About the Fricke-Macbeath curve

TL;DR

This paper explores the Fricke-Macbeath Hurwitz curve, analyzing its structure via fiber products of Fermat curves, and shows its Jacobian is isogenous to a product of elliptic curves, providing new geometric insights.

Contribution

It offers a geometric interpretation of the Fricke-Macbeath curve using fiber products of Fermat curves and details its Jacobian's isogeny to elliptic curve products.

Findings

Jacobian of the surface is isogenous to a product of seven elliptic curves.

Provides a geometric explanation of the three elliptic curves in Wiman's description.

Shows the Jacobian of the Fricke-Macbeath curve is isogenous to E^7 for a specific elliptic curve E.

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 $S_{\mu}$ admitting a group $G_{\mu} \cong {\mathbb Z}_{2}^{3}$ 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 Wiman's description. We also observe that the jacobian variety of the surface $S_{\mu}$ 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 $E$.

Equations

$$\left\{\begin{array}[]{lcl}y_{1}^{2}=(x-1)(x-\rho^{3})(x-\rho^{5})(x-\rho^{6})\\ \\ y_{2}^{2}=(x-\rho^{2})(x-\rho^{4})(x-\rho^{5})(x-\rho^{6})\\ \\ y_{3}^{2}=(x-\rho)(x-\rho^{3})(x-\rho^{4})(x-\rho^{5})\end{array}\right\}\subset{\mathbb{C}}^{4}.$$

$$1+7xy+21x^2{y}^2+35x^3{y}^3+28x^4{y}^4+2x^7+2y^7=0,$$

$$\left\{\begin{array}[]{rll}-x_{1}x_{2}+x_{1}x_{0}+x_{2}x_{6}+x_{3}x_{4}-x_{3}x_{5}-x_{3}x_{0}-x_{4}x_{6}-x_{5}x_{6}&=&0\\ x_{1}x_{3}+x_{1}x_{6}-x_{2}^{2}+2x_{2}x_{5}+x_{2}x_{0}-x_{3}^{2}+x_{4}x_{5}-x_{4}x_{0}-x_{5}^{2}&=&0\\ x_{1}^{2}-x_{1}x_{3}+x_{2}^{2}-x_{2}x_{4}-x_{2}x_{5}-x_{2}x_{0}-x_{3}^{2}+x_{3}x_{6}+2x_{5}x_{0}-x_{0}^{2}&=&0\\ x_{1}x_{4}-2x_{1}x_{5}+2x_{1}x_{0}-x_{2}x_{6}-x_{3}x_{4}-x_{3}x_{5}+x_{5}x_{6}+x_{6}x_{0}&=&0\\ x_{1}^{2}-2x_{1}x_{3}-x_{2}^{2}-x_{2}x_{4}-x_{2}x_{5}+2x_{2}x_{0}+x_{3}^{2}+x_{3}x_{6}+x_{4}x_{5}+x_{5}^{2}-x_{5}x_{0}-x_{6}^{2}&=&0\\ x_{1}x_{2}-x_{1}x_{5}-2x_{1}x_{0}+2x_{2}x_{3}-x_{3}x_{0}-x_{5}x_{6}+2x_{6}x_{0}&=&0\\ -2x_{1}x_{2}-x_{1}x_{4}-x_{1}x_{5}+2x_{1}x_{0}+2x_{2}x_{3}-2x_{3}x_{0}+2x_{5}x_{6}-x_{6}x_{0}&=&0\\ 2x_{1}^{2}+x_{1}x_{3}-x_{1}x_{6}+3x_{2}x_{0}+x_{4}x_{5}-x_{4}x_{0}-x_{5}^{2}+x_{6}^{2}-x_{0}^{2}&=&0\\ 2x_{1}^{2}-x_{1}x_{3}+x_{1}x_{6}+x_{2}^{2}+x_{2}x_{0}+x_{3}^{2}-2x_{3}x_{6}+x_{4}x_{5}-x_{4}x_{0}+x_{5}^{2}-2x_{5}x_{0}+x_{6}^{2}+x_{0}^{2}&=&0\\ x_{1}^{2}+x_{1}x_{3}-x_{1}x_{6}+2x_{2}x_{5}-3x_{2}x_{0}+2x_{3}x_{6}+x_{4}^{2}+x_{4}x_{5}-x_{4}x_{0}+x_{6}^{2}+3x_{0}^{2}&=&0\end{array}\right\}.$$

$$\Omega :=\left\{\mu =({\mu }_4,{\mu }_5,{\mu }_6,{\mu }_7)\in {\mathbb{C}}^4:{\mu }_j\notin \{0,1\},{\mu }_i\ne {\mu }_j,\text{mbox}fori\ne j\right\}\subset {\mathbb{C}}^4.$$

$$\widehat{S}_{\mu}=\left\{\begin{array}[]{rclrcl}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}&=&0,&\mu_{4}\;x_{1}^{2}+x_{2}^{2}+x_{4}^{2}&=&0,\\ \mu_{5}\;x_{1}^{2}+x_{2}^{2}+x_{5}^{2}&=&0,&\mu_{6}\;x_{1}^{2}+x_{2}^{2}+x_{6}^{2}&=&0,\\ \mu_{7}\;x_{1}^{2}+x_{2}^{2}+x_{7}^{2}&=&0\end{array}\right\}\subset{\mathbb{P}}_{\mathbb{C}}^{6}.$$

