ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 8 (2015) 275-289
Edmonds maps on the Fricke-Macbeath curve
Rubén A. Hidalgo *
Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54-D, 4780000 Temuco, Chile
Received 4 June 2013, accepted 27 December 2014, published online 4 February 2015
In 1985, L. D. James and G. A. Jones proved that the complete graph Kn defines a clean dessin d'enfant (the bipartite graph is given by taking as the black vertices the vertices of Kn and the white vertices as middle points of edges) if and only if n = pe, where p is a prime. Later, in 2010, G. A. Jones, M. Streit and J. Wolfart computed the minimal field of definition of them. The minimal genus g > 1 of these types of clean dessins d'enfant is g = 7, obtained for p = 2 and e = 3. In that case, there are exactly two such clean dessins d'enfant (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus 7) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over Q, but both Edmonds maps cannot be defined over Q; in fact they have as minimal field of definition the quadratic field Q(%/—7). It seems that no explicit models for the Edmonds maps over Q( V—7) are written in the literature. In this paper we start with an explicit model X for the Fricke-Macbeath curve provided by Macbeath, which is defined over Q(e2ni/7), and we construct an explicit birational isomorphism L : X ^ Z, where Z is defined over Q( V—7), so that both Edmonds maps are also defined over that field.
Keywords: Riemann surface, algebraic curve, dessin d'enfant. Math. Subj. Class.: 30F20, 30F10, 14Q05, 14H45, 14E05
1 Introduction
A dessin d'enfant D on a closed orientable surface is given by a bipartite map on it (vertices will be colored black and white). The dessin d'enfant is called clean if the white vertices have all valence 2.
A Belyi curve is an irreducible non-singular complex projective algebraic curve (i.e. a closed Riemann surface) S admitting a non-constant meromorphic map fi : S ^ C with
* Partially supported by Project Fondecyt 1150003. E-mail address: ruben.hidalgo@ufrontera.cl (Rubén A. Hidalgo)
Abstract
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
276
ArsMath. Contemp. 8(2015)275-289
at most three branch values; which we assume to be inside the set {to, 0,1}; we say that (S, ft) is a Belyi pair. Two Belyi pairs (Si, ^i) and (S2, ft2) are called equivalent, denoted this by the symbol (S1, ft^ = (S2, ft2), if there is an isomorphism f : S1 ^ S2 so that ft2 ◦ f = ft. _
A subfield K of Q is called a field of definition of a Belyi pair (S, ft) if there an equivalent Belyi pair (S, ft) where S and ft are both defined over K. As a consequence of Belyi's theorem [1], the field of algebraic numbers Q is a field of definition of every Belyi pair.
Each Belyi pair (S, ft) defines a dessin d'enfant on S by taking the edges as the components of ft-1((0,1)), the black vertices as the points in ft-1(0) and the white vertices as the points in ft-1(1). Conversely, as a consequence of the uniformization theorem, every dessin d'enfant on a closed orientable surface induces a (unique up to isomorphisms) Riemann surface structure (being a Belyi curve) and a Belyi map so that the original dessin d'enfant is homotopic to the one associated to the Belyi pair [11, 15].
A field of definition of a dessin d'enfant is a field of definition of the corresponding Belyi pair.
As there is a natural (faithful) action of the absolute Galois group Gal(Q/Q) on the collection of Belyi pairs [13], it also provides a (faithful) action on dessins d'enfant. This action may help in the study of the internal structure of the group Gal(Q/Q) in terms of combinatorial data.
Let us consider a dessin d'enfant D, which is defined by the Belyi pair (S, ft). By Belyi's theorem, we may assume that both S and ft are defined over Q. The field of moduli of D is then defined as the fixed field of the subgroup {a g Gal(Q/Q) : (S, ft) = (S, fta)}. The field of moduli of D is always contained in any field of definition of it, but it may be that the field of moduli is not a field of definition of it. Both, the computation of the field of moduli of a dessin d'enfant and to decide if the dessin d'enfant can be defined over it, are in general difficult problems. If the dessin d'enfant is regular, that is, the Belyi map ft is a Galois branched cover, then J. Wolfart [19] proved that D can be defined over its field of moduli. Also, if the dessin d'enfant has no non-trivial automorphisms, then it is definable over its field of moduli as a consequence of Weil's descent theorem [16]. So, the problem to decide if the field of moduli is a field of definition appears when it has non-trivial automorphisms but it is non-regular.
In 1985, L. D. James and G. A. Jones [10] proved that the complete graph Kn defines a clean dessin d'enfant (the bipartite graph is given by taking as the black vertices the vertices of Kn and the white vertices as middle points of edges) if and only if n = pe, where p is a prime. Later, in 2010, G. A. Jones, M. Streit and J. Wolfart [12] computed the minimal field of definition of such clean dessins d'enfant. The minimal genus g > 1 of these types of clean dessins d'enfants is g = 7, obtained for p = 2 and e = 3. In that case, there are exactly two (non-equivalent) such dessins (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus 7) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over Q, but each of the two Edmonds maps cannot be defined over Q; they have as minimal field of definition the quadratic field Q( V—7) [12]. No explicit models for the Edmonds maps over Q( V—7) seems to be written in the literature.
