# One-Dimensional Hurwitz Spaces,  Modular Curves, and Real Forms of  Belyi Meromorphic Functions

Authors: Antonio F. Costa; Milagros Izquierdo; Gonzalo Riera
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

Hurwitz spaces are spaces of pairs(S,f) where S is a Riemann surface and f:S→ℂ^ a meromorphic
function. In this work, we study 1-dimensional Hurwitz spaces ℋDp of meromorphic p-fold functions with four branched points, three
of them fixed; the corresponding monodromy representation over each branched
point is a product of (p−1)/2 transpositions and the monodromy group
is the dihedral group Dp. We prove that the completion ℋDp¯ of the Hurwitz space ℋDp is
uniformized by a non-nomal index p+1 subgroup of a triangular group with
signature (0;[p,p,p]). We also establish the relation of the meromorphic
covers with elliptic functions and show that ℋDp is a
quotient of the upper half plane by the modular group Γ(2)∩Γ0(p). Finally, we study the real forms of the Belyi projection
ℋDp¯→ℂ^ and show that
there are two nonbicoformal equivalent such real forms which are topologically conjugated.


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## Body

## 1. Introduction

Hurwitz spaces are spaces of pairs(S,f) where S is a Riemann surface and f:S→ℂ^ is a meromorphic function, that is, a
covering. These spaces have a natural complex structure and were introduced by
Clebsch and Hurwitz in the nineteenth century. In 1873, Clebsch [1] showed that the Hurwitz
space parametrizing simple n-fold coverings is connected  and Severi used this
result to show the irreducibility of the moduli of curves. See the recent
exposition by Eisenbud et al. [2]. In 1891, Hurwitz [3] gave a complex structure to
the set of pairs (S,f) having a fixed topological type. In 1969,
Fulton [4] showed
again the theorems of Clebsch and Severi using tools of algebraic geometry. He
showed how to produce Hurwitz spaces in positive characteristic. There are many
recent works studying Hurwitz spaces by
Fried, Völklein, Wewers, Bouw (see, e.g., [5, 6]).Another reason for the new attention to Hurwitz spaces
is that they provide examples of Frobenius manifolds in the sense of Dubrovin
[7].In this work, we study 1-dimensional Hurwitz spaces.
In 1989, Diaz et al. [8] showed that any covering of the Riemann sphere
branched on three points, that is, a Belyi curve [9], is a connected component of a 1-dimensional Hurwitz
space. The Belyi curves appear as Hurwitz spaces of meromorphic functions with
four branching points, three of them fixed. Hence, via Hurwtiz spaces, there is
a way to associate a Belyi curve and then a real algebraic curve to a type of
meromorphic function with four branching points. The correspondence between
types of meromorphic functions branched on four points and real algebraic
curves is not known in general. In this work, we will determine the real
algebraic curve describing the Hurwitz space of irregular dihedral coverings.
As a result, we obtain that there are two nonequivalent real forms for these
Hurwitz spaces.LetS be a Riemann surface and f:S→ℂ^ a meromorphic function branched on the set of
points B(f)={0,1,∞,λ:λ∉{0,1,∞}}.
Let p be a prime integer, we define an irregular p-fold dihedral covering as a meromorphic
function having a monodromy:(1.1)ω:π1(ℂ^−B(f),O)→Σp,suchthatthe monodromy group ω(π1(ℂ^−B(f),O))=Dp, given  byb∈B(f), ω(mb) is a product of (p−1)/2 transpositions, where mb is free homotopic to the boundary of a disc
neighborhood of b in ℂ^−(B(f)−{b}).LetℋDp be the Hurwitz space of irregular p-fold dihedral branched coverings and let π:ℋDp→ℂ^−{0,1,∞} be the covering defined by (f:ℂ^→ℂ^)→λ∈B(f)−{0,1,∞}.
Then ℋDp and π can be extended, in the Deligne-Munford
compactification, to a branched covering π¯:ℋDp¯→ℂ^ which is a Belyi function.In Section2, we present the uniformization of π¯:ℋDp¯→ℂ^ by a non-normal, index p+1,
subgroup of an hyperbolic (Euclidean for p=3) triangular group. Let Δ be the triangular Fuchsian (Euclidean, for p=3) group acting on the hyperbolic plane ℍ with signature (0;[p,p,p]) and canonical presentation:(1.2)〈x1,x2,x3:x1p=x2p=x3p=1;x1x2x3=1〉.We defineρ:Δ→PSL(2,p) by (1.3)ρ(x1)=[1−22−3],ρ(x2)=[1201],ρ(x3)=[10−21].Ifϕ:PSL(2,p)→Σp+1 is the natural map given by the geometrical
action of PSL(2,p) on ℙ1(ℤp) and δ=ϕ∘ρ,
then ℍ/δ−1(Stab(1)) is isomorphic to ℋDp¯ and the orbifold covering ℍ/δ−1(Stab(1))→ℍ/Δ is conformally equivalent to the covering π¯:ℋDp¯→ℂ^.
Therefore, the surface ℋDp¯ is a Riemann surface of genus (p−3)/2 which is the quotient of the underlying
surface of a regular hypermap of type (p,p,p) with automorphism group PSL(2,p) by the action of the stabilizer of infinity in PSL(2,p) (see [10]).In Section3, we establish the relation between
irregular p-fold dihedral coverings and elliptic curves.
We show that the space ℋDp is isomorphic to the quotient of the
hyperbolic plane by the modular group Γ(2)∩Γ0(p).
Some authors (see e.g., [11]) use different modular groups and curves in connection with
Hurwitz spaces. Our model is based on Definition 2.1 below, a concept consistent
with Diaz et al. [8].We end this section with a complete analysis of the
casep=3 establishing the relation between modular
groups, Belyi curves, modular equations, and euclidean crystallographic groups.Finally, in Section4 we study the real forms for the
Belyi function π¯:ℋDp¯→ℂ^.
A real form for a meromorphic function f:S→ℂ^ is a reflection r of ℂ^ and an anticonformal involution r˜ of S such that r˜ is the lift by f of r.
Two real forms (r1,r˜1) and (r2,r˜2) of a meromorphic function f:S→ℂ^ are conformally equivalent if there is an
automorfism α of ℂ^ and a lift of α by f to an automorphism α˜ of S,
such that r1=α−1∘r2∘α and r˜1=α˜−1∘r˜2∘α˜.
We establish that the meromorphic function ℋDp¯→ℂ^ admits two nonequivalent real forms: (r1,r˜1) and (r2,r˜2).
The set of real points for the anticonformal involutions r˜1 and r˜2 is connected and nonseparating. Hence r˜1 and r˜2 are topologically conjugate (see [12]).