$$\left[\begin{array}[]{rllllll}2x_{1}&2x_{2}&2x_{3}&0&0&0&0\\ 2\mu_{4}x_{1}&2x_{2}&0&2x_{4}&0&0&0\\ 2\mu_{5}x_{1}&2x_{2}&0&0&2x_{5}&0&0\\ 2\mu_{6}x_{1}&2x_{2}&0&0&0&2x_{6}&0\\ 2\mu_{7}x_{1}&2x_{2}&0&0&0&0&2x_{7}\\ \end{array}\right]$$

$$H_{\mu }=⟨a_1,\dots ,a_6⟩\cong {\mathbb{Z}}_2^6$$

$$A(x)=\frac{(1+\rho )(x-\rho )}{\rho (x-1)}.$$

$$P_{\mu }:{\hat{S}}_{\mu }\to \hat{\mathbb{C}}:[x_1:x_2:x_3:x_4:x_5:x_6:x_7]\mapsto A^{-1}\left(-{\left(\frac{x_2}{x_1}\right)}^2\right)$$

$$B_{\mu }:=\{1,\rho ,{\rho }^2,A^{-1}({\mu }_4),A^{-1}({\mu }_5),A^{-1}({\mu }_6),A^{-1}({\mu }_7)\},$$

$${\hat{\theta }}_{\mu }:\mathrm{Aut}({\hat{S}}_{\mu })\to {\mathrm{PSL}}_2(\mathbb{C}):T\mapsto A_T$$

$$P_{\mu }\circ T=A_T\circ P_{\mu },T\in \mathrm{Aut}({\hat{S}}_{\mu }).$$

$${\Gamma }_{\mu }=⟨{\alpha }_1,...,{\alpha }_7:{\alpha }_1^2=\cdots ={\alpha }_7^2={\alpha }_1{\alpha }_2\cdots {\alpha }_7=1⟩$$

$${\theta }_{\mu }:\mathrm{Aut}({\hat{S}}_{\mu })\to \mathrm{Aut}(H_{\mu })=\mathrm{Aut}({\mathbb{Z}}_2^6):T\mapsto {\Lambda }_T,$$

$${\Lambda }_T(a_j)=T{a}_j{T}^{-1}\in \{a_1,\dots ,a_7\}.$$

$$A({\mu }_4,{\mu }_5,{\mu }_6,{\mu }_7)=\left(\frac{1}{{\mu }_4},\frac{1}{{\mu }_5},\frac{1}{{\mu }_6},\frac{1}{{\mu }_7}\right),B({\mu }_4,{\mu }_5,{\mu }_6,{\mu }_7)=\left(\frac{{\mu }_7}{{\mu }_7-1},\frac{{\mu }_7}{{\mu }_7-{\mu }_4},\frac{{\mu }_7}{{\mu }_7-{\mu }_5},\frac{{\mu }_7}{{\mu }_7-{\mu }_6}\right),$$

$$T_{\sigma }({\mu }_4,{\mu }_5,{\mu }_6,{\mu }_7)=({\mu }_{\sigma (4)},{\mu }_{\sigma (5)},{\mu }_{\sigma (6)},{\mu }_{\sigma (7)}).$$

$$\lambda (a_1)=a_2,\lambda (a_2)=a_3,\lambda (a_3)=a_4,\lambda (a_4)=a_5,\lambda (a_5)=a_6,\lambda (a_6)=a_7,\lambda (a_7)=a_1,$$

$$\zeta (a_1)=a_2,\zeta (a_2)=a_1,\zeta (a_3)=a_7,\zeta (a_7)=a_3,\zeta (a_4)=a_6,\zeta (a_6)=a_4,\zeta (a_5)=a_5.$$

$${\mu }^0=(A({\rho }^3),A({\rho }^4),A({\rho }^5),A({\rho }^6))\in \Omega ,$$

$${\mu }_6={\mu }_5^2/{\mu }_4,{\mu }_7={\mu }_5^2,{\mu }_5^6-5{\mu }_5^4-6{\mu }_5^2-1=0.$$

$$H_{\mu },K_{\mu }=⟨a_1{a}_3{a}_7,a_2{a}_3{a}_5,a_1{a}_2{a}_4⟩and{K}_{\mu }^{\ast }=⟨a_1{a}_2{a}_6,a_2{a}_3{a}_7,a_1{a}_3{a}_4⟩.$$

$$a_1,a_1{a}_2,a_1{a}_3,a_1{a}_4,a_1{a}_2{a}_3,a_1{a}_2{a}_4,a_1{a}_2{a}_6,a_1{a}_3{a}_4.$$

$$K_{\mu }=⟨a_1{a}_3{a}_7,a_2{a}_3{a}_5,a_1{a}_2{a}_4⟩\cong {\mathbb{Z}}_2^3$$

$$\begin{array}[]{llllllll}t_{1}=x_{1}^{2},&t_{2}=x_{2}^{2},&t_{3}=x_{3}^{2},&t_{4}=x_{4}^{2},&t_{5}=x_{5}^{2},&t_{6}=x_{6}^{2},&t_{7}=x_{1}x_{2}x_{5},&t_{8}=x_{1}x_{2}x_{3}x_{6},\end{array}$$

$$\begin{array}[]{lllll}t_{9}=x_{1}x_{4}x_{6},&t_{10}=x_{1}x_{3}x_{4}x_{5},&t_{11}=x_{2}x_{4}x_{5}x_{6},&t_{12}=x_{2}x_{3}x_{4},&t_{13}=x_{3}x_{5}x_{6}.\end{array}$$

$$F:{\hat{S}}_{\mu }^0\to {\mathbb{C}}^{13}:(x_1,x_2,x_3,x_4,x_5,x_6)\mapsto (t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,t_9,t_{10},t_{11},t_{12},t_{13}),$$