In Section 2 we recall an explicit model X for the Fricke-Macbeath curve provided by Macbeath, which is defined over Q(e2ni/7), and describe both Edmonds maps. We also provide (as matter of interest for specialists) two different equations, over Q, for the Fricke-Macbeath curve which were independently obtained by Bradley Brock (personal
R. A. Hidalgo: Edmonds maps on the Fricke-Macbeath curve
277
communication) and by Maxim Hendriks in his Ph.D. Thesis [7]. In Section 3 we provide an explicit birational isomorphism L : X ^ Z, where Z is defined over Q(%/—7). In this model we obtain that the two Belyi maps defining the two Edmonds maps are defined over Q; in particular, this provides explicit models for both Edmonds maps over Q(V-7) as desired. In Section 4 we provide an explicit birational isomorphism L : X ^ W, where W is defined over Q. Unfortunately, the explicit equations over Q are not very simple (they are long ones) and they can be found in [9]. In Section 5 we construct a generalized Fermat curve S of type (2,6) [5] that covers the Fricke-Macbeath curve and we provide a description of the three elliptic curves appearing in the equations of X given by Macbeath. Another model of the Fricke-Macbeath curve is also described.
2 Macbeath's equations of the Fricke-Macbeath curve and the two Edmonds maps
In this section we recall equations of the Fricke-Macbeath curve, obtained by Macbeath in [14], and we describe both Edmonds maps discovered in [12]. As a matter of interest to specialists, we also describe two different models over Q, one obtained by Bradley Brock (personal communication) and the other by Maxim Hendriks in his Ph.D. Thesis [7].
2.1 Hurwitz curves
It is well known that | Aut(S) | < 84(g -1) (Hurwitz upper bound) if S is a closed Riemann surface of genus g > 2. In the case that | Aut(S) | = 84(g -1), one says that S is a Hurwitz curve. In this last situation, the quotient orbifold S/Aut(S) has signature (0; 2,3,7), that is, S = H2/r, where r is a torsion free normal subgroup of finite index in the triangular Fuchsian group A = (x, y : x2 = y3 = (xy)7 = 1) acting as isometries of the hyperbolic plane H2.
Wiman [17] noticed that there is no Hurwitz curve in each genera g G {2,4, 5, 6} and there is exactly one Hurwitz curve (up to isomorphisms) of genus three, this being Klein's quartic x3y + y3z + z3x = 0; whose automorphisms group is the simple group PSL(2, 7) (of order 168).
2.2 Macbeath's equations of the Fricke-Macbeath curve
In [14] Macbeath 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 PSL(2, 8), consisting of 504 symmetries. In the same paper, Macbeath computed the following explicit equations over Q(p), where p = e2ni/7, for the Fricke-Macbeath curve (involving three particular elliptic curves):
X
y2 = (x — 1)(x — p3)(x — p5)(x — p6) y2 = (x — p2)(x — p4)(x — p5)(x — p6) C C4. y4 = (x — p)(x — p3)(x — p4)(x — p5)
(2.1)
In Section 5 we provide a rough explanation about the elliptic curves in the above equations (different from the approach given in [14]) in geometric terms of the highest regular branched Abelian cover of the orbifold X/G of signature (0; 2,2,2,2, 2, 2,2).
278
Ars Math. Contemp. 8 (2015) 235-244
Another interesting fact on the Fricke-Macbeath curve is that its jacobian variety is isogenous to E7 where E is the elliptic curve with j-invariant j(E) = 1792 (E does not have complex multiplication); see, for instance, [2]. There are not to many explicit examples of Riemann surfaces whose jacobian variety is isogenous to the product of elliptic curves (see [6]).
2.3 Equations over Q of the Fricke-Macbeath curve
The uniqueness up to isomorphisms of the Fricke-Macbeath curve asserts that its field of moduli is the field of rational numbers Q. As quasiplatonic curves can be defined over their fields of moduli [19] and Hurwitz curve are quasiplatonic curves, it follows that the Fricke-Macbeath curve can be defined over Q. When the author put a first version of this paper in Arxiv [9] we didn't know of explicit equations of the Fricke-Macbeath curve over Q. Later, Bradley Brock sent me an e-mail in which he told me that, using some suitable change of coordinates on the above equations for X, he was able to compute a plane equation for X over Q, with some simple nodes as singularities, given as
1 + 7xy + 21x2y2 + 35x3y3 + 28x4y4 + 2x7 + 2y7 = 0.
An automorphism of order 7 is given by b(x, y) = (px, p-1y) and one of order two is given by ai(x,y) = (y,x).
The following model over Q, for the Fricke-Macbeath curve, was recently computed by Maxim Hendriks in his Ph.D. Thesis [7]
X1X2 + X1X0 + X2X6 + X3X4 - X3X5 - X3X0 - X4X6 - X5X6 = 0,
3 + x4x5 - x4x0 - x5
X1X3 + X1X6 - x2 + 2x2X5 + X2X0 - x3 + X4X5 - X4X0 - x2 = 0
X1X4 - 2x1x5 + 2x1x0 - x2x6 - x3x4 - x3x5 + x5x6 + x6x0 = 0, x2 - 2x1x3 - x2 - x2x4 - x2x5 + 2x2x0 + x3 + x3x6 + x4x5 + x5 - x5x0 - xg = 0, x1x2 - x1x5 - 2x1x0 + 2x2x3 - x3x0 - x5x6 + 2x6x0 = 0, -2x1x2 - x1x4 - x 1x5 + 2x1x0 + 2x2x3 - 2x3x0 + 2x5x6 - x6x0 = 0, 2x2 + x1x3 - x 1x6 + 3x2x0 + x4x5 - x4x0 - x2 + x2 - x0 = 0, 2x2 - x1x3 + x1x6 + x2 + x2x0 + x2 - 2x3x6 + x4x5 - x4x0 + x5 - 2x5x0 + x2 + x0 = 0, x2 + x 1x3 - x1x6 + 2x2x5 - 3x2x0 + 2x3x6 + x2 + x4x5 - x4x0 + x6 + 3x0 = 0
C J
x 12 - x 1 x 3 + x 22 - x 2 x 4 - x 2 x 5 - x 2 x 0 - x 23 + x 3 x 6 + 2 x 5 x 0 - x 02 = 0
In Section 4 we provide an explicitbirational isomorphism L : X ^ W, where W is defined over Q. The explicit form of L may be used to compute explicit equation for W; this can be done with MAGMA [3].
