Double coverings of Klein surfaces by a given Riemann surface

✍ Scribed by E. Bujalance; M.D.E. Conder; J.M. Gamboa; G. Gromadzki; M. Izquierdo

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 142 KB
• Volume: 169
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(01)00082-2

✦ Synopsis

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 ramiÿed coverings between compact Riemann surfaces X → Y of degree 2; where X has genus g. Moreover, for inÿnitely many values of g, we compute the maximum number U (g) of inequivalent unramiÿed coverings X → Y of degree 2 where X has genus g and admits no ramiÿed 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 .