## 2. Hurwitz Spaces of Irregular Dihedral Coverings

Hurwitz spaces are spaces of pairs(S,f) where S is a Riemann surface and f:S→ℂ^ is a meromorphic function. We will consider
the case when f has four branching points 0,1,∞,λ.
Definition 2.1.
Two
meromorphic functionsf1 and f2 are considered equivalent if there is an
automorphism g:S→S satisfying f1=f2∘g.Let(S1,f1) and (S2,f2) be two pairs of Riemann surfaces S1 and S2 and meromorphic functions f1:S1→ℂ^ and f2:S2→ℂ^ with four branching points. We say that (S1,f1) and (S2,f2) are of the same topological type if there are
homeomorphisms φ:S1→S2 and ψ:ℂ^→ℂ^ such that f2∘φ=ψ∘f1 and ψ(0)=0, ψ(1)=1 and ψ(∞)=∞.Lett be a class of topologically equivalent meromorphic
functions; ℋ(t) denotes the set of topological classes of
pairs (S,f) with f of topological type t.Given(S,f),
the representative of a point in ℋ(t),
we denote the branching set of f by B(f)={0,1,∞,λ}. Following [13], the pair (S,f) is given by B(f) and the monodromy representation of the
covering f:S→ℂ^: (2.1)ω:π1(ℂ^−B(f),O)→Σn.The groupω(π1(ℂ^−B(f),O)) is called the monodromy group of the n-fold covering f.Fixingω,
the variation of the point λ gives an 1-dimensional complex structure on the set of
pairs (S,f).LetB(f) be the branching set of f and b∈B(f).
Let Db be a disc in ℂ centered in b and such that Db¯∩B(f)={b}.
A meridian of b in ℂ^−B(f) based in O∈ℂ^ is a path starting and finishing at O and free homotopically equivalent to ∂Db,
where ∂Db is positively oriented. We will denote mb the homotopy class in π1(ℂ^−B(f),O) represented by a meridian of b.
Then we have the following
presentation of π1(ℂ^−B(f),O):(2.2)〈mb,b∈B(f):∏b∈B(f)mb=1〉.Definition 2.2.
Define an irregularp-fold dihedral covering as a covering  having a
monodromy ω:π1(ℂ^−B(f),O)→Σp,
such that the monodromy group ω(π1(ℂ^−B(f),O))=Dp,ω(mb) is a product of (p−1)/2 transpositions.We will denote the Hurwitz space of irregularp-fold dihedral branched coverings f:S→ℂ^ whose branching set consists exactly of 0,1,∞ and a variable point λ∈ℂ^−{0,1,∞} by ℋDp.There is a coveringπ:ℋDp→ℂ^−{0,1,∞},
defined by (f:S→ℂ^)→λ∈B(f)−{0,1,∞}.
Then ℋDp and π can be extended to a branched covering π¯:ℋDp¯→ℂ^ that is a Belyi function. We will determine π¯ and ℋDp¯.First of all we need to know the degree ofπ.
The degree of π is the number of different meromorphic
functions f:S→ℂ^ of degree p that are dihedral irregular coverings branched
on four fixed points. In other words, we look for the number of irregular
dihedral p-fold coverings S→ℂ^ with monodromy representation as in Definition 2.2.
Proposition 2.3.
There arep+1 classes of monodromies ω:π1(ℂ^−{0,1,∞,λ},i)→Σp of irregular p-fold dihedral coverings.Proof.
A monodromy is given by(ω(m0),ω(m1),ω(m∞)) (up to conjugacy in Σp). Let (2.3)s=(0)(1,p−1)(2,p−2)⋯(p−12,p+12)∈Σp,r=(0,1,2,…,p−1)∈Σp.
By conjugation inΣp,
we can assume that ω(m0)=s.
Now, eitherω(m0)=ω(m1)=s or s=ω(m0)≠ω(m1).
Ifω(m0)=ω(m1)=s,
by an automorphism of Dp,
we can assume that ω(m∞)=sr,
and so ω(mλ)=sr.
Ifs=ω(m0)≠ω(m1),
again by an automorphism of the group Dp,
we can assume that ω(m1)=sr.
Now each value of ω(m∞) gives a class of monodromies. Then we have (2.4)(ω(m0),ω(m1),ω(m∞))=(s,sr,sri),i=0,…,p.
Thus we havep+1 classes of monodromy representations.We have found that the degree ofπ and π¯ is p+1.We can establish a bijection between monodromy classes
and points ofℙ1(ℤp).
This bijection will be very useful in determining the monodromy representation
of π: (2.5)ℙ1(ℤp)ω(m0)ω(m1)ω(m∞)ω(mλ)(0:1)sssrsr(1:i)ssrsrisri−1.Since the degree ofπ is p+1,
the monodromy (2.6)δ:π1(ℂ^−{0,1,∞},−1)→Σp+1 associated to