2.4 A description of the two Edmonds maps
In the above model X of the Fricke-Macbeath curve it is easy to see a group Zf = G =
(A1, A2, A3) < Aut(X) where
A1(x,y1,y2,y4) = (x, -y1,y2,y4),
A2(x,y1,y2,y4) = (x,y1, -y2,y4),
A3(x,y1,y2,y4) = (x,y1,y2, -y4).
The quotient Riemann orbifold X/G has signature (0; 2, 2, 2,2,2,2, 2), that is, is the Riemann sphere with exactly 7 cone points of order 2.
R. A. Hidalgo: Edmonds maps on the Fricke-Macbeath curve
279
An automorphism of order 7 of the Fricke-Macbeath curve is given in such a model by
Ut \ ( 2 2 2 V1V2 \ B(x,yi,y2,y4) = px,p V2,p va,p -^-^ .
V	(x - p )(x - P ) J
The automorphism B normalizes G and it induces, on the orbifold X/G = C, the rotation T(x) = px. Moreover, X/(G, B) has signature (0; 2, 7,7), that is, the group (G, B) defines aregular dessin d'enfant (X, fi), where fi(x, y1,y2 ,y4) = x7 (this is one of the two Edmonds maps, but is defined over Q(p)).
We may also see that X admits the following anticonformal involution
j,	\ (l yi p5y2 p3y4\
J(x,yi,y2,y4)^x,x2^,^2-J .
It can be seen that JB J = B and JAj J = Aj, for j = 1,2,3. In this way, one gets another regular dessin d'enfant (X, S), where S(x,yi,y2,y4) = 1/x7 (this is the other Edmonds map, again defined over Q(p)).
As S = C o fi o J, where C(x) = x, we have that the two regular dessins d'enfant described above are chirals.
3 An explicit model of the Edmonds maps over Q(V-7)
In this section we will construct an explicit biregular isomorphism L : X ^ Z, where Z is defined over Q( V—7), so that both Edmonds maps are defined over such a field.
Note that Q(V—7) = Q(p + p2 + p4) since p + p2 + p4 = 2 (V—7 - 1). Most of the computations have been carried out with MAGMA [3] or with MATHEMATICA [20].
3.1
Let N = Gal(Q(p)/Q(V—7)) = (t) = Z3, where t(p) = p2. If we set x = (xi,x2,x3,x4) = (x, yi,y2,y4),
then it is not difficult to check that {fe = I, fT, fT2} is a Weil datum (i.e., they satisfies the Weil co-cycle condition in Weil's descent theorem [16]) with respect to the Galois extension Q(p)/Q(%/—7), where I denotes the identity and
y2y4
fT(x) = x,yi,y4, ■
(x — p4)(x — p5)
f	i	V2y4
f 2 (x) = (x,yl, (x — p4)(x — p5) ,y2
$1 : X -)• C12
3.2
Let us consider the rational map
i
(x,yi,y2,y4) ^ (x,w,v),
where
w = (wi,w2,w3, W4) = fT (x),
280
Ars Math. Contemp. 8 (2015) 235-244
V = (V1,V2, V3, V4) = fr2 (X).
We may see that $1 produces a birational isomorphism between X and $1(X) (its inverse is just the projection on the ¿-coordinate). Equations defining the algebraic curve $1 (X) are the following ones
= (X1 - 1)(X1 - P3)(X1 - P5)(X1 - p6) x3 = (X1 - p2)(x1 - p4)(X1 - p5)(x1 - p6) x2 = (X1 - p)(x1 - p3)(x1 - p4)(x1 - p5)
$1(X) =
W1 = X1, W2 = X2, W3 = X4, W4
X3X4
(X1 - p4)(x1 - p5)'
V1 = X1, V2 = X2, V3 =
X3X4
(X1 - p4)(x1 - p5), V4 X3
(3.1)
3.3
We consider the linear permutation action of N on the coordinates of C12 defined by
©1(t)(X, w, V) = (w, V,x).
Let us notice that the stabilizer of $1 (X), with respect to the above permutation action, is trivial since
{n € N : ©1(n)($1(X)) = $1(X)} = {n € N : Xn = X} = {e}.
3.4
Each 0 € Gal(C) induces a natural bijection
0 : C12 ^ C12 : (y1,..., y^) ^ (0(y1),..., 0(y12)). It is not hard to see that 0(X) = X6.
3.5
If 0 € Gal(C/Q(V-7)), then we denote by 0N is projection in N. With this notation, we see that the following diagram commutes (see also [8])
X
i-feN
$1 (X) 4-ei (6n )
X 6n	©1(0N )($1(X )) = $?N (X 6N ) = $1(X )6N
X il $1 (X) and, for every n, 0 e Gal(C/Q(V-7))), that
(*) ©1(nN) ◦ 0 = 0o ©1(nN).