$$S_{\mu}^{0}:=\left\{\begin{array}[]{llll}t_{1}+t_{2}+t_{3}=0,&\mu_{4}\;t_{1}+t_{2}+t_{4}=0,&\mu_{5}\;t_{1}+t_{2}+t_{5}=0,&\mu_{6}\;t_{1}+t_{2}+t_{6}=0,\\ \mu_{7}\;t_{1}+t_{2}+1=0,&t_{6}t_{10}=t_{9}t_{13},&t_{6}t_{7}t_{12}=t_{8}t_{11},&t_{5}t_{9}t_{12}=t_{10}t_{11},\\ t_{5}t_{8}=t_{7}t_{13},&t_{5}t_{6}t_{12}=t_{11}t_{13},&t_{4}t_{8}=t_{9}t_{12},&t_{4}t_{7}t_{13}=t_{10}t_{11},\\ t_{4}t_{6}t_{7}=t_{9}t_{11},&t_{3}t_{11}=t_{12}t_{13},&t_{3}t_{6}t_{7}=t_{8}t_{13},&t_{3}t_{5}t_{9}=t_{10}t_{13},\\ t_{3}t_{5}t_{6}=t_{13}^{2},&t_{3}t_{4}t_{7}=t_{10}t_{12},&t_{2}t_{10}=t_{7}t_{12},&t_{2}t_{9}t_{13}=t_{8}t_{11},\\ t_{2}t_{5}t_{9}=t_{7}t_{11},&t_{2}t_{4}t_{13}=t_{11}t_{12},&t_{2}t_{4}t_{5}t_{6}=t_{11}^{2},&t_{2}t_{3}t_{9}=t_{8}t_{12},\\ t_{2}t_{3}t_{4}=t_{12}^{2},&t_{1}t_{12}t_{13}=t_{8}t_{10},&t_{1}t_{11}=t_{7}t_{9},&t_{1}t_{6}t_{12}=t_{8}t_{9},\\ t_{1}t_{5}t_{12}=t_{7}t_{10},&t_{1}t_{4}t_{13}=t_{9}t_{10},&t_{1}t_{4}t_{6}=t_{9}^{2},&t_{1}t_{3}t_{4}t_{5}=t_{10}^{2},\\ t_{1}t_{2}t_{13}=t_{7}t_{8},&t_{1}t_{2}t_{5}=t_{7}^{2},&t_{1}t_{2}t_{3}t_{6}=t_{8}^{2}.\end{array}\right\}\subset{\mathbb{C}}^{13}$$

$$(t_1,\dots ,t_{13})\mapsto (t_1,t_2,t_3,t_4,t_5,t_6,-t_7,-t_8,-t_9,-t_{10},t_{11},t_{12},t_{13}),$$

$$(t_1,\dots ,t_{13})\mapsto (t_1,t_2,t_3,t_4,t_5,t_6,-t_7,-t_8,t_9,t_{10},-t_{11},-t_{12},t_{13}),$$

Full text

Partially supported by Project Fondecyt 1150003 and Project Anillo ACT 1415 PIA-CONICYT

1. Introduction

If $S$ is a closed Riemann surface of genus $g\geq 2$, then its group ${\rm Aut}(S)$, of conformal automorphisms, has order at most $84(g-1)$ (Hurwitz upper bound). One says that $S$ is a Hurwitz curve if $|{\rm Aut}(S)|=84(g-1)$; in this situation the quotient orbifold $S/{\rm Aut}(S)$ is the Riemann sphere $\widehat{\mathbb{C}}$ with exactly three cone points of respective orders $2,3$ and $7$, and $S={\mathbb{H}}^{2}/\Gamma$, where $\Gamma$ is a torsion free normal subgroup of finite index in the triangular Fuchsian group $\Delta=\langle x,y:x^{2}=y^{3}=(xy)^{7}=1\rangle$.

In 1895, A. Wiman [12] noticed that there is no Hurwitz curve in each genera $g\in\{2,4,5,6\}$ and that there is exactly one Hurwitz curve (up to isomorphisms) of genus three, this being given by Klein’s quartic $x^{3}y+y^{3}z+z^{3}x=0$; whose group of conformal automorphisms is the simple group ${\rm PSL}_{2}(7)$ of order $168$. Later, in 1965, A. Macbeath [10] observed that in genus seven there is exactly one (up to isomorphisms) Hurwitz curve, called the Fricke-Macbeath curve; its automorphisms group is the simple group ${\rm PSL}_{2}(8)$ of order $504$. In the same paper, an explicit equation for the Fricke-Macbeath curve over ${\mathbb{Q}}(\rho)$, where $\rho=e^{2\pi i/7}$, is given as the fiber product of three (isomorphic) elliptic curves as follows

[TABLE]

In [11] Y. Prokhorov classified the finite simple non-abelian subgroups of the Cremona group of rank $3$, that is, the group of birational automorphisms of the $3$-dimensional complex projective space; these groups being isomorphic to ${\textswab A}_{5}$, ${\textswab A}_{6}$, ${\textswab A}_{7}$, ${\rm PSL}_{2}(7)$, ${\rm PSL}_{2}(8)$ and ${\rm PSp}_{4}(3)$ (where ${\textswab A}_{n}$ denotes the alternating group in $n$ letters). In his classification, the group ${\rm PSL}_{2}(8)$ is seen to act on some smooth Fano threefold in ${\mathbb{P}}_{\mathbb{C}}^{8}$ of genus $7$, this being the dual Fano threefold of the Fricke-Macbeath curve.

