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Numerical calculation of three-point branched covers of the projective line

Part of: Computational aspects in algebraic geometry Riemann surfaces Curves Arithmetic algebraic geometry Birational geometry

Published online by Cambridge University Press:  01 September 2014

Michael Klug ,
Michael Musty ,
Sam Schiavone  and
John Voight
Michael Klug
Affiliation:
Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT 05401, USA email mklug@uvm.edu
Michael Musty
Affiliation:
Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT 05401, USA email michaelmusty@gmail.com
Sam Schiavone
Affiliation:
Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT 05401, USA email samuel.schiavone@uvm.edu
John Voight
Affiliation:
Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT 05401, USA Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA email jvoight@gmail.com

Abstract

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We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.

MSC classification

Secondary: 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 11G32: Dessins d'enfants, Belyĭ theory 14E20: Coverings 14H45: Special curves and curves of low genus 14H50: Plane and space curves 14Q05: Curves 30F10: Compact Riemann surfaces and uniformization
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 379 - 430
Copyright
© The Author(s) 2014 

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