(3.2)
9
R. A. Hidalgo: Edmonds maps on the Fricke-Macbeath curve
281
3.6
A generating set of invariant polynomials for the linear action ©1 (N) can be obtained with MAGMA as
¿1	= X1 + W1 + V1,	¿2	= x2 + w2 + V2
is	= xs + ws + vs,	t4	= x4 + w4 + V4
¿5	= x2 + w2 + v2,	¿6	= x2 + w^ + v|
¿7	= Xg + Wg + Vg,	¿8	= x| + w2 + v4
¿9	= xS + wS + Vg,	¿10	= xS + wS + vS
¿11	= xS + wS + Vg,	Í12	= xS + wS + v4
The map
: C12 ^ C12
(x,w, v) ^ (ti,..., 112) clearly satisfies the following properties:
=*1, j = 0,1, 2;
o ©1(Tj) = ^1, j = 0,1, 2.
(3.3)
Also (as we have chosen a set of generators of the invariant polynomials for the action of ©1(N)), it holds that is a branched regular cover with Galois group N. It turns out that, if we set Z1 = ^1($1(X)) and L1 = o , then
L1 : X ^ Z1
is a birational isomorphism (since the stabilizer of (X) is trivial). 3.7
If n € N, then
l (X n) = ◦ (xn) = ◦ ©onx^x)) =
zn

O $1(X) = L1(X) = Z1,
so Z1 can be defined by polynomials with coefficient over Q(%/—7). 3.8
Next, we proceed to compute explicit equations for Z1 and the inverse L-
Z1 ^ X.
The following equalities hold:
t1
X1 = "3 ,
X2
t2
¥, i4 =iS
(*) X4 =
(t3 - X3)(tj - P4)(ti - P5)
X3 + (f" - P4)(f" - P5)
t5 =31, t6 = "s,
= Í7
1
2
2
282	Ars Math. Contemp. 8 (2015) 235-244
(**) x2
= (tr - x3)( 3 - p4)2( - p5)2
4	x2 + ( - p4)2( - p5)2
^TU r ) H
11 ^2 t9 = -9 > ti0 = g2, ti2 =t11
(tii - x|)( ^ - p4)3( ^ - p5 )3
/ \ 3 _ m x-3)( 3_
X4 = x3 + (f- - p4)3( 3 - p5)3
Equality (*) permits to obtain x4 uniquely in terms of t i and x3 and the equation
x2 = (xi - 1)(xi - p3)(xi - p5)(xi - p6)
provides a polynomial equation (relating t i and t2) given by P1(t1,t2,t3,t7,t11) = 0, where
Pi(ii,i2,i3,ir,iii)
II
ti -ti +9^2
Equation
-81+27(1 + (p+p2 +p4))ti +9t2-3(p+p2 +p4)t3-ti +9t2 e Q(V-7)[ii,t2,¿3,tr,tii].
x3 = (xi - p )(xi - p )(xi - p )(xi - p ) permits to obtain the new equation
(1)	x2 = (ti - 3p2)(ti - 3p4)(ti - 3p5)(ti - 3p6)/81,
and the equation
x2 = (xi - p)(xi - p3)(xi - p4)(xi - p5)
provides the equation
(2)	x4 = (ti - 3p)(ti - 3p3)(ti - 3p4)(ti - 3p5)/81.
In this way, by replacing the above values for x3 and x2 (obtained in (1) and (2)) in the equality (**), we obtain the polynomial equation P2 (t i, t2, t3, tr, t ii) =0, where
P2(ti,t2,t3,tr,tii)
II
27 + 27(p + p2 + p4) - 18ti - 3(1 + (p + p2 + p4))t2 - 2t3 - t4 + 27tr e
Q(V=7)[ti,t2,t3,tr,tii].
Now, if we replace, inequality (***), x|| by x3(x2 - p2)(x2-p4)(x2 -p5)(x2 -p6)/81 and x4 by x4(ti - 3p)(ti - 3p3)(ti - 3p4)(ti - 3p5)/81, where x4 is given in (*), then we obtain a polynomial which is of degree one in the variable x3.