the covering π is determined as follows.The meridianμ∞ in π1(ℂ^−{0,1,∞},−1) is represented by a closed path (2.7)γ:[0,1]→ℂ^−{0,1,∞} around ∞,
with base point −1,
together with the marking λ=γ(t).
If we start with a monodromy ω for λ0=γ(0)=−1,
then at λ1=γ(1)=−1 the monodromy ω transforms in a new monodromy ω′.
The monodromy ω′ is precisely ω∘σ3∗2,
where σ3∗2 is the isomorphism of π1(ℂ^−{0,1,∞},−1) induced by the braid σ32∈B4 acting on ℂ^−{0,1,∞,−1}.
We say that the effect of μ∞ on the monodromies is given by the braid σ32∈B4.In the same way, the effect on the monodromies of the
meridianμ1 is given by the braid σ3σ22σ3−1∈B4 and the effect of the meridian μ0 by σ3σ2σ12σ2−1σ3−1∈B4.The value ofδ(μ∞) (resp., δ(μ0), δ(μ1)) is given by the transformation of the
monodromies when λ moves along μ∞ (resp., μ0, μ1). Since B4 acts on the meridians by (2.8)σ3:(m0,m1,m∞,mλ)→(m0,m1,mλ,m∞mλ),σ2:(m0,m1,m∞,mλ)→(m0,m∞,m1m∞,mλ),σ1:(m0,m1,m∞,mλ)→(m1,m0m1,mλ,mλ), we obtain that
the monodromy δ is defined by the following action on the
monodromies of the meromorphic functions: δ(μ∞)(s,sr,sri)=(s,sr,sri−2) and δ(μ∞)(s,s,sr)=(s,s,sr).The bijection between monodromies and points ofℙ1(ℤp) yields us(2.9)δ(μ∞)=[10−21],δ(μ1)=[1201](sinceμ1=σ3σ22σ3−1),δ(μ0)=[1−22−3](sinceμ0isσ3σ2σ12σ2−1σ3−1).Hence, the monodromy group ofπ¯:ℋsDp→ℂ^ is PSL(2,p),
(see [10]). The
function π¯ is a (p+1)-fold covering with three branching points: 0,1,∞ (a Belyi function). The preimage of each
branching point contains a ramification point of local degree p and a pseudoramification point of local degree
one. In terms of the monodromy δ: δ(μ∗)=(s1,…,sp)(sp+1).Summarizing, we can describeπ¯:ℋDp¯→ℂ^ as follows in Theorem 2.4.Theorem 2.4.
Letp be a prime integer, p>3.
Let Δ be a triangular Fuchsian group with signature (0;[p,p,p]) and canonical presentation (2.10)〈x1,x2,x3:x1p=x2p=x3p=1;x1x2x3=1〉.
Defineρ:Δ→PSL(2,p) by (2.11)ρ(x1)=[1−22−3],ρ(x2)=[1201],ρ(x3)=[10−21].
Ifϕ:PSL(2,p)→Σp+1 is the natural map given by the geometrical
action of PSL(2,p) on ℙ1(ℤp) and δ=ϕ∘ρ,
then
(1)
ℋDp¯ is uniformized by δ−1(Stab(1))≤Δ,
that is, ℍ/δ−1(Stab(1)) is isomorphic to ℋDp¯,(2)
the orbifold covering𝔻/δ−1(Stab(1))→ℍ/Δ is analytically equivalent to the covering π¯:ℋDp¯→ℂ^.A similar result it is obtained in [11] for some different types of
Hurwitz spaces.In Figure1 we can see a fundamental region for the
triangular group Δ and its subgroup for p=5.
In Section 3, we obtain a fundamental region for all p.
Remark 2.5.
The signature of the Fuchsian groupδ−1(Stab(1)) is ((p−3)/2;[p,p,p]).
See Section 3.Figure 1Remark 2.6.
Forp=3,
there is a completely analogous description using the Euclidean
crystallographic group (0;[3,3,3]) (the group p3 in crystallographic notation) instead of (0;[p,p,p]).Remark 2.7.
Let us
consider the regular coveringR=ℍ/kerδ→ℍ/Δ,
the Riemann surface R is the underlying surface to a regular
hypermap of type (p,p,p) with automorphism group PSL(2,p).
Then ℋDp¯ is the quotient of R by a subgroup of PSL(2,p) isomorphic to the semidirect product of Cp with Cp−1 (the stabilizer of infinity [10]).Remark 2.8.
The points inℋDp¯−ℋDp are of two types.
Points ofℋDp¯ where π¯:ℋDp¯→ℂ^ is a local homeomorphism: there are noded
Riemann surfaces consisting in p+1 Riemann spheres joined by p nodes.
Singular poins ofπ¯:ℋDp¯→ℂ^ corresponding to meromorphic functions f¯:ℂ^→ℂ^ of degree p having three branching points and monodromy ω:π1(ℂ^−B(f¯),O)→Σp,
such that the monodromy group ω(π1(ℂ^−B(f),O)) is Dp,ω(mb) is a product of (p−1)/2 transpositions for two branching points and a p-cycle for the remaining one.