The uniqueness of the Fricke-Macbeath curve asserts, by results due to J. Wolfart [13], that it can be also represented by a curve over ${\mathbb{Q}}$. Recently, the following affine plane algebraic model of the Fricke-Macbeath curve was obtained by Bradley Brock (personal communication)

[TABLE]

and the following projective algebraic curve model in ${\mathbb{P}}^{6}$ by Maxim Hendriks in his Ph.D. Thesis [5]

[TABLE]

In 1965, W. Edge [3] obtained the Fricke-Macbeath curve, via classical projective geometry, by using the fact that it admits a group $K=G\rtimes\langle T\rangle\cong{\mathbb{Z}}_{2}^{3}\rtimes{\mathbb{Z}}_{7}$ of conformal automorphisms. Edge first constructed a family of genus seven Riemann surfaces $S_{\mu}$, each one admitting the group $G_{\mu}\cong{\mathbb{Z}}_{2}^{3}$ as a group of conformal automorphisms so that $S_{\mu}/G_{\mu}$ has genus zero, and then he was able to isolate one of them in order to admit and extra order seven automorphism; this being the Fricke-Macbeath curve.

In this paper, we provide a different approach (but somehow related to Edge’s) to obtain another set of equations and to explain, in a geometric manner, the three elliptic curves appearing in equation (1.1). Part of the results presented in here were already obtained in [6], where certain regular dessins d’enfants (Edmonds maps) were explicitly described over their minimal field of definition [8]. We also describe explicitly an isogenous decomposition of the jacobian variety of $S_{\mu}$ as a product of seven (explicit) elliptic curves. Then we explicitly find the parameter $\mu=\mu^{0}$ so that $S_{\mu^{0}}$ is the Fricke-Macbeath curve and observe that these elliptic curves are isomorphic, obtaining the well known fact that its jacobian is isogenous to $E^{7}$, where $E$ is an elliptic curve.

2. A $4$-dimensional family of genus seven Riemann surfaces

As already noticed, the Fricke-Macbeath curve $S$ admits an abelian group $G\cong{\mathbb{Z}}_{2}^{3}$ as a group of conformal automorphism with quotient orbifold $S/G$ being the Riemann sphere (with exactly seven cone points of order two). We proceed to construct a $4$-dimensional of genus seven Riemann surfaces $S_{\mu}$, each one admitting a group $G_{\mu}\cong{\mathbb{Z}}_{2}^{3}$ as a group of conformal automorphisms so that $S_{\mu}/G_{\mu}$ has genus zero. Moreover, we will observe that each of these genus seven Riemann surfaces have a completely decomposable jacobian variety.

2.1. A $4$-dimensional family of Riemann surfaces of genus $49$

Let us consider the $4$-dimensional domain

[TABLE]

As introduced in [1, 4], a closed Riemann surface $R$ is called a generalized Fermat curves of type $(2,6)$ if it admits a group $H\cong{\mathbb{Z}}_{2}^{6}$ of conformal automorphisms so that $R/H$ has genus zero. By the Riemann-Hurwitz formula, $R$ has genus $g=49$ and $R/H$ has exactly seven cone points, each one of order two. As we may identify $S/H$ with the Riemann sphere $\widehat{\mathbb{C}}$ (by the uniformization theorem) and any three different points can be sent to $\infty,0,1$ by a (unique) Möbius transformation, we may assume these seven cone points to be $\infty$, [math], $1$, $\mu_{4}^{R}$, $\mu_{5}^{R}$, $\mu_{6}^{R}$ and $\mu_{7}^{R}$, where $\mu^{R}:=(\mu_{4}^{R},\mu_{5}^{R},\mu_{6}^{R},\mu_{7}^{R})\in\Omega$.

Now, for each $\mu:=(\mu_{4},\mu_{5},\mu_{6},\mu_{7})\in\Omega$, we may consider the projective algebraic set

[TABLE]

As the matrix

[TABLE]

has maximal rank for $(x_{1},\ldots,x_{7})\in{\mathbb{C}}^{7}-\{(0,\ldots,0)\}$ satisfying the above system of equations (2.1), it turns out that $\widehat{S}_{\mu}$ is a smooth projective algebraic curve, so (by the implicit function theorem) a closed Riemann surface.

If $j\in\{1,\ldots,7\}$, then denote by $a_{j}\in{\rm GL}_{7}({\mathbb{C}})$ the order two linear transformation defined by multiplication of the coordinate $x_{j}$ by $-1$. It can be seen that each $a_{j}$ keeps $\widehat{S}_{\mu}$ invariant; so it induces a conformal automorphism of order two on it (which we will still be denoting as $a_{j}$). In this way, the group

[TABLE]

is a group of conformal automorphisms of $\widehat{S}_{\mu}$ and $a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}=1.$

Let $A$ be the Möbius transformation so that $A(1)=\infty$, $A(\rho)=0$ and $A(\rho^{2})=1$, that is,

[TABLE]

It is not difficult to check that the map

[TABLE]

is a regular branched cover, whose branch locus is the set

[TABLE]

and whose deck group is $H_{\mu}$. In particular, $\widehat{S}_{\mu}$ is an example of a generalized Fermat curve of type $(2,6)$. In [1, 4] it was observed that there is an isomorphism $\phi:R\to\widehat{S}_{\mu^{R}}$ conjugating $H$ to $H_{\mu^{R}}$.