x3 = (-9p2(-162ti - 18tf + 4t5 - 243(1 + tii) + ti(27 - 54t3) + 6t4t3) + 3(729 + 18t4 -6t5 -27t3(-6+13) -ti(-2+t3) + 243ti(3+13) + 81t2(2 +tn +13))+ p3(2187-
R. A. Hidalgo: Edmonds maps on the Fricke-Macbeath curve	283
*1 + 27*1(-6 + is) + 9ti(-3 + *3) + 486*1*3 + 81tf(1+ *3) + 729*i (1 + 2*3)) + p5(2187 + 27*1 + 12*1 + *1 - 729*i(-1 + in - ¿3) + 729*2*3 + 81*1(5 + ¿3) + 9*5(1 + 2*3)) + p(2916*i + 3*1 - *7 - 81*1(-6 + *3) - 2187(-2 + *3) - 27*4(-2 + £3) + 9*1(2 + *3) + 243*2(5 + 2*3)) + p4(2187 + *7 - 729*i(-3 + *n -2*3) - 81*3(-1+ *3)+ 27*1(1+ *3) + 9*5(-1 + 2*3) + 243*2(1 + 3*3)))/(9(*1 - 243*ii + 27*1 (-1 + *3) + 81*1*3 + 9*3*3 + 3*4*3 + p(3 + *i)(—81 + 18*2 - 9*i + 2*4 + 27*i*3) + 27p2*i(3 + *2 + *i(3 + *3)) + p4*i (243 + 3*3 + *4 + 9*i(-1 + *3) + 27*i(3 + *3)) + p5(-6*i + *5 + 243(1 + *3) + 81*i(2 + *3) + 9*3(2 + *3)+ 27*2(3 + *3))+ p3*i(162 + 36*2 + 6*1 + 2*1 + 27*i(4 + *3))))
Then, using (*), we obtain
x4 = -((3p4 - *i)(3p5 - *i)(-p3(-2187 - 729*i + *7 + 243*i*3(2 + *3) + 9*i(3 + *3) + 27*4(6 + *3)+81*i(-1 + 3*3)) + p4 (2187 + 27*1 + * i + 9*i(- 1+ *3) -729*i(-3 +
*	i ^ *3) - 243* i (-1+ *3) - 81*i (-1+ *2))+ p(4374 +486*3 + 54* i + 3*6 - * i - 9*5(-2 + *3) - 243*2(-5 + *2) - 729* i(- 4 - *3 + *3)) - 3(* i(-2 + *3) + 3*5(2 + *3 ) - 729(1 +
*	i i *3) - 81*2(2 + * 11 + 2*3 - *2) + 9*4(—2 + *3) + 243* i(-3 - *3 + *2) + 27*3(-6 + *3 + *3)) - 9p2(4*5 - 243(1 + * 11) + 81* i(-2 + *3) + 6*4*3 + 9*3(-2 + 3*3) + 27* 1(1 + *3 + *§)) + p5(12*6 + *7 - 243*1*2 + 9*5(1 + *3) + 27*1 (1 + 2*3) - 81*3(-5 + *3 + *3) - 2187(-1 + *3 + *2) - 729* 1 (-1 + * 11 + *3 + *2))))/(9(567*1 + 6*6 + *7 + p(-3 +
*	1)(-54*3 + *6 + 9*1 (-4 + *3) + 729(-2 + *3) + 243* 1 (-2 + *3) - 81* 1 (-2 + *3)) + 27*4 (-7 + *3) + 9*5 (—5 + *3) + 2187(2 + *3) + 243*1 (-1 + 2*3) + 729*i(1 + 2*3) + p5(2187 +216*4 + 3*6 + 2*1 + 729*1*3 + 729*1(1+ *3) + 18*1 (2 + *3) + 81*1(16 + *3)) + P3*i (9*5 + *6 + 27*1 (-4 + *3) + 9*4 (3 + *3) + 851*1 (5 + *3) + 729(-5 + 2*3) + 243*i(-3 + 2*3)) + p4(2187 + 6*1 + 2*7 - 81*3( —14 + *3) + 18*5(-2 + *3) + 1458*1*3 + 27*4(5 + *3) + 243*2 (1 + 3*3)) - 9p2 (-243 + 243* 1 - 27*1 + *5 - 54*2 (- 5 + *3) + *4 (- 9 + 6*3)))).
Now, using such values for x3 and x4, and replacing them in (1) and (2) above, we obtain another two polynomials identities P3 (* 1, *3, *7, * 11) = 0 and P4 (* 1, *3, *7, * 11) = 0, where these two new polynomials are defined over Q(p) (see [9] for these long polynomials). In this way, we have obtained the following equations over Q(p) for Z1:
*4 = *3, 3*5 = *1, 3*6 = *2, *8 = *9 9*9 = *i, 9*10 = *3, * 12 = *11 Pi(*1,*2,*3,*7,*1i) =0 P2(*1,*2,*3,*7,*1i) =0 P3(*1,*2,*3,*7,*1i) =0 P4(*1,*2,*3,*7,*1i) =0
Notice that, by the above computations, we have explicitly the inverse of L1 given as
L-1 : Zi ^ X
(*1, ..., * 12) ^ (xi,X2,X3,X4),
where x1, x2, x3 and x4 are in terms of *1, *2, *3, *7 and *11.
As the variables *1,..., *12 are uniquely determined only by the variables *1, *2, *3, *7 and *11, if we consider the projection
n : C12 ^ C5
(*1, ..., * 12) ^ (*1, *2,*3,*7, *1i),
Zi =
C
12
284
Ars Math. Contemp. 8 (2015) 235-244
then
L = n o L1 : X ^ Z
Li (x, V\,V2,V4)
II
f	22	y^y2	3 3	\
3yi, + VA + (x-p4)(X-p5), y2 + y4 + (x-p4)2(X-p5)2 , V3 + y4 + (x-p4)3(i-p5)3 J
is a birational isomorphism, where
f Pi(ii,i2,is,ir,tii)=0 Z I P2(ii,i2,is,ir,tii)=0 I
Z ] Ps(ii,i2,i3,ir,tii)=0 C [ P4(ti,i2,i3,ir,tii) = 0
The inverse L- : Z ^ X is given as
L-i(ti,t2,t3,t7,tii) = (xi,X2,X3,X4).
