Articulo
About the Fricke-Macbeath curve
European Journal of Mathematics
Registro en:
1150003
1150003
Autor
Hidalgo-Ortega, Rubén Antonio
Institución
Resumen
A closed Riemann surface of genus g >= 2 is called a Hurwitz curve if its group of conformal automorphisms has order 84(g - 1). In 1895, Wiman noticed that there is no Hurwitz curve of genus g = 2, 4, 5, 6 and, up to isomorphisms, there is a unique Hurwitz curve of genus g = 3 this being Klein's plane quartic curve. Later, in 1965, Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus g = 7 this known as the Fricke-Macbeath curve. Equations were also provided that being the fiber product of suitable three elliptic curves. In the same year, Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a 4-dimensional family of genus seven closed Riemann surfaces S-mu admitting a group G(mu) congruent to Z(2)(3) of conformal automorphisms so that S-mu/G(mu) has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath's description. We also observe that the jacobian variety of each S-mu is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke-Macbeath curve, we obtain the well-known fact that its jacobian variety is isogenous to E-7 for a suitable elliptic curve E. Regular 2015 FONDECYT FONDECYT