As a consequence of the results in [7], the group $H_{\mu}$ is a normal subgroup of ${\rm Aut}(\widehat{S}_{\mu})$ (in fact, in [7] it was proved that $H_{\mu}$ is the unique subgroup $L\cong{\mathbb{Z}}_{2}^{6}$ of ${\rm Aut}(\widehat{S}_{\mu})$ with quotient orbifold $\widehat{S}_{\mu}/L$ being of genus zero). The normality property of $H_{\mu}$ asserts the existence of a natural homomorphism

[TABLE]

whose kernel is $H_{\mu}$ and satisfies

[TABLE]

Remark 2.1*.*

As a consequence of the Klein-Koebe-Poincaré uniformization theorem, there is a Fuchsian group

[TABLE]

acting on the hyperbolic plane ${\mathbb{H}}^{2}$ uniformizing the orbifold $\widehat{S}_{\mu}/H_{\mu}$. The derived subgroup $\Gamma_{\mu}^{\prime}$ of $\Gamma_{\mu}$ is torsion free and $\widehat{S}_{\mu}={\mathbb{H}}^{2}/\Gamma_{\mu}^{\prime}$; $H_{\mu}=\Gamma_{\mu}/\Gamma_{\mu}^{\prime}$ (see, for instance, [1]). The generator $\alpha_{j}$ induces the involution $a_{j}$, for $j=1,\ldots,7$. This in particular asserts that the image of $\widehat{\theta}_{\mu}$ is exactly the stabilizer of the set $B_{\mu}$ in the group of Möbius transformations.

In [1] it was observed that the only non-trivial elements of $H_{\mu}$ acting with fixed points on $\widehat{S}$ are $a_{1},\ldots,a_{6}$ and $a_{7}$. This property, together the normality of $H_{\mu}$, asserts the existence of another natural homomorphism

[TABLE]

where

[TABLE]

It can be seen that the kernel of $\theta_{\mu}$ is again $H_{\mu}$ (just note that if $T$ belongs to the kernel of $\theta_{\mu}$, then $A_{T}=I$ as it will necessarily fix each of the points in $B_{\mu}$).

Remark 2.2*.*

Clearly, every automorphism of $H_{\mu}$ in the image of $\theta_{\mu}$ must keep invariant the set $\{a_{1},\ldots,a_{7}\}$. If $\eta\in{\rm Aut}(H_{\mu})$ is so that $\eta(a_{j})\in\{a_{1},\ldots,a_{7}\}$, for every $j=1,\ldots,7$, then there is some orientation-preserving homeomorphism $F:\widehat{S}_{\mu}\to\widehat{S}_{\mu}$ normalizing $H_{\mu}$ and inducing $\eta$. Sometimes, one may chose $\mu$ in order to assume $F$ to be a conformal automorphism of $\widehat{S}_{\mu}$, but this is not in general true. For instance, if $\eta(a_{1})=a_{2}$, $\eta(a_{2})=a_{3}$, $\eta(a_{3})=a_{1}$, $\eta(a_{4})=a_{5}$, $\eta(a_{5})=a_{6}$, $\eta(a_{6})=a_{7}$ and $\eta(a_{7})=a_{4}$, then the existence of $T\in{\rm Aut}(\widehat{S}_{\mu})$ so that $\eta=\Lambda_{T}$ will assert the existence of a Möbius transformation $A_{T}\in{\rm PSL}_{2}({\mathbb{C}})$ such that $A_{T}(1)=\rho$, $A_{T}(\rho)=\rho^{2}$, $A_{T}(\rho^{2})=1$, $A_{T}(A^{-1}(\mu_{4}))=A^{-1}(\mu_{5})$, $A_{T}(A^{-1}(\mu_{5}))=A^{-1}(\mu_{6})$, $A_{T}(A^{-1}(\mu_{6}))=A^{-1}(\mu_{7})$ and $A_{T}(A^{-1}(\mu_{7}))=A^{-1}(\mu_{4})$, which is clearly impossible.

If we set

[TABLE]

then ${\mathbb{G}}=\langle A,B\rangle\cong{\mathfrak{S}}_{7}$ is a group of holomorphic automorphisms of $\Omega$ and, in [4] it was observed that, for $\mu,\widetilde{\mu}\in\Omega$, the Riemann surfaces $\widehat{S}_{\mu}$ and $\widehat{S}_{\widetilde{\mu}}$ are isomorphic if and only if they are ${\mathbb{G}}$-equivalent. Let us also observe that, for each $\sigma\in{\rm Perm}\{4,5,6,7\}\cong{\mathfrak{S}}_{4}$, $T_{\sigma}\in{\mathbb{G}}$, where

[TABLE]

2.2. A $4$-dimensional family of genus seven Riemann surfaces

For each $\mu\in\Omega$, let $\lambda,\zeta\in{\rm Aut}(H_{\mu})$ be defined as:

[TABLE]

[TABLE]

The subgroup $\langle\lambda,\zeta\rangle$ of ${\rm Aut}(H_{\mu})$ is isomorphic to the dihedral group of order $14$.

It can be checked (see also Section 3) that $\lambda$ belongs the image of $\theta_{\mu^{0}}$, where

[TABLE]

and $\zeta$ belongs to the image of $\theta_{(\mu_{4},\mu_{5},\mu_{6},\mu_{7})}$, where

[TABLE]

Note that $\mu^{0}$ satisfies these last conditions; so $\zeta$ also belongs to the image of $\theta_{\mu^{0}}$.

Lemma 2.3**.**

The only subgroups of $H_{\mu}$ which are $\lambda$-invariant are

[TABLE]

Moreover, the automorphism $\zeta$ permutes $K_{\mu}$ with $K_{\mu}^{*}$.

Proof.