We have obtained equations for Z over Q(p). But, as Z^ = Zi, for every n € N, and n is defined over Q, we may see that Zn = Z, for every n € N, that is, Z can be defined by polynomials over Q(V—7). To obtain such equations over Q(V—7), we replace each polynomial Pj (j = 3,4) by the new polynomials (with coefficients in Q( V—7))
Qj,i = Tr(Pj), Qj,2 = Tr(pPj), Qj,3 = Tr(p2Pj)
that is
Z =
Pi(ii,i2,i3,ir,in) = 0 P2(i1,i2,i3,ir,in) = 0 P3(tl, ¿2, t3, t7, til) + P3(tl, ¿2, t3, ¿7, ¿ll)" + P3^l, ¿2, ¿3, ¿7, ¿ll)"' = 0 Pf3(il, ¿2, t3, ¿7, tll) + P2f3(tl, ¿2, ¿3, ¿7, tll)T + P4f3(tl, ¿2, ¿3, ¿7, ¿ll)T' = 0 p2P3(tl, ¿2, ¿3, ¿7, ¿ll) + p4P3(il, ¿2, ¿3, ¿7, ¿ll)T + pPj^l, ¿2, ¿3, ¿7, ¿ll)T ' = 0
Pl(il, ¿2,¿3,¿7,¿ll) + P4(¿l,¿2,¿3,¿7,¿ll)T + P4(il, ¿2,¿3,¿7,¿ll)T' = 0 pP4(il, ¿2, ¿3, ¿7, ¿ll) + p2Pt(il, ¿2, ¿3, ¿7, ¿ll)T + p4Pt (¿l, ¿2, ¿3, ¿7, ¿ll)T ' = 0 p2P4 (¿l, ¿2, ¿3, ¿7, ¿ll) + p4Pt(il, ¿2, ¿3, ¿7, ¿ll)T + pPt (¿l, ¿2, ¿3, ¿7, ¿ll)T ' = 0
C C5
We have obtained an explicit model Z for the Fricke-Macbeath curve over Q(%/—7) together explicit birational isomorphisms L : X ^ Z and L-:L : Z ^ X.
3.9
Finally, notice that the regular dessin d'enfant (X, ^), given before, is isomorphic to that provided by the pair (Z, ,0*), where ^*(il, ¿2, ¿3, ¿7, ¿ll) = P O L l(il,Î2,Î3, ¿7, ¿ll) — (¿l/3)7; that is, the dessin d'enfant is defined over Q( V—7).
R. A. Hidalgo: Edmonds maps on the Fricke-Macbeath curve
285
4 An explicit isomorphism L : X ^ W where W is defined over Q
Next we explain how to construct an explicit birational isomorphism L : X ^ W, where W is known to be defined over Q.
Let us consider the explicit model Z c C5 over Q(%/—7) constructed above. Let M = Gal(Q(V—7)/Q) = (n) = Z2, where n is the complex conjugation. As already noticed, since X admits the anticonformal involution J (defined previously), the curve Z admits the anticonformal involution T = L o Jo L-1. It is not difficult to see that by setting ge = I and = S o T, where S(t1, ¿2, ¿3, ¿7, ¿n) = (¿^¿^¿^ ¿7, ¿n), we obtain a Weil datum for the Galois extension Q(%/—7)/Q. Now, identically as done above, we consider the rational map
$2 : Z ^ C10
(¿1, ¿2, ¿3, ¿7, ¿11) ^ (¿1, ¿2, ¿3, ¿7, ¿11, «1, «2, «3, «7, S1i)
where (¿1, ¿2, ¿3, i7,i11) = (s1, s2, s3, s7, s11). We may see that $2 induces a birational isomorphism between Z and $2(Z). In this case,
^ (Z ) = J Q1,1 (¿1, ¿2, ¿3, ¿7, ¿11) = ••• = ^4,3^1,¿2,¿3,¿7,¿11 ) = 0 1 ^ c10 2( ) \	gn(¿1,¿2,¿3,¿7,¿11) = («1, «2, «3, «7, «11)	J	'
The Galois group M induces the permutation action ©2(M) defined as
©2 (^(¿1, ¿2, ¿3, ¿7, ¿11, «1, «2, «3, «7, «11) = («1, «2, «3, «7, «11, ¿1, ¿2, ¿3, ¿7, ¿11)
A set of generators for the invariant polynomials (with respect to the previous permutation action) is given by
91 = ¿1 + «1, q2 = ¿2 + «2, 93 = ¿3 + «3,
22
94 = ¿7 + «7, 95 = ¿11 + «11, 96 = ¿1 + «1,
97 = ¿2 + «2, 98 = ¿3 + «3, 99 = ¿7 + «7,
2 2 910 = ¿11 + sii
Then the rational map
: c10 C10
(¿1, ¿2, ¿3, ¿7, ¿11, «1, «2, «3, «7, «11) ^ (91, 92, 93, 94, 95, 96, 97, 98, 99, 91o) satisfies the following properties:
= *2;	(4 1)
^2 o ©i(n) = ^1.
There are two possibilities:
1.	$2(Z) = ©2(n)($2(Z)); in which case Zn = Z and Z will be already defined over Q (which seems not to be the case); and
2.	the stabilizer of $2 (Z) under ©2 (M) is trivial.
286
ArsMath. Contemp. 8(2015)275-289
Under the assumption (2) above, we have that : $2(Z) ^ W = ^2($2(Z)) is a biregular isomorphism and that, as before, W is defined over Q. That is, the map L1 = o $2 : Z ^ W is an explicit biregular isomorphism and W is defined over Q. In this way, L = L1 o L : X ^ W is an explicit birational isomorphism as desired.
