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

Volume 2008, Article ID 609425,18pages doi:10.1155/2008/609425

Research Article

One-Dimensional Hurwitz Spaces,

Modular Curves, and Real Forms of

Belyi Meromorphic Functions

Antonio F. Costa,1_{Milagros Izquierdo,}2 _{and Gonzalo Riera}3

1_{Departamento de Matem´aticas, Facultad de Ciencias, Universidad Nacional de Educaci´on}

a Distancia (UNED), Senda del rey, 9, 28040 Madrid, Spain

2_{Matematiska Institutionen, Link¨opings Universitet 581 83 Link¨oping, Sweden} 3_{Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile,}

Avenida Vicu ˜na Mackenna 4860, Santiago, Chile

Correspondence should be addressed to Antonio F. Costa,[email protected]

Received 9 June 2008; Accepted 13 October 2008

Recommended by Heinrich Begehr

Hurwitz spaces are spaces of pairsS, f where S is a Riemann surface and f : S → C a meromorphic function. In this work, we study 1-dimensional Hurwitz spacesHDp_{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 groupDp. We prove that the completionHDpof the Hurwitz

spaceHDp_{is uniformized by a non-nomal indexp}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 thatHDp_{is a quotient of the upper half plane by the modular group}Γ₂∩Γ_{0p. Finally, we}

study the real forms of the Belyi projectionHDp → C_{and show that there are two nonbicoformal}

equivalent such real forms which are topologically conjugated.

Copyrightq2008 Antonio F. Costa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

spaces in positive characteristic. There are many recent works studying Hurwitz spaces by Fried, V ¨olklein, Wewers, Bouwsee, 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 Dubrovin7.

In this work, we study 1-dimensional Hurwitz spaces. In 1989, Diaz et al.8showed that any covering of the Riemann sphere branched on three points, that is, a Belyi curve9, 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.

LetSbe a Riemann surface andf : S→C a meromorphic function branched on the set of pointsBf {0,1,∞, λ:λ /∈ {0,1,∞}}. Letpbe a prime integer, we define an irregular p-fold dihedral covering as a meromorphic function having a monodromy:

ω:π1C−Bf, O−→Σp, such that 1.1

the monodromy groupωπ1C −Bf, O Dp,given byb ∈Bf,ωmbis a product of p−1/2 transpositions, wherembis free homotopic to the boundary of a disc neighborhood ofbinC−Bf− {b}.

Let HDp _{be the Hurwitz space of irregular p-fold dihedral branched coverings and}

letπ : HDp→C − {_0,1,∞}_{be the covering defined by}f _: C→C→_λ ∈ Bf− {_0,1,∞}_.

ThenHDp _{andπ can be extended, in the Deligne-Munford compactification, to a branched}

coveringπ :HDp→C_{which is a Belyi function.}

InSection 2, we present the uniformization ofπ : HDp→C _{by a non-normal, index}

p1, subgroup of an hyperbolicEuclidean forp3triangular group. LetΔbe the triangular Fuchsian Euclidean, for p 3 group acting on the hyperbolic plane H with signature 0;p, p, pand canonical presentation:

x1, x2, x3:x₁pxp₂ xp₃ 1;x1x2x31

. 1.2

We defineρ:Δ→PSL2, pby

ρx1

1 −2 2 −3

, ρx2

1 2 0 1

, ρx3

1 0

−2 1

. 1.3

Ifφ : PSL2, p→Σp1is the natural map given by the geometrical action of PSL2, p onP1_{Zpandδφ◦ρ, thenH/δ}−1_{Stab1is isomorphic toH}Dp_{and the orbifold covering}

H/δ−1_{Stab1→H/Δis conformally equivalent to the coveringπ:H}Dp→C. Therefore, the

surfaceHDp _{is a Riemann surface of genus}p−₃/_{2 which is the quotient of the underlying}

surface of a regular hypermap of typep, p, pwith automorphism group PSL2, pby the action of the stabilizer of infinity in PSL2, p see10.

plane by the modular groupΓ2∩Γ0p. Some authorssee e.g.,11use different modular groups and curves in connection with Hurwitz spaces. Our model is based onDefinition 2.1

below, a concept consistent with Diaz et al.8.

We end this section with a complete analysis of the casep3 establishing the relation between modular groups, Belyi curves, modular equations, and euclidean crystallographic groups.