It is clear that the subgroups $H_{\mu}$, $K_{\mu}$ and $K_{\mu}^{*}$ are $\lambda$-invariants and that $\zeta$ permutes $K_{\mu}$ with $K_{\mu}^{*}$. We need to check that these are the only $\lambda$-invariant subgroups. If $K$ is a $\lambda$-invariant subgroup of $H_{\mu}$, then it must contain at least one of the following elements:

[TABLE]

If $a_{1}a_{2}a_{4}\in K$, then $K=K_{\mu}$ and if $a_{1}a_{2}a_{6}\in K$, then $K=K_{\mu}^{*}$. Next, we proceed to check that $K=H_{\mu}$ for the other cases. (i) If $a_{1}\in K$, then $K=H_{\mu}$. (ii) If $a_{1}a_{2}\in K$, then $a_{3}a_{4},a_{6}a_{7}=a_{1}a_{2}a_{3}a_{4}a_{5}\in K$; so $a_{5}\in K$ and $K=H_{\mu}$. (iii) If $a_{1}a_{3}\in K$, then $a_{2}a_{4},a_{5}a_{7}=a_{1}a_{3}a_{2}a_{4}a_{6}\in K$; so $a_{6}\in K$ and $K=H_{\mu}$. (iv) If $a_{1}a_{4}\in K$, then $a_{2}a_{5},a_{3}a_{6},a_{4}a_{7}=a_{1}a_{2}a_{5}a_{3}a_{6}\in K$; so $a_{1}\in K$ and $K=H_{\mu}$. (v) If $a_{1}a_{2}a_{3}\in K$, then $a_{5}a_{6}a_{7}=a_{1}a_{2}a_{3}a_{4}\in K$; so $a_{4}\in K$ and $K=H_{\mu}$. (vi) If $a_{1}a_{2}a_{5}\in K$, then $a_{3}a_{4}a_{7}=a_{1}a_{2}a_{5}a_{6}\in K$; so $a_{6}\in K$ and $K=H_{\mu}$. (vii) If $a_{1}a_{3}a_{5}\in K$, then $a_{2}a_{4},a_{6},a_{3}a_{5}a_{7}=a_{1}a_{2}a_{4}a_{6}\in K$; so $a_{1}\in K$ and $K=H_{\mu}$. ∎

As the $\lambda$-invariant subgroup

[TABLE]

does not contains any of the involutions $a_{j}$, it acts freely on $\widehat{S}_{\mu}$ (the same holds for $K_{\mu}^{*}$).

As a consequence of the Riemann-Hurwitz formula, the quotient $S_{\mu}=\widehat{S}_{\mu}/K_{\mu}$ is a closed Riemann surface of genus seven with $G_{\mu}=H_{\mu}/K_{\mu}\cong{\mathbb{Z}}_{2}^{3}$ as a group of conformal automorphisms and with $S_{\mu}/G_{\mu}=\widehat{S}_{\mu}/H_{\mu}$ being the orbifold whose underlying Riemann surface is $\widehat{\mathbb{C}}$ and whose cone points are the points in $B_{\mu}$, each one of order $2$.

Let us denote by $a_{j}^{*}$ the conformal involution of $S_{\mu}$ induced by the involution $a_{j}$, for $j=1,\ldots,7$.

Remark 2.4*.*

Let $\Gamma_{\mu}$ be the Fuchsian group as in remark 2.1. As $G_{\mu}$ is abelian group, there is a torsion free finite index subgroup $N_{\mu}$ of $\Gamma_{\mu}$ so that $\Gamma_{\mu}^{\prime}$ is contained in $N_{\mu}$, $S_{\mu}={\mathbb{H}}^{2}/N_{\mu}$ and $G_{\mu}=\Gamma_{\mu}/N_{\mu}$. In fact, $N_{\mu}=\llangle\Gamma_{\mu}^{\prime},\alpha_{1}\alpha_{3}\alpha_{7},\alpha_{2}\alpha_{3}\alpha_{5},\alpha_{1}\alpha_{2}\alpha_{4}\rrangle_{\Gamma_{\mu}}.$

2.3. Algebraic equations for $S_{\mu}$

Using the equations for $\widehat{S}_{\mu}$ and the explicit group $K_{\mu}$, classical invariant theory permits to obtain a set of equations for $S_{\mu}$. For it, we consider the affine model of $\widehat{S}_{\mu}$, say by taking $x_{7}=1$, which we denote by $\widehat{S}_{\mu}^{0}$. In this affine model the automorphism $a_{7}$ is given by $a_{7}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})=(-x_{1},-x_{2},-x_{3},-x_{4},-x_{5},-x_{6})$. A set of generators for the algebra of invariant polynomials in ${\mathbb{C}}[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}]$ under the natural linear action induced by $K_{\mu}$ is given by

[TABLE]

[TABLE]

If we set

[TABLE]

then $F(\widehat{S}_{\mu}^{0})=S_{\mu}^{0}$ will provide the following affine model for $S_{\mu}$:

[TABLE]

Of course, one may see that the variables $t_{2}$, $t_{3}$, $t_{4}$, $t_{5}$ and $t_{6}$ are uniquely determined by the variable $t_{1}$. Other variables can also be determined in order to get a lower dimensional model. The group $G_{\mu}$ is, in this model, generated by the following three involutions:

[TABLE]

[TABLE]

[TABLE]

2.4. Some elliptic curves associated to $S_{\mu}$

Let us observe that

[TABLE]

and that

[TABLE]

In this way, each of the seven involutions $a_{r}^{*}\in G_{\mu}$ is induced by exactly one of the involutions of $H_{\mu}$ with fixed points and, in particular, it acts with exactly $4$ fixed points on $S_{\mu}$. By the Riemann-Hurwitz formula, the quotient orbifold $S_{\mu}/\langle a_{r}^{*}\rangle$ has genus three and exactly four cone points, each of order $2$ (over that orbifold there is the conformal action of the group $G_{\mu}/\langle a_{r}^{*}\rangle\cong{\mathbb{Z}}_{2}^{2}$ for which the cone points form one orbit).

