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Fundamental domains for genus-zero and genus-one congruence subgroups

Part of: Other groups of matrices Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 July 2010

C. J. Cummins
C. J. Cummins*
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, Québec H3G 1M8, Canada (email: cummins@mathstat.concordia.ca)

Abstract

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In this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups of PSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integer f, which determines a maximal discrete subgroup Γ0(f)+ of SL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+ is in a subgroup G; and the index of G in Γ0(f)+. The output consists of: a fundamental domain for G, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators of G. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.

MSC classification

Secondary: 20H05: Unimodular groups, congruence subgroups 11F30: Fourier coefficients of automorphic forms 11F22: Relationship to Lie algebras and finite simple groups 20H10: Fuchsian groups and their generalizations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 222 - 245
Copyright
Copyright © London Mathematical Society 2010

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