Finally, in Section 4we study the real forms for the Belyi functionπ : HDp→C. A

real form for a meromorphic functionf :S→C is a reflectionr ofC and an anticonformal involution r of S such that r is the lift by f of r. Two real forms r1,r1 and r2,r2 of a meromorphic function f : S→C are conformally equivalent if there is an automorfism α of C and a lift ofαbyf to an automorphismα ofS, such that r1 α−1◦ r2◦αand r1 α−1_◦r

2◦α. We establish that the meromorphic functionHDp→C admits two nonequivalent real forms:r1,r1andr2,r2. The set of real points for the anticonformal involutionsr1and r2is connected and nonseparating. Hencer1andr2are topologically conjugatesee12.

2. Hurwitz spaces of irregular dihedral coverings

Hurwitz spaces are spaces of pairsS, fwhereSis a Riemann surface andf : S→C is a meromorphic function. We will consider the case whenfhas four branching points 0,1,∞, λ.

Definition 2.1. Two meromorphic functionsf1 andf2are considered equivalent if there is an automorphismg:S→Ssatisfyingf1f2◦g.

LetS1, f1andS2, f2be two pairs of Riemann surfacesS1andS2and meromorphic functionsf1 : S1→C andf2 : S2→ C with four branching points. We say thatS1, f1and S2, f2are of the same topological type if there are homeomorphismsϕ : S1→S2 andψ :

C→Csuch thatf2◦ϕψ◦f1andψ0 0, ψ1 1 andψ∞ ∞.

Lettbe a class of topologically equivalent meromorphic functions;Htdenotes the set of topological classes of pairsS, fwithfof topological typet.

GivenS, f, the representative of a point inHt, we denote the branching set off byBf {0,1,∞, λ}. Following13, the pairS, fis given byBfand the monodromy representation of the coveringf:S→C:

ω:π1C−Bf, O

−→Σn. 2.1

The groupωπ1C−Bf, Ois called the monodromy group of then-fold covering f.

Fixingω, the variation of the pointλgives an 1-dimensional complex structure on the set of pairsS, f.

LetBfbe the branching set offandb∈Bf. LetDbbe a disc inCcentered inband such thatDb∩Bf {b}. A meridian ofbinC −Bfbased inO∈C is a path starting and finishing atOand free homotopically equivalent to∂Db, where∂Dbis positively oriented. We will denotemb the homotopy class inπ1C −Bf, Orepresented by a meridian ofb. Then we have the following presentation ofπ1C−Bf, O:

mb, b∈Bf:

b∈Bf

mb 1

Definition 2.2. Define an irregular p-fold dihedral covering as a covering having a monodromyω:π1C−Bf, O→Σp, such that the monodromy groupωπ1C−Bf, O Dp, ωmbis a product ofp−1/2 transpositions.

We will denote the Hurwitz space of irregularp-fold dihedral branched coveringsf : S→C whose branching set consists exactly of 0,1,∞and a variable pointλ∈C− {0,1,∞}by

HDp_.

There is a coveringπ : HDp→C − {_0,1,∞}_{, defined by} f _{: S}→C→_λ ∈ Bf−

{0,1,∞}. ThenHDp _{and π can be extended to a branched covering π :} HDp→C _{that is a}

Belyi function. We will determineπandHDp_.

First of all we need to know the degree ofπ. The degree ofπis the number of different meromorphic functionsf :S→C of degreepthat are dihedral irregular coverings branched on four fixed points. In other words, we look for the number of irregular dihedral p-fold coveringsS→C with monodromy representation as inDefinition 2.2.

Proposition 2.3. There arep1 classes of monodromiesω:π1C− {0,1,∞, λ}, i→Σpof irregular

p-fold dihedral coverings.

Proof. A monodromy is given byωm0, ωm1, ωm_∞ up to conjugacy inΣp. Let

s 01, p−12, p−2· · ·

p−1 2 ,

p1 2

∈Σp, r 0,1,2, . . . , p−1∈Σp.

2.3

By conjugation inΣp, we can assume thatωm0 s. Now, eitherωm0 ωm1 sorsωm0/ωm1.

Ifωm0 ωm1 s, by an automorphism ofDp, we can assume thatωm∞ sr,

and soωmλ sr.

Ifsωm0/ωm1, again by an automorphism of the groupDp, we can assume that ωm1 sr. Now each value ofωm_∞gives a class of monodromies. Then we have

ωm0

, ωm1

, ωm_∞s, sr, sri, i0, . . . , p. 2.4

Thus we havep1 classes of monodromy representations.

We have found that the degree ofπandπisp1.