If $a_{i}^{*},a_{j}^{*}\in G_{\mu}$ are two different involutions and $L_{ij}=\langle a_{i}^{*},a_{j}^{*}\rangle\cong{\mathbb{Z}}_{2}^{2}$, then (by the Riemann-Hurwitz formula) the quotient orbifold $S_{\mu}/L_{ij}$ is a closed Riemann surface $T_{ij}$ of genus one with exactly six cone points, all of them of order $2$. Further, $L^{*}_{ij}=G_{\mu}/L_{ij}\cong{\mathbb{Z}}_{2}$ acts as a group of automorphisms of $T_{ij}$ permuting these six cone points (and fixing none of them). These six cone points are projected onto three of the cone points of $S_{\mu}/G_{\mu}=T_{ij}/L^{*}_{ij}$; they being $\mu_{i}$, $\mu_{j}$ and $\mu_{r}$, where $a_{r}^{*}=a_{i}^{*}a_{j}^{*}$ and $\mu_{1}=\infty$, $\mu_{2}=0$ and $\mu_{3}=1$. In this way,

[TABLE]

where, in the case $k=1$ the factor $(x-\mu_{1})$ is deleted from the previous expression.

Let us observe that there are exactly seven different subgroups $L_{ij}$ of $G_{\mu}$, these being given by

[TABLE]

so there are only seven elliptic curves as above;

[TABLE]

It can be seen that another equation for $S_{\mu}$ is provided by the fiber product of the three elliptic curves $T_{23},T_{12}$ and $T_{13}$ as follows

[TABLE]

In this model, the group $G_{\mu}$ is generated by the involutions

[TABLE]

[TABLE]

[TABLE]

2.5. An elliptic decomposition of the jacobian variety $JS_{\mu}$

If $\mu=(\mu_{4},\mu_{5},\mu_{6},\mu_{7})\in\Omega$, then in [2] we obtained that the jacobian variety $J\widehat{S}_{\mu}$ of $\widehat{S}_{\mu}$ is isogenous to a product

[TABLE]

where each of the $E_{j,\mu}$ is an elliptic curve (corresponding to the choice of $4$ points in $B_{\mu}$) and each of the $X_{j,\mu}$ is a genus two Riemann surface (these are provided by the choice of $6$ points in $B_{\mu}$). On the other hand, $J\widehat{S}_{\mu}$ is also isogenous to a product $JS_{\mu}\times{\mathcal{P}}_{\mu}$, where ${\mathcal{P}}_{\mu}$ is a polarized abelian variety of dimension $42$. In particular, one obtains that

[TABLE]

Unfortunately, the above isogeny does not provide more expliciy information on $JS_{\mu}$. In order to get a decomposition of it we proceed as follows. First, observe that, for the seven subgroups $L_{12},L_{13},L_{15},L_{23},L_{1,7},L_{34},L_{47}$, it holds the following: (i) $L_{ij}L_{kr}=L_{kr}L_{ij}$, (ii) each quotient $S_{\mu}/L_{ij}$ has genus one and (iii) $S_{\mu}/L_{ij}L_{kr}=S_{\mu}/G_{\mu}$ has genus zero. So, from Kani-Rosen decomposition theorem [9], one obtains the following.

Theorem 2.5**.**

For each $\mu\in\Omega$, it holds that

[TABLE]

The above asserts that the jacobian variety $JS_{\mu}$ of $S_{\mu}$ is completely decomposable (see also [2]). Observe that the seven elliptic factors are some of the $35$ elliptic factors of $J\widehat{S}_{\mu}$.

2.6. A remark about $K^{*}_{\mu}$

As $K^{*}_{\mu}$ also acts freely on $\widehat{S}_{\mu}$, we may consider the Riemann surface $S_{\mu}^{*}=\widehat{S}_{\mu}/K^{*}_{\mu}$, which admits the group ${\mathbb{Z}}_{2}^{3}\cong G_{\mu}^{*}=H_{\mu}/K^{*}_{\mu}=\langle b_{1},b_{2},b_{3}\rangle$, where $b_{j}$ is induced by $a_{j}$. In this case,

[TABLE]

Working in a similar fashion as for $K_{\mu}$, one can obtain the following set of equations for $S_{\mu}^{*}$

[TABLE]

In this model, the group $G_{\mu}^{*}$ is again generated by the involutions

[TABLE]

[TABLE]

[TABLE]

Direct computations permits to see that

[TABLE]

when

[TABLE]

In fact, just use the change of variable induced by the transformation $T(x)=x/\mu_{5}$ to $S_{\mu}$ to obtain $S^{*}_{\widetilde{\mu}}$.

3. The Fricke-Macbeath curve

In this section we observe that the Fricke-Macbeath curve is one of the members of the previous $4$-dimensional family $S_{\mu}$ of genus seven Riemann surfaces.

First, as already observed in the introduction, the Fricke-Macbeath curve $S$ admits a group $K=G\rtimes\langle T\rangle\cong{\mathbb{Z}}_{2}^{3}\rtimes{\mathbb{Z}}_{7}$ of conformal automorphisms. Let us consider the orbifold $S/G$, which has some number of cone points, each one of order $2$. Moreover, the automorphism $T$ induces an automorphism $A_{T}$ of $S/G$, of order seven and keeping invariant these cone points.

Proposition 3.1**.**

The quotient orbifold $S/G$ is the Riemann sphere with exactly seven cone points of order $2$.

Proof.

