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Double coverings of Klein surfaces by a given Riemann surface. (English) Zbl 0997.30032
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).

##### MSC:
 30F10 Compact Riemann surfaces and uniformization 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 14H99 Curves in algebraic geometry
##### Keywords:
degree 2 coverings; real forms of algebraic curves
Magma
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##### References:
 [1] Accola, D.M., On lifting the hyperelliptic involution, Proc. amer. math. soc., 122, 341-347, (1994) · Zbl 0859.14012 [2] Alling, N.L.; Greenleaf, N., Foundations of the theory of Klein surfaces, Lecture notes in mathematics, Vol. 219, (1971), Springer Berlin [3] W. Bosma, J. Cannon, Handbook of Magma Functions, University of Sydney, Sydney, 1994. · Zbl 0964.68595 [4] Bujalance, E., A classification of unramified double coverings of hyperelliptic Riemann surfaces, Arch. math., 47, 93-96, (1986) · Zbl 0599.30070 [5] E. Bujalance, F.J. Cirre, J.M. Gamboa, G. Gromadzki, Symmetry types of hyperelliptic Riemann surfaces, Memo. Soc. Math. France, to appear. · Zbl 1078.14044 [6] Bujalance, E.; Gromadzki, G., On ramified double covering maps of Riemann surfaces, J. pure appl. algebra, 146, 29-34, (2000) · Zbl 0952.30036 [7] Bujalance, E.; Gromadzki, G.; Izquierdo, M., On real forms of a complex algebraic curve, J. austral. math. soc. ser. A, 69, 1-9, (2000) [8] Cirre, F.J., Complex automorphism groups of real algebraic curves of genus 2, J. pure appl. algebra, 157, 2-3, 157-181, (2001) · Zbl 0985.14013 [9] F.J. Cirre, J.M. Gamboa, Klein surfaces and real algebraic curves, Proceedings of the Conference on Riemann surfaces, Madrid, 1998, Lecture Notes of the London Mathematical Society, Vol. 287, Cambridge Univ. Press, Cambridge, 2001, pp. 113-131. · Zbl 1028.14011 [10] A. Duma, Zur Konkretisierung Kompakter Riemannscher Flächen, Bayer Akad. Wiss. Math-Natur II (1974) 87-100. [11] Farkas, H.M., Unramified double coverings of hyperelliptic surfaces, J. anal. math., 20, 150-155, (1976) · Zbl 0348.32006 [12] Farkas, H.M., Unramified double coverings of hyperelliptic surfaces II, Proc. amer. math. soc., 101, 3, 470-474, (1987) · Zbl 0634.30040 [13] H.M. Farkas, Unramified coverings of hyperelliptic Riemann surfaces, in: Complex Analysis I, Lecture Notes in Mathematics, Vol. 1275, Springer, Berlin, 1987, pp. 113-130. [14] G. Gromadzki, Symmetries of Riemann surfaces from a combinatorial point of view, Proceedings of the Conference on Riemann surfaces, Madrid, 1998, Lecture Notes of the London Math. Soc., Vol. 287, Cambridge Univ. Press, Cambridge, 2001. · Zbl 1085.14027 [15] Horiuchi, R., Normal coverings of hyperelliptic Riemann surfaces, J. math. Kyoto univ., 19, 3, 497-523, (1979) · Zbl 0427.30038 [16] Kato, T., On the realization problem of compact Riemann surfaces, Hokkaido math. J., 10, 336-347, (1981) · Zbl 0487.30031 [17] A.M. Macbeath, Discontinous groups and birational transformations, Proceedings of Dundee Summer School, University of St. Andrews, St. Andrews, 1961. [18] Maclachlan, C., Smooth coverings of hyperelliptic surfaces, Quart. J. math. Oxford ser. 2, 22, 117-123, (1971) · Zbl 0208.10101 [19] Mednykh, A.D., Hurwitz problem on the number of nonequivalent coverings of compact Riemann surfaces, Siber. math. J., 23, 3, 415-420, (1983) · Zbl 0513.30036 [20] Mednykh, A.D., Nonequivalent coverings of Riemann surfaces with a prescribed ramification type, Siber. math. J., 25, 4, 606-625, (1984) · Zbl 0598.30058 [21] Mednykh, A.D.; Pozdnyakova, G.G., Number of nonequivalent coverings of compact Riemann surfaces over a non-orientable compact surface, Siber. math. J., 27, 1, 99-106, (1986) · Zbl 0598.30059 [22] Th. Meis, Die minimale Blätlerzahl der Konkretisierungen einer kompakten Riemannscher Flaäche, Schr. Math. Inst. Univ. Münster, Vol. 16, 1960. [23] Turbek, P., A necessary and sufficient condition for lifting the hyperelliptic involution, Proc. amer. math. soc., 125, 9, 2615-2625, (1997) · Zbl 0942.30025 [24] Turbek, P., The full automorphism group of the kulkarni surface, Rev. mat. univ. complut. Madrid, 10, 2, 265-276, (1997) · Zbl 0892.30033
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