We can establish a bijection between monodromy classes and points ofP1_{Zp. This} bijection will be very useful in determining the monodromy representation ofπ:

P1_{Zp ωm} 0

ωm1

ωm_∞ ωmλ 0 : 1 s s sr sr 1 :i s sr sri _sri−1_.

2.5

Since the degree ofπisp1, the monodromy

δ:π1C− {0,1,∞},−1−→Σp1 2.6

The meridianμ∞inπ1C − {0,1,∞},−1is represented by a closed path

γ:0,1−→C− {0,1,∞} 2.7

around ∞, with base point −1, together with the marking λ γt. If we start with a monodromyωforλ0 γ0 −1, then atλ1 γ1 −1 the monodromyωtransforms in a new monodromyω. The monodromyωis preciselyω◦σ₃2_∗, whereσ₃2_∗is the isomorphism of π1C− {0,1,∞},−1induced by the braidσ₃2 ∈B4acting onC− {0,1,∞,−1}. We say that the effect ofμ_∞on the monodromies is given by the braidσ2

3 ∈B4.

In the same way, the effect on the monodromies of the meridianμ1 is given by the braidσ3σ₂2σ₃−1∈B4and the effect of the meridianμ0byσ3σ2σ₁2σ₂−1σ₃−1∈B4.

The value of δμ_∞ resp., δμ0, δμ1 is given by the transformation of the monodromies whenλmoves alongμ∞resp.,μ0, μ1. SinceB4acts on the meridians by

σ3:m0, m1, m∞, mλ

−→m0, m1, mλ, mm_∞λ

,

σ2:m0, m1, m∞, mλ

−→m0, m∞, mm1∞, mλ

,

σ1:

m0, m1, m∞, mλ

−→m1, mm01, mλ, mλ

,

2.8

we obtain that the monodromyδis defined by the following action on the monodromies of the meromorphic functions:δμ∞s, sr, sri s, sr, sri−2andδμ∞s, s, sr s, s, sr.

The bijection between monodromies and points ofP1_{Zpyields us}

δμ∞

1 0

−2 1

,

δμ1

1 2

0 1 sinceμ1σ3σ 2 2σ3−1

,

δμ0

1 −2

2 −3 sinceμ0 isσ3σ2σ 2 1σ2−1σ3−1

.

2.9

Hence, the monodromy group ofπ:HDsp→Cis PSL2, p,see10. The functionπis ap1-fold covering with three branching points: 0,1,∞a Belyi function. The preimage of each branching point contains a ramification point of local degreepand a pseudoramification point of local degree one. In terms of the monodromyδ:δμ∗ s1, . . . , spsp1.

Summarizing, we can describeπ:HDp→C _{as follows inTheorem 2.4.}

Theorem 2.4. Letpbe a prime integer,p > 3. LetΔbe a triangular Fuchsian group with signature

0;p, p, pand canonical presentation

x1, x2, x3:x₁pxp₂ xp₃ 1;x1x2x31

. 2.10

Defineρ:Δ→PSL2, pby

ρx1

1 −2 2 −3

, ρx2 1 2 0 1

, ρx3

1 0

−2 1

. 2.11

A

B B

C D E

A

F

F

C E

D

Figure 1

1HDp_{is uniformized byδ}−1_Stab1≤Δ_{, that is,}H_/δ−1_{Stab1is isomorphic to}HDp_,

2the orbifold coveringD/δ−1_{Stab1→H/Δis analytically equivalent to the coveringπ:}

HDp→C_.

A similar result it is obtained in11for some different types of Hurwitz spaces. In Figure 1 we can see a fundamental region for the triangular group Δ and its subgroup forp5. InSection 3, we obtain a fundamental region for allp.

Remark 2.5. The signature of the Fuchsian group δ−1_{Stab1 is p− 3/2;p, p, p. See} Section 3.

Remark 2.6. For p 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 coveringRH/kerδ→H/Δ, the Riemann surface R is the underlying surface to a regular hypermap of type p, p, p with automorphism group PSL2, p. ThenHDp _{is the quotient ofR by a subgroup of PSL2}, p_{isomorphic to}

the semidirect product ofCpwithCp−1the stabilizer of infinity10.

Remark 2.8. The points inHDp− HDp _{are of two types.}

Points ofHDp_{whereπ :}HDp→C_{is a local homeomorphism: there are noded Riemann}

surfaces consisting inp1 Riemann spheres joined bypnodes.