The automorphism $T$ must permute the seven elements of order two in $G$. There are only two possibilities: (i) these seven elements form one orbit or (ii) each element of order two commutes with $T$. The quotient orbit $S/G$, by the Riemann-Hurwitz formula, is either a torus with exactly three cone points of order $2$ or the Riemann sphere $\widehat{\mathbb{C}}$ with exactly $7$ cone points of order $2$. If $S/G$ has genus one, as there are only three cone points and such a set is invariant under $A_{T}$, it follows that $A_{T}$ must have fixed points; a contradiction as there is no genus one Riemann surface admitting a conformal automorphism of order seven with fixed points. It follows that $S/G$ is the Riemann sphere (with exactly seven cone points of order $2$). In this last situation, $A_{T}$ only has two fixed points, so $T$ only satisfies condition (i) above. ∎

Let us consider a regular branched cover $Q:S\to\widehat{\mathbb{C}}$ whose deck group is $G\cong{\mathbb{Z}}_{2}^{3}$. Up to postcomposing $Q$ by a suitable Möbius transformation, we may assume that $A_{T}(z)=\rho z$, where $\rho=e^{2\pi i/7}$. As the seven cone points of order $2$ are cyclically permuted by $A_{T}$, we may also assume that they are given by the points

[TABLE]

Let us observe that

[TABLE]

As $S/G={\widehat{S}}_{{\mu}^{0}}/H_{{\mu}^{0}}$ and $G$ is abelian (recall that $\widehat{S}_{{\mu}^{0}}$ is the highest abelian branched cover), by classical covering theory, there should be a subgroup $K<H_{\mu^{0}}$, $K\cong{\mathbb{Z}}_{2}^{3}$, acting freely on $\widehat{S}_{\mu^{0}}$ so that there is an isomorphism $\phi:S\to\widehat{S}_{\mu^{0}}/K$ with $\phi G\phi^{-1}=H_{\mu^{0}}/K$.

Let us also observe that the rotation $A_{T}(z)=\rho z$ lifts under $P_{\mu^{0}}$ to an automorphism $\widehat{T}$ of $\widehat{S}_{\mu^{0}}$ of order $7$ of the form [4]

[TABLE]

where

[TABLE]

One has that $\widehat{T}a_{j}\widehat{T}^{-1}=a_{j+1}$, for $j=1,...,6$ and $\widehat{T}a_{7}\widehat{T}^{-1}=a_{1}$. It follows that, for this explicit value of $\mu=\mu^{0}$, one has that $\lambda=\Lambda_{\widehat{T}}$.

The subgroup $K$ above must satisfy that $\widehat{T}K\widehat{T}^{-1}=K$ (since $R$ also lifts to the Fricke-Macbeath curve $S$), that is, $K$ is a $\lambda$-invariant subgroup of $H_{\mu^{0}}$; it follows (by Lemma 2.3) that $K\in\{K_{\mu^{0}},K^{*}_{\mu^{0}}\}$. But, for this value of $\mu=\mu^{0}$, the automorphism $\zeta$ is induced by an automorphism of $\widehat{S}_{\mu^{0}}$ given by

[TABLE]

where

[TABLE]

In particular, both genus seven Riemann surfaces unifomized by $K_{\mu^{0}}$ and $K_{\mu^{0}}^{*}$ are isomorphic. In this way, we may assume that

[TABLE]

Remark 3.2*.*

Another way to see that we may assume $K=K_{\mu^{0}}$ is as follows. The subgroup

[TABLE]

acts freely on $\widehat{S}_{\mu^{0}}$ and it is normalized by $\widehat{T}$. In particular, $S^{*}=\widehat{S}_{\mu^{0}}/K_{\mu_{0}}$ is a closed Riemann surface of genus $7$ admitting the group $L=H_{\mu^{0}}/K_{\mu^{0}}\cong{\mathbb{Z}}_{2}^{3}$ as a group of conformal automorphisms. It also admits an automorphism $T^{*}$ of order $7$ (induced by $\widehat{T}$) permuting cyclically the involutions $a_{j}^{*}$. As $S^{*}/\langle L,T^{*}\rangle=\widehat{S}_{\mu^{0}}/\langle H_{\mu^{0}},\widehat{T}\rangle$ has signature $(0;2,7,7)$, and the Fricke-Macbeath curve is the only closed Riemann surface of genus seven, up to isomorphisms, admitting a group of automorphisms isomorphic to ${\mathbb{Z}}_{2}^{3}\rtimes{\mathbb{Z}}_{7}$ with corresponding quotient orbifold of the previous signature, we obtain the desired result.

Summarizing all the above is the following.

Theorem 3.3**.**

The Fricke-Macbeath curve is isomorphic to the Riemann surface $S_{\mu^{0}}$, where

[TABLE]

In particular, the Fricke-Macbeath curve is given by the fiber product equation

[TABLE]

Remark 3.4*.*

It can be seen that the fiber product equation (3.1) is isomorphic to Wiman’s fiber product (1.1) using the change of variable $u=A(x)$.

In this case, i.e., for $\mu=\mu^{0}$, all the seven elliptic curves $T_{ij}$ are isomorphic to

[TABLE]

in particular, as a consequence of Theorem 2.5, we obtain the following well known fact.

Corollary 3.5**.**

If $S$ is the Fricke-Macbeath curve, then

[TABLE]

Remark 3.6*.*

We also note that

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Also, observe that $E_{3}\cong E$.

Acknowledgments

The author is grateful to the referee whose suggestions, comments and corrections done to the preliminary versions helped to improve the presentation of the paper. I also want to thanks I. Cheltsov for the reference to the work of Y. Prokhorov.