Bujalance, Emilio; Javier Cirre, Francisco; Manuel Gamboa, José; Gromadzki, Grzegorz On the number of ovals of a symmetry of a compact Riemann surface. (English) Zbl 1172.30014 Rev. Mat. Iberoam. 24, No. 2, 391-405 (2008). Summary: Let \(X\) be a symmetric compact Riemann surface, the full group of conformal automorphisms of which is cyclic. We derive a formula for counting the number of ovals of the symmetries of \(X\) in terms of a few data of the monodromy of the covering \(X\rightarrow X/G\), where \(G=\text{Aut}^\pm X\) is the full group of conformal and anticonformal automorphisms of \(X\). MSC: 30F10 Compact Riemann surfaces and uniformization 14H37 Automorphisms of curves Keywords:Riemann surface; symmetries; ovals × Cite Format Result Cite Review PDF Full Text: DOI Euclid EuDML References: [1] Bujalance, E. and Conder, M. D. E.: On cyclic groups of automorphisms of Riemann surfaces. J. London Math. Soc. (2) 59 (1999), no. 2, 573-584. · Zbl 0922.20054 · doi:10.1112/S0024610799007115 [2] Bujalance, E., Costa, A. F. and Singerman, D.: Application of Hoare’s theorem to symmetries of Riemann surfaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 307-322. · Zbl 0794.30029 [3] Bujalance, E., Etayo, J. J., Gamboa, J. M. and Gromadzki, G.: Automorphism Groups of Compact Bordered Klein Surfaces. Lecture Notes in Math. 1439 . Springer-Verlag, Berlin, 1990. · Zbl 0709.14021 · doi:10.1007/BFb0084977 [4] Gromadzki, G.: On a Harnack-Natanzon theorem for the family of real forms of Riemann surfaces. J. Pure Appl. Algebra 121 (1997), no. 3, 253-269. · Zbl 0885.14026 · doi:10.1016/S0022-4049(96)00068-0 [5] Gromadzki, G.: On ovals on Riemann surfaces. Rev. Mat. Iberoamericana 16 (2000), no. 3, 515-527. · Zbl 1087.14507 · doi:10.4171/RMI/282 [6] Harnack, A.: Über die Vieltheiligkeit der ebenen algebraischen Kurven. Math. Ann. 10 (1876), 189-198. · doi:10.1007/BF01442458 [7] Klein, F.: Über Realitätsverhältnisse bei der einem beliebigen Geschlechte zugehörigen Normalkurve der \(\varphi\). Math. Ann. 42 (1893), no. 1, 1-29. · JFM 25.0689.03 [8] Izquierdo, M. and Singerman, D.: Pairs of symmetries of Riemann surfaces. Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 3-24. · Zbl 0914.30029 [9] Meleko\uglu, A.: Symmetries of Riemann Surfaces and Regular Maps. Doctoral thesis, Faculty of Mathematical Studies, University of Southampton, 1998. [10] Nakamura, G.: The existence of symmetric Riemann surfaces determined by cyclic groups. Nagoya Math. J. 151 (1998), 129-143. · Zbl 0914.30030 [11] Natanzon, S. M.: Finite groups of homeomorphisms of surfaces, and real forms of complex algebraic curves. (Russian). Trudy Moskov. Mat. Obshch. 51 (1988), 3-53, 258. Translation in Trans. Moscow Math. Soc. (1989), 1-51. · Zbl 0692.14020 [12] Natanzon, S. M.: On the total number of ovals of real forms of complex algebraic curves. Uspekhi Mat. Nauk (1) 35 , 1980, 207-208. ( Russian Math. Surveys (1) 35 , 1980, 223-224.) · Zbl 0467.14009 [13] Singerman, D.: On the structure of non-euclidean crystallographic groups. Proc. Cambridge Philos. Soc. 76 (1974), 233-240. · Zbl 0284.20053 · doi:10.1017/S0305004100048891 [14] Singerman, D.: Mirrors on Riemann surfaces. In Second International Conference on Algebra (Barnaul, 1991) , 411-417. Contemp. Math. 184 . Amer. Math. Soc., Providence, RI, 1995. · Zbl 0851.30025 [15] Weichold, G.: Über symmetrische Riemannsche Flächen und die Periodizitätsmodulen der zugerhörigen Abelschen Normalintegrale erstes Gattung. Dissertation, Leipzig, 1883. · JFM 15.0434.01 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.