Singular poins ofπ :HDp→C _{corresponding to meromorphic functionsf :}C→C _of

3. The Hurwitz spacesHDp _{uniformized by modular groups}

We establish first the relation between the irregularp-fold dihedral coverings ofCand elliptic curves. As before, letf :C→C be a rational function of degreepwith branching points at 0,1,∞, λ given by a monodromy representation as inDefinition 2.2. The Galois covering, given by the kernel of the monodromy, is a torusT∗whereDpacts by a translation of orderp and the elliptic involution. The quotient ofT∗by the translation group is again a torusTand the natural projectionηgives the following conmutative diagram:

T η T∗

C f C.

3.1

Both vertical arrows are 2 to 1 maps. The horizontal arrows arepto 1 maps. On the other hand we may start with a torusT, an elliptic involutionε, and the 2 to 1 projection with branching points 0,1,∞, λ. We obtainp1 different coveringsfas follows. Let1, τ, Imτ > 0, be the group of translations onCso thatTC/1, τ. Consider the group epimorphisms

α:Z⊕Zτ −→Zp. 3.2

The kernel ofαdefines a subgroup of indexpand the quotient ofCby this subgroup defines the torusT∗; there arep1 homomorphisms with different kernels given by

αj1,0 j, 0≤j≤p−1, αj0, τ 1,

αp1,0 1, αp0, τ 0.

3.3

If℘denotes the classical Weierstrass elliptic function, the arrows in3.1are obtained by

zmod1,τ id zmodkerαj

℘(z; 1,τ) f ℘z; 1 + (p−i)τ,pτ

3.4

fori≤j ≤ p−1 and℘z;p, τforip.℘z; 1, τis an even elliptic function for kerαj. Thus ℘z; 1, τis a rational function of℘z; 1 p−iτ, pτgiving us an explicit formula forf. It may be worthwhile noticing that, for eachτ, we get a discrete group acting onC, depending analytically onτ and uniformizing an orbifold of genus 0 and four conic points of order 2, that is, an Euclidean crystallographic group with signature0;2,2,2,2,namely,

Gτ{z−→ ±znmτ}. 3.5

A

C

1 0

−1

Figure 2

i Γ PSL2,Z, the modular group acting on the upper half planeH; ii Γ2 {gτ aτb/cτdinΓ:a, d≡1 mod 2, b, c≡0 mod 2}.

The groupΓ2is a normal subgroup ofΓof index 6 given by the kernel of the natural map fromΓ to PSL2,Z2 Σ3. A fundamental region forΓ2that we will use is given in

Figure 2.

In this figure the fundamental region is divided into twelve parts, each two adjacent parts being a fundamental region forΓ. The free generators forΓ2are

Aτ τ2, Cτ τ

−2τ1, 3.6 withBτ τ−2/2τ−3.Bfixes 1. We have the relation CBAId.

The functionλ is the universal covering map fromHto C − {0,1,∞} with a group of covering automorphismsΓ2, that is,λ∞ 0,λ0 1,λ1 ∞. In terms of elliptic functions,

λ e3−e1 e2−e1

, 3.7

wheree1℘1/2; 1, τ, e2 ℘1/2τ/2; 1, τ, e3℘τ/2; 1, τ. We also need to consider the following groups:

Γ0p

gτ aτb

cτd inΓ:c≡0 modp

,

Γ0_p

gτ aτb

cτd inΓ:b≡0 modp

.

3.8

LetXbe a fixed Riemann surface andf1:X→X1a 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. Teichm ¨uller spaceTXis the set of equivalence classesf1.

The setQCXof quasiconformal homeomorphisms ofXacts onTXvia

g∗f1 f1◦g. 3.9

IfQC0Xis the normal subgroup consisting of those maps homotopic to the identity relative to the ideal boundary, then the modular group isMX QCX/QC0X.

Proposition 3.1. The modular group of the four times punctured sphere is a semidirect product of

Γ PSL2,Zwith Klein’s group of order four.

The groupΓ2is isomorphic to the subgroup formed by the elements that give the identity on the punctures.

Proof. LetGbe the group of transformations generated byz→z1, z→zi,z→ −z. Gacts properly discontinously onC C−1/2Ziwith quotient surfaceX C/Gisomorphic to the Riemann sphere with the set{−1,0,1,∞}deleted. An explicit isomorphism is given by the restriction toCof the elliptic function

℘z−℘1i/2

℘1/2−℘1i/2. 3.10

An elementMin SL2, Zacts onCas a linear mapping:

Mx, y axby, cxdy 3.11

porducing an element of the modular group. Observe thatMand −M∗ provide the same action onX. The homeomorphism induced byMonXpermutes in general the three points

{−1,0,1}. Together with elements of Klein’s group of order four such asz→1i/2−z, they fully generate the modular group and induce the groupΣ4of permutations of{−1,0,1,∞}.

