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Computational Approach to Riemann Surfaces pp 183–193Cite as

Uniformizing Real Hyperelliptic M-Curves Using the Schottky–Klein Prime Function

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2013))

Abstract

It is a consequence of the uniformization theorem of Koebe and Poincaré that any smooth complex algebraic curve C of genus g 1 is conformally equivalent to H/GF where H is the upper-half complex plane and GF Є PSL2(R) is a Fuchsian group. On the other hand, C is also known to be equivalent to Г/GS where Г is some domain in the Riemann sphere and GS is a Schottky group GF Є PSL2(C) where the fundamental region associated with GS is multiply connected.

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, United Kingdom

    Darren Crowdy & Jonathan S. Marshall

Authors
  1. Darren Crowdy
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  2. Jonathan S. Marshall
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Correspondence to Darren Crowdy .

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Editors and Affiliations

  1. Institute of Mathematics, Technical University of Berlin, Straße des 17.Juni 136, Berlin, 10623, Germany

    Alexander I. Bobenko

  2. Institut de Mathématiques, Université de Bourgogne, av. Alain Savary 9, Dijon Cedex, 21078, France

    Christian Klein

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Crowdy, D., Marshall, J.S. (2011). Uniformizing Real Hyperelliptic M-Curves Using the Schottky–Klein Prime Function. In: Bobenko, A., Klein, C. (eds) Computational Approach to Riemann Surfaces. Lecture Notes in Mathematics(), vol 2013. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17413-1_5

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