## 3. The Hurwitz SpacesℋDp¯ Uniformized by Modular Groups

We establish first the relation between the irregularp-fold dihedral coverings of ℂ^ and elliptic curves. As before, let f:ℂ^→ℂ^ be a rational function of degree p with branching points at 0,1,∞,λ given by a monodromy representation as in
Definition 2.2. The Galois covering, given by the kernel of the monodromy, is a
torus T∗ where Dp acts by a translation of order p and the elliptic involution. The quotient of T∗ by the translation group is again a torus T and the natural projection η gives the following conmutative diagram:
(3.1)Both vertical arrows are2 to 1 maps. The horizontal arrows are p to 1 maps. On the other hand we may start with a
torus T,
an elliptic involution ε,
and the 2 to 1 projection with branching points 0,1,∞,λ.
We obtain p+1 different coverings f as follows. Let 〈1,τ〉,
Im τ>0,
be the group of translations on ℂ so that T=ℂ/〈1,τ〉.
Consider the group epimorphisms (3.2)α:ℤ⊕ℤτ→ℤp.The kernel ofα defines a subgroup of index p and the quotient of ℂ by this subgroup defines the torus T∗;
there are p+1 homomorphisms with different kernels given by (3.3){αj(1,0)=j,0≤j≤p−1αj(0,τ)=1,,{αp(1,0)=1,αp(0,τ)=0.If℘ denotes the classical Weierstrass elliptic
function, the arrows in (3.1) are obtained by (3.4)for i≤j≤p−1 and ℘(z;p,τ) for i=p. ℘(z;1,τ) is an even elliptic function for kerαj.
Thus ℘(z;1,τ) is a rational function of ℘(z;1+(p−i)τ,pτ) giving us an explicit formula for f.
It may be worthwhile noticing that, for each τ,
we get a discrete group acting on ℂ,
depending analytically on τ and uniformizing an orbifold of genus 0 and four conic points of order 2,
that is, an Euclidean crystallographic group with signature (0;[2,2,2,2]), namely, (3.5)Gτ={z→±z+n+mτ}. We obtain
corresponding subgroups for kerαj.
Each mapping f may be visualized through
appropriate fundamental regions for the group Gτ and its (non normal) subgroups G(kerαj).The theory of the automorphic functionλ(τ) is classical and well known; we recall here
the necessaries to fix the notations:(i)
Γ=PSL(2,ℤ),
the modular group acting on the upper half plane ℍ;(ii)
Γ(2)={g(τ)=(aτ+b)/(cτ+d) in Γ:a,d≡1mod2,b,c≡0mod2}.The groupΓ(2) is a normal subgroup of Γ of index 6 given by the kernel of the natural map from Γ to PSL(2,ℤ2)≃Σ3.
A fundamental region for Γ(2) that we will use is given in Figure 2.Figure 2In this figure the fundamental region is divided into
twelve parts, each two adjacent parts being a fundamental region forΓ.
The free generators for Γ(2) are (3.6)A(τ)=τ+2,C(τ)=τ−2τ+1,
with B(τ)=(τ−2)/(2τ−3).
B fixes 1. We have the relation CBA=Id.The functionλ is the  universal covering map from ℍ to ℂ^−{0,1,∞} with a group of covering automorphisms Γ(2),
that is, λ(∞)=0, λ(0)=1, λ(1)=∞.
In terms of elliptic functions, (3.7)λ=e3−e1e2−e1, where e1=℘(1/2;1,τ),e2=℘(1/2+τ/2;1,τ),e3=℘(τ/2;1,τ).We also need to consider the following groups:(3.8)Γ0(p)={g(τ)=aτ+bcτ+dinΓ:c≡0modp},Γ0(p)={g(τ)=aτ+bcτ+dinΓ:b≡0modp}.In order to explain whyΓ(2) (and not Γ) is our main group it is necessary to review
some basic facts of Teichmüller theory of Riemann surfaces. See [14] for complete details.LetX be a fixed Riemann surface and f1:X→X1 a quasiconformal homeomorphism. Two such maps f1, f2 are considered equivalent if there is a
conformal isomorphism g:X1→X2 such that f2−1∘g∘f1 is homotopic to the identity relative to the
ideal boundary. Teichmüller space T(X) is the set of equivalence classes [f1].The setQC(X) of quasiconformal homeomorphisms of X acts on T(X) via (3.9)g∗[f1]=[f1∘g].IfQC0(X) is the normal subgroup consisting of those
maps homotopic to the identity relative to the ideal boundary, then the modular
group is M(X)=QC(X)/QC0(X).
Proposition 3.1.
The
modular group of the four times punctured sphere is a semidirect product ofΓ=PSL(2,ℤ) with Klein's group of order four.
The groupΓ(2) is isomorphic to the subgroup formed by the
elements that give the identity on the punctures.Proof.
LetG be the group of transformations generated by z→z+1, z→z+i, z→−z. G acts properly discontinously on ℂ′=ℂ−(1/2)ℤ[i] with quotient surface X′=ℂ′/G isomorphic to the Riemann sphere with the set {−1,0,1,∞} deleted. An explicit isomorphism is given by
