372 KB33 KB200820252015Hidalgo, Rubén A.številka:2letnik:8str. 275-289COBISSID:17376601ISSN:1855-3966URN:URN:NBN:SI:doc-0VYU4G4IenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaalgebraic curvealgebraična krivuljadessin d'enfantotroška risbaRiemann surfaceRiemannova ploskevEdmonds maps on the Fricke-Macbeath curve|In 1985, L. D. James and G. A. Jones proved that the complete graph ?$K_n$? defines a clean dessin d'enfant (the bipartite graph is given by taking as the black vertices the vertices of ?$K_n$? and the white vertices as middle points of edges) if and only if ?$n = p^e$?, where ?$p$? is a prime. Later, in 2010, G. A. Jones, M. Streit and J. Wolfart computed the minimal field of definition of them. The minimal genus ?$g > 1$? of these types of clean dessins d'enfant is ?$g = 7$?, obtained for ?$p = 2$? and ?$e = 3$?. In that case, there are exactly two such clean dessins d'enfant (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus 7) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over ?$\mathbb{Q}$?, but both Edmonds maps cannot be defined over ?$\mathbb{Q}$?; in fact they have as minimal field of definition the quadratic field ?$\mathbb{Q}(\sqrt{- 7})$?. It seems that no explicit models for the Edmonds maps over ?$\mathbb{Q}(\sqrt{- 7})$? are written in the literature. In this paper we start with an explicit model ?$X$? for the Fricke-Macbeath curve provided by Macbeath, which is defined over ?$\mathbb{Q}(e^{2\pi i/7})$?, and we construct an explicit birational isomorphism ?$L \colon X \to Z$?, where ?$Z$? is defined over ?$\mathbb{Q}(\sqrt{- 7})$?, so that both Edmonds maps are also defined over that fieldLeta 1985 so L. D. James in G. A. Jones dokazali, da polni graf ?$K_n$? definira čisto otroško risbo (za črna vozlišča dvodelnega grafa vzamemo vozlišča grafa ?$K_n$?, za bela vozlišča pa središčne točke njegovih povezav) če in samo če je ?$n = p^e$?, kjer je ?$p$? praštevilo. Kasneje, leta 2010, so G. A. Jones, M. Streit in J. Wolfart izračunali minimalno definicijsko območje takšnih čistih otroških risb. Minimalni rod ?$g > 1$? teh tipov čistih otroških risb je ?$g = 7$?, dobljen pa je pri ?$p = 2$? in ?$e = 3$?. V tem primeru obstajata natanko dve takšni čisti otroški risbi, (doslej znani pod imenom Edmondsova zemljevida), ki določata t.i. Fricke-Macbeathovo krivuljo (edino Hurwitzovo krivuljo roda 7) in tvorita kiralni par. Enoličnost Fricke-Macbeathove krivulje zagotavlja, da jo je mogoče definirati nad ?$\mathbb{Q}$?, toda oba Edmondsova zemljevida ne moreta biti definirana nad ?$\mathbb{Q}$?; v resnici je njuno minimalno definicijsko območje kvadratni obseg ?$\mathbb{Q}(\sqrt{- 7})$?. Zdi se, da v literaturi niso zabeleženi nobeni eksplicitni modeli Edmondsovih zemljevidov nad ?$\mathbb{Q}(\sqrt{- 7})$?. V tem članku vzamemo za izhodišče ekspliciten Macbeathov model ?$X$? za Fricke-Macbeathovo krivuljo, ki je definirana nad ?$\mathbb{Q}(e^{2\pi i/7})$?, potem pa konstruiramo ekspliciten biracionalen izomorfizem ?$L \colon X \to Z$?, kjer je ?$Z$? definiran nad ?$\mathbb{Q}(\sqrt{- 7})$?, nad istim obsegom pa sta definirana tudi oba Edmondsova zemljevidaTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije