About the Fricke–Macbeath curve

Abstract

A closed Riemann surface of genus $$g \geqslant 2$$ 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 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.