the restriction to ℂ′ of the elliptic function (3.10)℘(z)−℘((1+i)/2)℘(1/2)−℘((1+i)/2).
An elementM in SL(2,Z) acts on ℂ′ as a linear mapping:(3.11)M(x,y)=(ax+by,cx+dy) porducing an
element of the modular group. Observe that M and −M∗ provide the same action on X′.
The homeomorphism induced by M on X′ permutes in general the three points {−1,0,1}.
Together with elements of Klein's group of order four such as z→(1+i)/2−z,
they fully generate the modular group and induce the group Σ4 of permutations of {−1,0,1,∞}.
To prove that the elements ofΓ(2) fix the punctures, it is enough to check this
for the generators A and C given above. Now, A acts as the linear map that sends the pair (1,0) and (0,1) to (1,0) and (2,1),
thus it sends 1/2 to itself, (1+i)/2 to (3+i)/2≡(1+i)/2 and i/2 to (2+i)/2≡i/2.
In the same manner, C(1/2)=(1−2i)/2≡1/2, C(i/2)=i/2,
and C((1+i)/2)=(1−i)/2=(1+i)/2.
Finally,Γ/Γ(2) is isomorphic to the group Σ3 of permutations of {−1,0,1} so that if an element of Γ fixes them, it belongs to Γ(2).Theorem 3.2.
The Hurwitz spaceℋDp of irregular p-fold dihedral branched coverings of the
sphere with four marked points is isomorphic to ℍ/Γ(2)∩Γ0(p).Proof.
Givenτ in ℍ,
we consider the linear map (3.12)fτ[xy]=(1r0s)[xy],whereτ=r+si,s>0. It sends the
lattice 〈1,i〉 to the lattice 〈1,τ〉 in ℂ and gives a quasiconformal homeomorphism from ℂ^−{0,1,∞,−1} to ℂ^−{0,1,∞,λ(τ)}.
We define a left action ofPSL(2,ℤ) on [fτ] via (3.13)M([fτ])=[fτ∘M−1]=[fτ∗], which is given
explicitely by (3.14)τ∗=aτ−b−cτ+difM=(abcd).
Observe thatM∈Γ(2) if and only if λ(τ)=λ(τ∗).
Now,M also acts on the right of
the epimorphisms (3.15)α:ℤ⊕ℤi→ℤp via M(α)=α∘M.
In particular, for α0 in (3.3), we have (3.16)M(α0)(1,0)=c,M(α0)(0,i)=a, so that kerα0=kerM(α0) if and only if c≡0modp.
Givenτ in ℍ,
we have a p-covering of the lattice 〈1,τ〉 by α0∘fτ−1,
therefore a covering of ℂ^−{0,1,∞,λ(τ)}.
Two such coverings will be equivalent in the sense of Definition 2.1 if and only
if c≡0modp.
The p+1 cosets of Γ(2)∩Γ0(p) in Γ(2) correspond to the p+1  homomorphisms αj and to the monodromy representations ω of Proposition 2.3: (3.17)kerM(α0)=αjifa≠0,j≡ca−1,kerM(α0)=αpifa≡0.An  explicit set of coset representatives will be given
next.Lemma 3.3.
Letφ:Γ(2)→PSL(2,ℤp) be the natural homomorphism that sends a
matrix to its class modulo p: (3.18)φ(A)=(1201),φ(C)=(10−21)inPSL(2,ℤp).
LetP denote the subgroup of matrices (ab0a−1) modulo p of order p((p−1)/2) and index p+1.
Then (3.19)kerφ=Γ(2)∩Γ(p),φ−1(P)=Γ(2)∩Γ0(p).Proof.
We have to establish thatφ is surjective. Since (3.20)φ(A−((p−1)/2))=(1101),φ(C−((p−1)/2))=(10−11) it is enough to
prove that these two matrices generate PSL(2,ℤp).
Consider A∗(τ)=τ+1, C∗(τ)=τ/(−τ+1) in Γ=PSL(2,ℤ).
Then C∗A∗(τ) has order three and fixes (−1+i3)/2, whereas C∗A∗C∗(τ) has order two and fixes i.
It is well known that A∗,C∗A∗,C∗A∗C∗ generate Γ.
Since the natural homomorphism Γ→PSL(2,ℤp) is surjective, so is φ.
The definitions of Γ(p) and Γ0(p) give kerφ and φ−1(P).Observe that we have the following inclusions:(3.21)Γ(2)∩Γ(p)⊲Γ(2)indexp(p2−12),Γ(2)∩Γ(p)⊲Γ(2)∩Γ0(p)<Γ(2),[Γ(2):Γ(2)∩Γ0(p)]=p+1,Γ(2)∩Γ0(p)⊲Γ0(p),index6.Proposition 3.4.
One has the right coset decomposition(3.22)Γ(2)=⋃k=0p−1(Γ(2)∩Γ0(p))Ck∪(Γ(2)∩Γ0(p))D, where D=CmB and 3−4m≡0modp.Proof.
We establish the decomposition(3.23)PSL(2,ℤp)=⋃k=0p−1Pφ(C)k∪Pφ(D).
Now(3.24)(abcd)(102k1)=(a+2bkbc+2dkd).
Therefore, ifd≠0modp,
we define k by c+2dk≡0modp to obtain a matrix in P.
If d≡0modp,
then (3.25)(ab−b−10)(02n1)=(bn2a+b0−2b−1)∈P,φ(D)=φ(Cm)φ(B)=(1−22−2m4m−3).
Now, taking4m−3≡0modp, φ(D)=(02n1)−1,
as required. (n=2m−2).Corollary 3.5.
LetF be a fundamental region for Γ(2) as in Figure 2. Then ⋃k=0p−1Ck(F)∪D(F) is a fundamental region for Γ(2)∩Γ0(p) in ℍ.When we compactify this region by filling in the
punctures of orderp,
then F corresponds to the quadrilateral with angles (2π/p,π/p,2π/p,π/p), a fundamental region for the Fuchsian group Δ in Theorem 2.4. The correspondence between
generators is (3.26)A⟷x2,C⟷x3,B⟷x1.This explains Figure1 for p=5,
where D(F) has been separated into two triangles for
symmetry.


