Edmonds maps on Fricke-Macbeath curve

Rubén A. Hidalgo

Ars Mathematica Contemporanea, Tome 8 (2015) no. 2, pp. 275-289.

Résumé

In 1985, L. D. James and G. A. Jones proved that the complete graph Kn defines a clean dessin d’enfant (the bipartite graph is given by taking as the black vertices the vertices of Kn and the white vertices as middle points of edges) if and only if n = pe, 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 Q, but both Edmonds maps cannot be defined over Q; in fact they have as minimal field of definition the quadratic field Q(sqrt( − 7)). It seems that no explicit models for the Edmonds maps over 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 Q(e2πi / 7), and we construct an explicit birational isomorphismL: X → Z, where Z is defined over Q(sqrt( − 7)), so that both Edmonds maps are also defined over that field.

Détail

DOI : 10.26493/1855-3974.496.61a

Keywords: Riemann surface, algebraic curve, dessin d’enfant