Geometric structures on toroidal maps and elliptic curves
Geometric structures on toroidal maps and elliptic curves
From the work [JONES,~G.~A.---SINGERMAN,~D.: {\it Theory of maps on orientable surfaces\/}, Proc. London Math. Soc.~(3) {\bf 37} (1978), 273--307] or [GROTHENDIECK,~A.: {\it Esquisse d'un programme\/}. In: Geometric Galois Actions~1 (L.~Schneps, P.~Lochakeds, eds.). London Math. Soc. Lecture Note Ser.~242, Cambridge University Press, Cambridge, 1997] there is associated with every map on a surface, a geometric structure on the surface, which is either spherical, Euclidean, or hyperbolic. A surface of genus~1 necessarily has a hyperbolic structure, but the torus can have either a Euclidean or hyperbolic structure. We study the genus~1 maps which have a Euclidean structure, both from the viewpoint of graph embeddings and of elliptic curves. We also find an embedding of the complete graph K_6 which necessarily has a hyperbolic structure and where the edges are hyperbolic geodesics.
graph embedding, geometric structure on a map, toroidal map, algebraic curve, elliptic curve, riemann surface, euclidean structure, hyperbolic structure
273-307
Singerman, David
3eeb0783-c87c-4405-81d7-e80ae4c15f8b
Syddall, Robert I.
a123b76c-e1f2-4a02-8e0f-9410cac9e218
2000
Singerman, David
3eeb0783-c87c-4405-81d7-e80ae4c15f8b
Syddall, Robert I.
a123b76c-e1f2-4a02-8e0f-9410cac9e218
Singerman, David and Syddall, Robert I.
(2000)
Geometric structures on toroidal maps and elliptic curves.
Mathematica Slovaca, 50 (5), .
Abstract
From the work [JONES,~G.~A.---SINGERMAN,~D.: {\it Theory of maps on orientable surfaces\/}, Proc. London Math. Soc.~(3) {\bf 37} (1978), 273--307] or [GROTHENDIECK,~A.: {\it Esquisse d'un programme\/}. In: Geometric Galois Actions~1 (L.~Schneps, P.~Lochakeds, eds.). London Math. Soc. Lecture Note Ser.~242, Cambridge University Press, Cambridge, 1997] there is associated with every map on a surface, a geometric structure on the surface, which is either spherical, Euclidean, or hyperbolic. A surface of genus~1 necessarily has a hyperbolic structure, but the torus can have either a Euclidean or hyperbolic structure. We study the genus~1 maps which have a Euclidean structure, both from the viewpoint of graph embeddings and of elliptic curves. We also find an embedding of the complete graph K_6 which necessarily has a hyperbolic structure and where the edges are hyperbolic geodesics.
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Published date: 2000
Keywords:
graph embedding, geometric structure on a map, toroidal map, algebraic curve, elliptic curve, riemann surface, euclidean structure, hyperbolic structure
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Local EPrints ID: 29801
URI: http://eprints.soton.ac.uk/id/eprint/29801
ISSN: 0139-9918
PURE UUID: d17f4e80-42e4-4360-a1d9-02f0fdd179c5
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Date deposited: 14 May 2007
Last modified: 11 Dec 2021 15:15
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Author:
Robert I. Syddall
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