### 3.1. The Casep=3

The theory presented above is of course valid forp=3 but there are aspects in this particular case
that make it worthwhile to examine it in more detail.We start with the description of the rational
functionsf:ℂ^→ℂ^ of degree three with four branching points. If
such a function has simple points at 0,1,∞ and double ramification points, then it must
be of the form (3.27)f(x)=x(ax+bcx+d)2.The reader may easily check that the function(3.28)f(x)=x((2z−1)2x−3z(z−2)3(2z−1)x+(z−2)2)2 has double
ramifications points at (3.29)3z(z−2)(2z−1)2,(z−22z−1)2,−13(z−2)22z−1,z(z−2)2z−1, lying over the
branching points 0,1,∞,λ with (3.30)λ=z3z−21−2z and simple
points at 0,1,∞,λ∗ with (3.31)λ∗=z(z−21−2z)3, lying over the
same branching points.We recover Proposition2.3 since, for each value of λ,
there are four possible covers by (3.30). On the other hand, given τ in ℍ,
the 2 to 1 mapping from T=ℂ/〈1,τ〉 to ℂ^ is given by y=℘(u;1,τ) and the mapping from T∗=ℂ/〈1,3τ〉 to ℂ^ is given by the corresponding function y=℘(u;1,3τ).
Given a point u1 in ℂ modulo 〈1,τ〉,
there are three preimages: u1, u2=(1+τ)−u1 and u3=u1+τ modulo 〈1,3τ〉.
Branching will happen when y is branched but x is not. Thus y is a 3 to 1 rational function of x with simple points at e1(3τ),e2(3τ),e3(3τ),∞ lying over e1(τ),e2(τ),e3(τ),∞.
Normalizing these points we obtain the values (3.30) and (3.31): (3.32)λ=λ(τ),λ∗=λ(3τ), therefore, (3.33)z−22z−1=e2−e3e2−e1℘(1/2+τ/2)−e1℘(1/2+τ/2)−e3(3τ).We consider now a fundamental region for the groupΓ(2)∩Γ0(3) of index 4 in Γ(2) in Figure 3.Figure 3If the sides are numbered from 1 to 10
counterclockwise starting at the vertical side on the left, the pairing of the
sides and the group generators are as follows:(3.34)A(τ)=τ+2:1⟷10,C3(τ)=−τ6τ−1:5⟷6,H1(τ)=5τ+212τ+5:4⟷7,H2(τ)=7τ+412τ+7:3⟷8,H3(τ)=5τ+46τ+5:2⟷9.We observe that at the puncture at0,λ has a triple value 1 and a simple value at the puncture at 2/3.
It has a simple 0 at ∞ and a triple 0 at 1/2 and a simple pole at 1/3 and a triple pole at 1.
This gives us a Belyi map (3.35)ℍΓ(2)∩Γ0(3)→ℍΓ(2), determined by λ as  a function of z as in (3.30). But the values of λ(3τ) yield also an interesting configuration. This
function is automorphic with respect to the group (3.36)Γ∗={aτ+bcτ+d,ad−bc=1,a,d∈ℤ,b∈(2/3)ℤ,c∈6ℤ} with [Γ∗:Γ(2)∩Γ0(3)]=4.
Indeed, this group is conjugated to Γ(2) via the transformation τ→3τ and Γ(2)∩Γ0(3) to Γ(2)∩Γ0(3).
Multiplying by 3 sends the fundamental region in Figure 3 to a
region bounded by arcs at −3, −2, −3/2, −1, 0, 1, 3/2,2,3.
The fundamental region for Γ(2),
as in Figure 2, pulls back to a fundamental region for Γ∗.
We observe now that λ(3τ) has a simple value 1 at the puncture at 0 and a triple value at 2/3.
Similar configurations are obtained at the other punctures; we have then a
Belyi map (3.37)ℍΓ(2)∩Γ0(3)→ℍΓ∗ determined by λ∗=λ(3τ) as a function of z as in (3.31). If we fill in the punctures of
Figure 3 we obtain the Euclidean crystallographic group (0;[3,3,3]); as shown in Figure 4.Figure 4We sumarize all this in Theorem3.6.
Theorem 3.6.
There is an isomorphism between the following spaces:
(a)
the completionℋD3¯ of the Hurwitz space of irregular 3-fold dihedral coverings of ℂ^;(b)
the quotient spaceℍ/Γ(2)∩Γ0(3)¯;(c)
the quotient spaceℂ/G where G is the group generated by g(u)=ρu, h(u)=ρu+(1−ρ), (ρ=(−1+i3)/2);(d)
the curve with Belyi mapx=z3((z−2)/(1−2z));(e)
the algebraic modular curve(3.38)x4+y4−256xy+384(x2y+xy2)−132(x3y+xy3)−762x2y2+384(x3y2+x2y3)−256x3y3=0, where x=λ(τ), y=λ(3τ).Proof.
Only (e) remains to be proven. The algebraic equation
is obtained by eliminatingz from (3.30) and (3.31). It can be done in a computer
algebra system via the instruction “Groebner basis of the ideal generated
by (3.39)x(1−2z)3+z(2−z)3,y(1−2z)+z3(2−z) and
lexicographic order z>x>y.”
We obtain a map fromℍ/Γ(2)∩Γ0(3)¯ into this modular curve. Since the quotient is
of genus 0,
this is the modular curve and the map is onto. Given now (x,y) we may determine z up to an eight root of unity by z8=x3/y.
For generic z these give eight different values of x.
But a generic one to one rational function from surfaces of genus 0 is one to one, proving the equivalence.