As R2 and Z are explicitly given, one may compute explicit equations for W over Q(%/—'7), say by the polynomials A1,..., Am e Q(%/—7)[q1, ...,q10] (this may be done with MAGMA [3] or by hands, but computations are heavy and very long). To obtain equations over Q we replace each Aj (which is not already defined over Q) by the traces Aj + An and iAj - iAn.
5 A remark on the elliptic curves in the model X
5.1 A connection to homology covers
Let us set A1 = 1, A2 = p, A3 = p2, A4 = p3, A5 = p4, A6 = p5 and A7 = p6, where p = e2ni/7. If S is the Fricke-Macbeath curve, then there is a regular branched cover Q : S ^ C having deck group G = Z3 and whose branch locus is the set {A1, A2, A3, A4, A5, A6, A7}. Let us consider a Fuchsian group
r = («1,..., «7 : a2
X7 = «1^2 • • • ®7 = 1}
acting on the hyperbolic plane H2 uniformizing the orbifold S/G.
If r' denotes the derived subgroup of r, then r' acts freely and S = H2/r' is a closed Riemann surface. Let H = r/r' = Zf; a group of conformal automorphisms of ¡3. Then there exists a set of generators of H, say a1,..., a6, so that the only elements of H acting with fixed points are these and a7 = a1a2a3a4a5a6. In [4, 5] it was noted that S corresponds to the generalized Fermat curve of type (2,6) (also called the homology cover of S/H)
x2 + x2 + x2 = 0 A3 - 1 1 2 3
r*2 I /-y*2 I /-y»2
-'1 + x2 + x4
A4-( A4 -
s =
A5 -( A5 -
A6 -A6 -
A7 -
0
X2 I yy»2 I ,-y»2
1 + x2 + X5
X2 I yy»2 I /-y»2 1 + x2 + x6
X2 I yy»2 I /-y»2
1 "I" x2 +1- X7
C PC,
that aj is just multiplication by -1 at the coordinate xj and that the regular branched cover P : S ^ C given by
P([x1 : X2 : X3 : X4 : X5 : X6 : X7]) =
X2 I /-y»2
2 x 1
X2 + A7X1
0
0
0
= z
has H has its deck group and branch locus given by the set of the 7th-roots of unity {A1,...,A7}.
R. A. Hidalgo: Edmonds maps on the Fricke-Macbeath curve
287
By classical covering theory, there should be a subgroup K < H, K = Zf, acting freely on S so that there is an isomorphism ^ : S ^ 5/K with ^G^-1 = H/K.
Let us also observe that the rotation R(z) = pz lifts under P to an automorphism T of S of order 7 of the form
T([xi : • • • : X7]) = [01x7 : c2xi : 03x2 : 04x3 : C5X4 : 06X5 : C7X6]
for suitable comples numbers cj. One has that TajT-1 = aj+1, for j = 1,..., 6 and Ta7T-1 = a1. The subgroup K above must satisfy that TKT-1 = K as R also lifts to the Fricke-Macbeath curve (as noticed in the Introduction).
5.2 About the elliptic curves in the Fricke-Macbeath curve
The subgroup
K* = (a!a3a7, a2a3a5, a^2a4) = Zf
acts freely on S and it is normalized by T. In particular, S* = 5/K * is a closed Riemann surface of genus 7 admitting the group L = H/K * = {e, a*,..., a* } = Zf (where a* is the involution induced by aj) as a group of automorphisms and it also has an automorphism T* of order 7 (induced by T) permuting cyclically the involutions a*. As S*/(L, T*) = S/(H, T) has signature (0; 2,7,7), we may see that S = S* and K = K*.
We may see that L = (a*,a*,a3) and a4 = a1af, a5 = a*a3, a* = a*a*a* and a* = a*a*. In this way, we may see that every involution of H/K is induced by one of the involutions (and only one) with fixed points; so every involution in L acts with 4 fixed points on S.
Let a*, a* G H/K be any two different involutions, so (a*, a*) = Zf. Then, by the Riemann-Hurwitz formula, the quotient surface S/(a*, a*) is a closed Riemann surface of genus 1 with six cone points of order 2. These six cone points are projected onto three of the cone points of S/H. These points are Aj, Aj and Ar, where a* = a* a*. In this way, the corresponding genus one surface is given by the elliptic curve
y2 = n (x - Ak)
So, for instance, if we consider i = 2 and j = 3, then r = 5 and the elliptic curve is
y2 = (x - l)(x - p3)(x - p5)(x - p6). If i = 1 and j = 2, then r = 4 and the elliptic curve is
y2 = (x - p2)(x - p4)(x - p5)(x - p6). If i = 1 and j = 3, then r = 7 and the elliptic curve is
y4 = (x - p)(x - p3 )(x - p4)(x - p5).
We have obtained the three elliptic curves appearing in the Fricke-Macbeath equation (2.1).
288
Ars Math. Contemp. 8 (2015) 235-244
5.3 Another model for the Fricke-Macbeath curve
The above description of the Fricke-Macbeath curve in terms of the homologycover S permits to obtain an explicit model. Let us consider now an affine model of 'S, say by taking x7 = 1, which we denote by S0. In this case the involution a7 is multiplication of all coordinates by -1. A set of generators for the algebra of invariant polynomials in C[xi, x2, x3, x4, x5, x6] under the natural linear action induced by K is
tl = x1,t2 = x2,is = x|,t4 = X2, t5 = x2,t6 = X2, = XiX2X5,t8 = XiX2,X3X6
tg = X1X4X6,ti0 = XiX3X4X5, tii = X2X4X5X6,112 = X2X3X4,ti3 = X3X5X6.
