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Erschienen in:
2011 | OriginalPaper | Buchkapitel
Computing Poincaré Theta Series for Schottky Groups
verfasst von : Markus Schmies
Erschienen in: Computational Approach to Riemann Surfaces
Verlag: Springer Berlin Heidelberg
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Common numerical methods represent Riemann surfaces through algebraic curves, see Chaps. 2 and 3. This seems natural because algebraic curves can be used to define a Riemann surface, but this approach has some serious disadvantages: if one is interested in the corresponding Riemann surface only, one needs to factorize algebraic curves with respect to birational maps. This complicates the corresponding parameterization of Riemann surfaces. Moreover, a representation of a Riemann surfaces as a ramified multi-sheeted covering complicates the description of homology and integration paths, which leads to complex algorithms. Schottky uniformization (see Chap. 1) is attractive alternative to describing Riemann surfaces in terms of algebraic curves.