Multiple prime covers of the riemann sphere

Aaron Wootton

Open Mathematics (2005)

  • Volume: 3, Issue: 2, page 260-272
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Aaron Wootton. "Multiple prime covers of the riemann sphere." Open Mathematics 3.2 (2005): 260-272. <http://eudml.org/doc/268774>.

@article{AaronWootton2005,
abstract = {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.},
author = {Aaron Wootton},
journal = {Open Mathematics},
keywords = {14H30; 14H37; 30F10; 30F60; 20H10},
language = {eng},
number = {2},
pages = {260-272},
title = {Multiple prime covers of the riemann sphere},
url = {http://eudml.org/doc/268774},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Aaron Wootton
TI - Multiple prime covers of the riemann sphere
JO - Open Mathematics
PY - 2005
VL - 3
IS - 2
SP - 260
EP - 272
AB - 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.
LA - eng
KW - 14H30; 14H37; 30F10; 30F60; 20H10
UR - http://eudml.org/doc/268774
ER -

References

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  1. [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 Zbl0212.42501
  2. [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 Zbl0174.37401
  3. [3] T. Breuer: Characters and Automorphism Groups of Compact Riemann Surfaces, Cambridge University Press, 2001. Zbl0952.30001
  4. [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 Zbl1019.20018
  5. [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 Zbl0134.16603
  6. [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 Zbl0223.30008
  7. [7] D. Singerman: “Finitely Maximal Fuchsian Groups”, J. London Math. Soc., Vol. 2 (6), (1972), pp. 29–38. Zbl0251.20052
  8. [8] A. Wootton: “Non-Normal Belyî p-gonal Surfaces”, In: Computational Aspects of Algebraic Curves, Lect. Notes in Comp., (2005), to appear. 
  9. [9] A. Wootton: “Defining Algebraic Polynomials for Cyclic Prime Covers of the Riemann Sphere”, Dissertation, (2004). 

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