If we set
F : S0 ^ C
i3
F (xi ,X2,X3,X4,X5,X6 ) = (ti,t2,t3,t4,t5,t6,t7,ts,tg,ti0,tii,ti2,ti3),
then F(Sj0) will provide a model for the Fricke-Macbeath (affine) curve S. Equations for such an affine model of S are
t6ti0 = t5t8 = t4t7ti3 t3t6t7 t3t4t7 : t2t5tg =
ti + t2 + t3 = 0
A3- i
A4-	i
A4-	i
A5-	i
A5-	i
A6-	i
A6	-i
A7-i tgti3, t6t7ti2 =
t7ti3, t5t6ti2 = ti0tii, t4t6t7
= t8ti3, t3t5tg = ti0ti2, t2ti0 = = t7tii, t2t4ti3 =
ti + t2 + t4 = 0 ti + t2 + t5 =0 ti + t2 + t6 = 0 ti + t2 + 1 = 0
t8tii, t5t9ti2
tiiti3, t4t8 =
= tgtii, t3tii =
ti0ti3, t3t5t6
t7ti2, t2tgti3
tiiti2, t2t4t5t6 2
ti0tii tgti2
ti2ti3
= t2 = ti3
t8tii
t
t2t3t9 = t8ti2, t2t3t4 = t12, titi2ti3 = t8ti0 titii = t7tg, tit6ti2 = t8tg, tit5ti2 = t7ti0
tit4ti3 = tgti0, tit4t6 = tg, tit3t4t5 = ti0
tit2ti3 = t7t8, tit2t5 = t7, tit2t3t6 = t8
C
i3
ii
Of course, one may see that the variables t2, t3, t4, t5 and t6 are uniquely determined by the variable ti. Other variables can also be determined in order to get a lower dimensional model.
2
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 J. Wolfart for many early discussions about the results obtained in here.
R. A. Hidalgo: Edmonds maps on the Fricke-Macbeath curve
289
References
[1]	G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267-276, 479.
[2]	K. Berry and M. Tretkoff, The period matrix of Macbeath's curve of genus seven, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990), 31-40, Contemp. Math., 136, Amer. Math. Soc., Providence, RI, 1992.
[3]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, I. The user language, Computational algebra and number theory (London, 1993), J. Symbolic Comput. 24 (1997), 235-265,http://dx.doi.org/10.1006/jsco.1996.0125
[4]	A. Carocca, V. Gonzalez-Aguilera, R. A. Hidalgo and R. E. Rodriguez, Generalized Humbert curves, Israel]. Math. 164 (2008), 165-192.
[5]	G. Gonzalez-Diez, R.A. Hidalgo and M. Leyton, Generalized Fermat curves, Journal of Algebra 321 (2009), 1643-1660.
[6]	T. Ekedahl and J.-P. Serre, Exemples de courbes algebriques a jacobienne completement decomposable, C. R. Acad. Sci. Pari Ser. I Math. 317 (1993), 509-513.
[7]	M. Hendriks, Platonic maps of low genus, Ph. D. Thesis, Technische Universiteit Eindhoven, 2013
[8]	R. A. Hidalgo and S. Reyes-Carocca, Towards a constructive proof of Weil's descent theorem, http://arxiv.org/abs/120 3.62 94
[9]	R. A. Hidalgo. A computational note about Fricke-Macbeath's curve. http://arxiv.org/ abs/1203.6314
[10]	L. D. James and G. A. Jones, Regular orientable imbeddings of complete graphs, ]. Combin. Theory Ser. B 39 (1985), 353-367.
[11]	G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. 37 (1978), 273-307.
[12]	G. A. Jones, M. Streit and J. Wolfart, Wilson's map operations on regular dessins and cyclotomic fields of definition, Proc. Lond. Math. Soc. 100 (2010), 510-532.
[13]	S. K. Lando and A. Zvonkin, Graphs on surfaces and their applications, with an appendix by Don B. Zagier, Encyclopaedia of Mathematical Sciences 141, Low-Dimensional Topology, II. Springer-Verlag, Berlin, 2004.
[14]	A. Macbeath, On a curve of genus 7, Proc. London Math. Soc. 15 (1965), 527-542.
[15]	D. Singerman. Automorphisms of maps, permutation groups and Riemann surfaces. Bull. London Math. Soc. 8 (1976), 65-68.
[16]	A. Weil, The field of definition of a variety, Amer. ]. Math. 78 (1956), 509-524.
[17]	A. Wiman, "Ueber die hyperelliptischen Curven und diejenigen vom Geschlechte p = 3, welche eindeutigen Transformationen in sich zulassen" and "Ueber die algebraischen Curven von den Geschlechtern p = 4, 5 und 6 welche eindeutigen Transformationen in sich besitzen", Bihang Till Kongl. Svenska Vetenskaps-Akademiens Hadlingar, Stockholm, 1895-96.
[18]	J. Wolfart (joint work with G. Jones and M. Streit), Wilson's operations on regular dessins and cyclotomic fields of definition, Talk in Ankara, March 2011.
[19]	J. Wolfart, ABC for polynomials, dessins d'enfants and uniformization - a survey. Elementare und analytische Zahlentheorie, 313-345, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006.
[20]	Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, IL (2010).