One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions
Hurwitz spaces are spaces of pairs (S,f) where S is a Riemann surface and f:S→ℂ^ a meromorphic function. In this work, we study 1-dimensional Hurwitz spaces ℋDp of meromorphic p-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branch...
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Wiley
2008-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2008/609425 |
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| author | Antonio F. Costa Milagros Izquierdo Gonzalo Riera |
| author_facet | Antonio F. Costa Milagros Izquierdo Gonzalo Riera |
| author_sort | Antonio F. Costa |
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| description | Hurwitz spaces are spaces of pairs (S,f) where S is a Riemann surface and f:S→ℂ^ a meromorphic function. In this work, we study 1-dimensional Hurwitz spaces ℋDp of meromorphic p-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of (p−1)/2 transpositions and the monodromy group is the dihedral group Dp. We prove that the completion ℋDp¯ of the Hurwitz space ℋDp is uniformized by a non-nomal index p+1 subgroup of a triangular group with signature (0;[p,p,p]). We also establish the relation of the meromorphic covers with elliptic functions and show that ℋDp is a quotient of the upper half plane by the modular group Γ(2)∩Γ0(p). Finally, we study the real forms of the Belyi projection ℋDp¯→ℂ^ and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated. |
| format | Article |
| id | doaj-art-503021b2f86b455fa13da949dfda8e8c |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-503021b2f86b455fa13da949dfda8e8c2025-02-03T07:26:04ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252008-01-01200810.1155/2008/609425609425One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic FunctionsAntonio F. Costa0Milagros Izquierdo1Gonzalo Riera2Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional de Educación a Distancia (UNED), Senda del rey, 9, 28040 Madrid, SpainMatematiska Institutionen, Linköpings Universitet, 581 83 Linköping, SwedenFacultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, ChileHurwitz spaces are spaces of pairs (S,f) where S is a Riemann surface and f:S→ℂ^ a meromorphic function. In this work, we study 1-dimensional Hurwitz spaces ℋDp of meromorphic p-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of (p−1)/2 transpositions and the monodromy group is the dihedral group Dp. We prove that the completion ℋDp¯ of the Hurwitz space ℋDp is uniformized by a non-nomal index p+1 subgroup of a triangular group with signature (0;[p,p,p]). We also establish the relation of the meromorphic covers with elliptic functions and show that ℋDp is a quotient of the upper half plane by the modular group Γ(2)∩Γ0(p). Finally, we study the real forms of the Belyi projection ℋDp¯→ℂ^ and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.http://dx.doi.org/10.1155/2008/609425 |
| spellingShingle | Antonio F. Costa Milagros Izquierdo Gonzalo Riera One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions International Journal of Mathematics and Mathematical Sciences |
| title | One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions |
| title_full | One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions |
| title_fullStr | One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions |
| title_full_unstemmed | One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions |
| title_short | One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions |
| title_sort | one dimensional hurwitz spaces modular curves and real forms of belyi meromorphic functions |
| url | http://dx.doi.org/10.1155/2008/609425 |
| work_keys_str_mv | AT antoniofcosta onedimensionalhurwitzspacesmodularcurvesandrealformsofbelyimeromorphicfunctions AT milagrosizquierdo onedimensionalhurwitzspacesmodularcurvesandrealformsofbelyimeromorphicfunctions AT gonzaloriera onedimensionalhurwitzspacesmodularcurvesandrealformsofbelyimeromorphicfunctions |