## 3.1. The Casep=3

The theory presented above is of course valid forp=3 but there are aspects in this particular case
that make it worthwhile to examine it in more detail.We start with the description of the rational
functionsf:ℂ^→ℂ^ of degree three with four branching points. If
such a function has simple points at 0,1,∞ and double ramification points, then it must
be of the form (3.27)f(x)=x(ax+bcx+d)2.The reader may easily check that the function(3.28)f(x)=x((2z−1)2x−3z(z−2)3(2z−1)x+(z−2)2)2 has double
ramifications points at (3.29)3z(z−2)(2z−1)2,(z−22z−1)2,−13(z−2)22z−1,z(z−2)2z−1, lying over the
branching points 0,1,∞,λ with (3.30)λ=z3z−21−2z and simple
points at 0,1,∞,λ∗ with (3.31)λ∗=z(z−21−2z)3, lying over the
same branching points.We recover Proposition2.3 since, for each value of λ,
there are four possible covers by (3.30). On the other hand, given τ in ℍ,
the 2 to 1 mapping from T=ℂ/〈1,τ〉 to ℂ^ is given by y=℘(u;1,τ) and the mapping from T∗=ℂ/〈1,3τ〉 to ℂ^ is given by the corresponding function y=℘(u;1,3τ).
Given a point u1 in ℂ modulo 〈1,τ〉,
there are three preimages: u1, u2=(1+τ)−u1 and u3=u1+τ modulo 〈1,3τ〉.
Branching will happen when y is branched but x is not. Thus y is a 3 to 1 rational function of x with simple points at e1(3τ),e2(3τ),e3(3τ),∞ lying over e1(τ),e2(τ),e3(τ),∞.
Normalizing these points we obtain the values (3.30) and (3.31): (3.32)λ=λ(τ),λ∗=λ(3τ), therefore, (3.33)z−22z−1=e2−e3e2−e1℘(1/2+τ/2)−e1℘(1/2+τ/2)−e3(3τ).We consider now a fundamental region for the groupΓ(2)∩Γ0(3) of index 4 in Γ(2) in Figure 3.Figure 3If the sides are numbered from 1 to 10
counterclockwise starting at the vertical side on the left, the pairing of the
sides and the group generators are as follows:(3.34)A(τ)=τ+2:1⟷10,C3(τ)=−τ6τ−1:5⟷6,H1(τ)=5τ+212τ+5:4⟷7,H2(τ)=7τ+412τ+7:3⟷8,H3(τ)=5τ+46τ+5:2⟷9.We observe that at the puncture at0,λ has a triple value 1 and a simple value at the puncture at 2/3.
It has a simple 0 at ∞ and a triple 0 at 1/2 and a simple pole at 1/3 and a triple pole at 1.
This gives us a Belyi map (3.35)ℍΓ(2)∩Γ0(3)→ℍΓ(2), determined by λ as  a function of z as in (3.30). But the values of λ(3τ) yield also an interesting configuration. This
function is automorphic with respect to the group (3.36)Γ∗={aτ+bcτ+d,ad−bc=1,a,d∈ℤ,b∈(2/3)ℤ,c∈6ℤ} with [Γ∗:Γ(2)∩Γ0(3)]=4.
Indeed, this group is conjugated to Γ(2) via the transformation τ→3τ and Γ(2)∩Γ0(3) to Γ(2)∩Γ0(3).
Multiplying by 3 sends the fundamental region in Figure 3 to a
region bounded by arcs at −3, −2, −3/2, −1, 0, 1, 3/2,2,3.
The fundamental region for Γ(2),
as in Figure 2, pulls back to a fundamental region for Γ∗.
We observe now that λ(3τ) has a simple value 1 at the puncture at 0 and a triple value at 2/3.
Similar configurations are obtained at the other punctures; we have then a
Belyi map (3.37)ℍΓ(2)∩Γ0(3)→ℍΓ∗ determined by λ∗=λ(3τ) as a function of z as in (3.31). If we fill in the punctures of
Figure 3 we obtain the Euclidean crystallographic group (0;[3,3,3]); as shown in Figure 4.Figure 4We sumarize all this in Theorem3.6.
Theorem 3.6.
There is an isomorphism between the following spaces:
(a)
the completionℋD3¯ of the Hurwitz space of irregular 3-fold dihedral coverings of ℂ^;(b)
the quotient spaceℍ/Γ(2)∩Γ0(3)¯;(c)
the quotient spaceℂ/G where G is the group generated by g(u)=ρu, h(u)=ρu+(1−ρ), (ρ=(−1+i3)/2);(d)
the curve with Belyi mapx=z3((z−2)/(1−2z));(e)
the algebraic modular curve(3.38)x4+y4−256xy+384(x2y+xy2)−132(x3y+xy3)−762x2y2+384(x3y2+x2y3)−256x3y3=0, where x=λ(τ), y=λ(3τ).Proof.
Only (e) remains to be proven. The algebraic equation
is obtained by eliminatingz from (3.30) and (3.31). It can be done in a computer
algebra system via the instruction “Groebner basis of the ideal generated
by (3.39)x(1−2z)3+z(2−z)3,y(1−2z)+z3(2−z) and
lexicographic order z>x>y.”
We obtain a map fromℍ/Γ(2)∩Γ0(3)¯ into this modular curve. Since the quotient is
of genus 0,
this is the modular curve and the map is onto. Given now (x,y) we may determine z up to an eight root of unity by z8=x3/y.
For generic z these give eight different values of x.
But a generic one to one rational function from surfaces of genus 0 is one to one, proving the equivalence.


