2017/03/06 by Rubén A. Hidalgo, Hidalgo, Ruben A.
Mathematics · #30F10 #30F20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1703.01869
openalex publication_date 2017/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Hurwitz curve is a closed Riemann surface of genus g ≥ 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 Sμ admitting a group Gμ ≅ \mathbb Z23 of conformal automorphisms so that Sμ/Gμ 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 Sμ 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 E7 for a suitable elliptic curve E.