A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C q of prime order q such that X/C q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.
@article{bwmeta1.element.doi-10_2478_BF02479202,
author = {Aaron Wootton},
title = {Multiple prime covers of the riemann sphere},
journal = {Open Mathematics},
volume = {3},
year = {2005},
pages = {260-272},
zbl = {1108.30030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02479202}
}
Aaron Wootton. Multiple prime covers of the riemann sphere. Open Mathematics, Tome 3 (2005) pp. 260-272. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02479202/
[1] R.D.M. Accola: “Strongly Branched Covers of Closed Riemann Surfaces”, Proc. of the AMS, Vol. 26(2), (1970), pp. 315–322. http://dx.doi.org/10.2307/2036396 | Zbl 0212.42501
[2] R.D.M. Accola: “Riemann Surfaces with Automorphism Groups Admitting Partitions”, Proc. Amer. Math. Soc., Vol. 21, (1969), pp. 477–482. http://dx.doi.org/10.2307/2037029 | Zbl 0174.37401
[3] T. Breuer: Characters and Automorphism Groups of Compact Riemann Surfaces, Cambridge University Press, 2001. | Zbl 0952.30001
[4] E. Bujalance, F.J. Cirre and M.D.E. Conder: “On Extendability of Group Actions on Compact Riemann Surfaces”, Trans. Amer. Math. Soc., Vol. 355, (2003), pp. 1537–1557. http://dx.doi.org/10.1090/S0002-9947-02-03184-7 | Zbl 1019.20018
[5] A.M. Macbeath: “On a Theorem of Hurwitz”, Proceedings of the Glasgow Mathematical Association, Vol. 5, (1961), pp. 90–96. http://dx.doi.org/10.1017/S2040618500034365 | Zbl 0134.16603
[6] B. Maskit: “On Poincaré's Theorem for Fundamental Polygons”, Advances in Mathematics, (1971), Vol. 7, pp. 219–230. http://dx.doi.org/10.1016/S0001-8708(71)80003-8 | Zbl 0223.30008
[7] D. Singerman: “Finitely Maximal Fuchsian Groups”, J. London Math. Soc., Vol. 2 (6), (1972), pp. 29–38. | Zbl 0251.20052
[8] A. Wootton: “Non-Normal Belyî p-gonal Surfaces”, In: Computational Aspects of Algebraic Curves, Lect. Notes in Comp., (2005), to appear.
[9] A. Wootton: “Defining Algebraic Polynomials for Cyclic Prime Covers of the Riemann Sphere”, Dissertation, (2004).