the computation of U(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 . c 2002 Elsevier Science B.V. All rights reserved.