About the Fricke-Macbeath curve

Abstract

A Hurwitz curve is a closed Riemann surface of genus $g \geq 2$ whose group of conformal automorphisms has order $84(g-1)$. In 1895, Wiman proved that for $g=3$ there is, up to isomorphisms, a unique Hurwitz curve; this being Klein's plane quartic curve. Moreover, he also proved that there is no Hurwitz curve of genus $g=2,4,5,6$. 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 closed Riemann surfaces $S_μ$ admitting a group $G_μ \cong {\mathbb Z}_{2}^{3}$ of conformal automorphisms so that $S_μ/G_μ$ 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 Wiman's description. We also observe that the jacobian variety of the surface $S_μ$ 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 $E$.