Riemann surfaces with symmetry: algorithms and applications
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Riemann surfaces frequently possess automorphisms which can be exploited to simplify calculations. However, existing computer software (Maple in particular) is designed for maximum generality and has not yet been extended to make use of available symmetries. In many calculations, the symmetries can be most easily used by choosing a speci c basis for 𝐻1( ,Z) under which the automorphism group acts neatly. This thesis describes a Maple library, designed to be used in conjunction with the existing algcurves, which allows such a choice to be made. In addition we create a visual tool to simplify the presentation of Riemann surfaces as sheeted covers of C and the creation of homology bases suitable for use in the Maple library. Two applications are considered for these techniques, rst Klein's curve and then Bring's. Both of these possess maximal symmetry groups for their genus, and this fact is exploited to obtain neat algebraic homology bases. In the Klein case the basis is novel; Bring's is derived from work in the hyperbolic setting by Riera. In both cases previous hyperbolic work and calculations are related to the algebraic results. Vectors of Riemann constants are calculated for both curves, again exploiting the symmetry. Finally this thesis moves back to simpler cases, and presents a general algorithm to convert results from general genus 2 curves into results based on a symmetric basis if one exists. This is applied to algebraic and numeric examples where we discover an elliptic surface covered in each case.