## 4. Real Forms of The Belyi MapℋDp¯→ℂ^

LetΔ be the triangular Fuchsian group with signature (0;[p,p,p]) and canonical presentation (4.1)〈x1,x2,x3:x1p=x2p=x3p=1;x1x2x3=1〉.We defineρ:Δ→PSL(2,p) by (4.2)ρ(x1)=[1−22−3],ρ(x2)=[1201],ρ(x3)=[10−21].Lemma 4.1.
For each
primep>3 there are non-Euclidean crystallographic (NEC)
groups Λ1 and Λ2,
such that Λ1+=Δ=Λ2+ with signatures (4.3)s(Λ1)=(0;+;[−];{(p,p,p)}),s(Λ2)=(0;+;[p];{(p)}). There are two
epimorphisms: (4.4)ρ1:Λ1→PGL(2,p),ρ2:Λ2→PGL(2,p) such that ρ1|Δ=ρ2|Δ=ρ.Proof.
By [15] (see also [16]), we obtain the existence of the groups Λ1 and Λ2 such that Λ1+=Δ=Λ2+.
Let(4.5)〈c0,c1,c2:c02=c12=c22=1;(c0c1)p=(c1c2)p=(c2c0)p=1〉 be a canonical
presentation for the NEC group Λ1 and let (4.6)〈x,c0,c1,e:c02=c12=xp=1;xe=(c0c1)p=1;c0=ec1e−1〉 be a canonical
presentation for the NEC group Λ2.
Then we defineρ1:Λ1→PGL(2,p) as (4.7)ρ1(c0)=[102−1],ρ1(c1)=[1−20−1],ρ1(c2)=[100−1], and we define ρ2:Λ2→PGL(2,p) by (4.8)ρ2(x)=[1201],ρ2(c0)=[0110].
It is clear thatρ1|Δ=ρ2|Δ=ρ.Remark 4.2.
Forp=3,
the two extensions of (0;[3,3,3]) (or p3) are the classical plane Euclidean
crystallographic groups p3m1 and p31m.An anticonformal involutionr of ℂ^ conjugated to the complex conjugation z→z¯ is called a reflection ofℂ^.
Definition 4.3 (real form of a meromorphic function; see [17]).
LetS be a Riemann surface and f:S→ℂ^ a meromorphic function. A real form for f:S→ℂ^ is a reflection r of ℂ^ and an anticonformal involution σ of S such that σ is the lift of r by f.Definition 4.4 (equivalence of real forms of a meromorphic function).
Two real forms(r1,σ1) and (r2,σ2) of a meromorphic funcion f:S→ℂ^ are biconformally equivalent if there are
automorfisms α of ℂ^ and α˜ of S,
such that (4.9)α∘f=f∘α˜,r1=α∘r2∘α−1,σ1=α˜∘σ2∘α˜−1.Proposition 4.5.
The meromorphic functionℋDp¯→ℍ/Δ=ℂ^ admits two nonequivalent real forms.Proof.
With the same notation as in Theorem2.4 and Lemma 4.1.
Let ϕ′:PGL(2,p)→Σp+1 be the natural representation given by the
geometrical action of PGL(2,p) on ℙ1(ℤp).
Let δi=ϕ′∘ρi, i=1,2.
The orbifold coverings(4.10)ℋDp¯=ℍδ−1(Stab(1))→ℍδi−1(Stab(1)),i=1,2, provide us the
existence of two anticonformal involutions σ1 and σ2 in ℋDp¯,
defining the two required real forms.
The fact that the signatures ofΛ1 and Λ2 are different implies that the two defined
real forms are not equivalent.Proposition 4.6.
Letp>3.
The set of real points for each of the two real forms of the meromorphic
function ℋDp¯→ℍ/Δ in the above proposition is connected and
nonseparating.Proof.
(1) The set of real points is connected.
In order to compute the number of connected components
of the set of real points of each real form, we need to compute the number of
period cycles in the signature ofδi−1(Stab(1)).
We will use the technics in [18, 19].
Following [18] (see also [19]), we construct the Schreier graph given by the action
of the canonical generators of Λ1 by ρi on the cosets of ϕ′−1(Stab(1))≤PGL(2,p).
Each connected component of this graph corresponds to a period cycle in s(δi−1(Stab(1))).
For each reflection cj in Λi we have that ρi(cj) has two fixed points in ℙ1(ℤp) and then the permutation δi(cj) left invariant two indices, so each reflection
gives rise to two vertices of the graph: vj1,vj2.
Since all periods in Λi are prime integers, then we connect the
vertices vjk with vj+1k by an edge and we have that δi−1(Stab(1)) has one or two cycles. Finally, the hyperbolic
generator with axis in the fixed point set of the reflection in δi−1(Stab(1)) is sent by δi to an element of order two: (4.11)[1−12−1](forδ1)or[0−110](forδ2).
So the verticesv31 and v02 for δ1 and v11 and v02 for δ2 are joined by an edge. Hence the graph is
connected and there is only one period cycle in s(δi−1(Stab(1))).
Therefore, the set of real points of each real form is connected.
(2) The real points are nonseparating
Forp≡1mod4,
there are square roots of −1 in ℤp by Wilson's theorem. For p≡3mod4,
there are ((p−1)/2)-roots of −1 in ℤp since (ℤp)∗ is cyclic. Let q be such a root of −1.
Consider the following element of PGL(2,ℤp): (4.12)[100q].
The above element inPGL(2,ℤp)−PSL(2,ℤp) has order 4 or p−1 and fixes (1:0)∈ℙ1(ℤp),
then there are orientation reversing transformations in δi−1(Stab(1)) of order at least 4.
Hence there is a − sign in s(δi−1(Stab(1))) and the real parts
of the two real forms are nonseparating.Remark 4.7.
With the notation in the previous
theorem, the complete signatures ofs(δi−1(Stab(1))),
for p>3 are (4.13)(p−32;−;[−];(p,p,p))forδ1,(p−32;−;[p];(p))forδ2.
Remark that there is only one exceptional case,p=3 when the two real parts are connected but
separating (in this case (p−3)/2=0), the signatures are
(i)
(0;+;[−];(3,3,3)) (the Euclidean crystallographic group p3m1),(ii)
(0;+;[3];(3)) (the Euclidean crystallographic group p31m).



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*Source: 609425-2008-12-17.xml*