GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS
We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded...
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| Published in | Glasgow mathematical journal Vol. 51; no. 2; pp. 289 - 299 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.05.2009
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| Subjects | |
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| Abstract | We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods. |
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| AbstractList | Abstract We consider families of quasiplatonic Riemann surfaces characterised by the fact that - as in the case of Fermat curves of exponent n - their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group - and graph - theoretic results by G. A. Jones, R. Nedela and M. Skoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods. [PUBLICATION ABSTRACT] Abstract We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph K n,n , where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs K n,n [ 7 ] and generalises the analogous results for maps obtained in [ 9 ], partly using different methods. We consider families of quasiplatonic Riemann surfaces characterised by the fact that - as in the case of Fermat curves of exponent n - their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group - and graph - theoretic results by G. A. Jones, R. Nedela and M. Skoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods. We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods. |
| Author | STREIT, MANFRED COSTE, ANTOINE D. JONES, GARETH A. WOLFART, JÜRGEN |
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| Cites_doi | 10.1017/CBO9780511758874.008 10.1016/S0393-0440(96)00027-7 10.4171/dm/96 10.1016/j.jalgebra.2006.10.009 10.1090/S0002-9947-02-03184-7 10.1093/qmath/hah054 10.1007/BF02101464 10.1070/IM1980v014n02ABEH001096 10.1112/jlms/s2-6.1.29 10.1007/s000130050453 10.1112/blms/28.6.561 10.1016/j.ejc.2005.07.021 |
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| References | S0017089509004972_ref11 S0017089509004972_ref10 S0017089509004972_ref15 S0017089509004972_ref16 Streit (S0017089509004972_ref12) 2000; 13 Grothendieck (S0017089509004972_ref6) 1997 S0017089509004972_ref1 Voevodsky (S0017089509004972_ref14) 1989; 39 S0017089509004972_ref8 S0017089509004972_ref9 Streit (S0017089509004972_ref13) 2001; 6 S0017089509004972_ref7 S0017089509004972_ref4 S0017089509004972_ref5 S0017089509004972_ref2 S0017089509004972_ref3 |
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| Snippet | We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying... Abstract We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their... Abstract We consider families of quasiplatonic Riemann surfaces characterised by the fact that - as in the case of Fermat curves of exponent n - their... We consider families of quasiplatonic Riemann surfaces characterised by the fact that - as in the case of Fermat curves of exponent n - their underlying... |
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| Title | GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS |
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