To prove that the elements ofΓ2fix the punctures, it is enough to check this for the generatorsA and Cgiven above. Now, Aacts as the linear map that sends the pair 1,0 and0,1to1,0and2,1, thus it sends 1/2 to itself,1i/2 to3i/2 ≡1i/2 and i/2 to2i/2 ≡ i/2. In the same manner,C1/2 1−2i/2 ≡ 1/2, Ci/2 i/2, and C1i/2 1−i/2 1i/2.

Finally,Γ/Γ2is isomorphic to the groupΣ3of permutations of{−1,0,1}so that if an element ofΓfixes them, it belongs toΓ2.

Theorem 3.2. The Hurwitz spaceHDp _{of irregularp-fold dihedral branched coverings of the sphere}

with four marked points is isomorphic toH/Γ2∩Γ0p.

Proof. GivenτinH, we consider the linear map

fτ

x y

1 r 0 s

x y

, whereτ rsi, s >0. 3.12

We define a left action of PSL2,Zonfτvia

Mfτ

fτ◦M−1

fτ∗, 3.13

which is given explicitely by

τ∗ aτ−b

−cτd ifM

a b c d

. 3.14

Observe thatM∈Γ2if and only ifλτ λτ∗. Now,Malso acts on the right of the epimorphisms

α:Z⊕Zi−→Zp 3.15

viaMα α◦M. In particular, forα0in3.3, we have

Mα0

1,0 c, Mα0

0, i a, 3.16

so that kerα0 kerMα0if and only ifc≡0 modp.

GivenτinH, we have ap-covering of the lattice1, τbyα0◦fτ−1, therefore a covering of C − {0,1,∞, λτ}. Two such coverings will be equivalent in the sense ofDefinition 2.1

if and only ifc ≡ 0 modp. Thep1 cosets of Γ2∩Γ0pinΓ2correspond to thep1 homomorphismsαjand to the monodromy representationsωofProposition 2.3:

kerMα0 αj ifa /0, j≡ca−1, kerMα0 αp ifa≡0.

3.17

An explicit set of coset representatives will be given next.

Lemma 3.3. Letϕ:Γ2→PSL2,Zpbe the natural homomorphism that sends a matrix to its class

modulop:

ϕA

1 2 0 1

, ϕC

1 0

−2 1

in PSL2,Zp. 3.18

LetP denote the subgroup of matricesa b

0a−1modulopof orderpp−1/2and indexp1.

Then

kerϕ Γ2∩Γp,

ϕ−1P Γ2∩Γ0p. 3.19

Proof. We have to establish thatϕis surjective. Since

ϕA−p−1/2

1 1 0 1

, ϕC−p−1/2

1 0

−1 1

3.20

Observe that we have the following inclusions:

Γ2∩ΓpΓ2indexp

p2−1 2

,

Γ2∩ΓpΓ2∩Γ0p<Γ2,

Γ2:Γ2∩Γ0pp1, Γ2∩Γ0pΓ0p, index 6.

3.21

Proposition 3.4. One has the right coset decomposition

Γ2 p−1

k0

Γ2∩Γ0pCk∪Γ2∩Γ0pD, 3.22

whereDCm_{Band 3−4m≡0 modp.}

Proof. We establish the decomposition

PSL2,Zp p−1

k0

P ϕCk_{∪P ϕD. 3.23}

Now

a b c d

1 0 2k 1

a2bk b c2dk d

. 3.24

Therefore, ifd /0 modp, we definekbyc2dk≡0 modpto obtain a matrix inP. If d≡0 modp, then

a b

−b−1 0

0 2 n 1

bn 2ab 0 −2b−1

∈P,

ϕD ϕCm_ϕB

1 −2 2−2m 4m−3

.

3.25

Now, taking 4m−3≡0 modp, ϕD 0 2 n1

−1

, as required.n2m−2.

Corollary 3.5. LetFbe a fundamental region forΓ2as inFigure 2. Thenp_k−1₀Ck_{F∪DFis a}

fundamental region forΓ2∩Γ0pinH.

When we compactify this region by filling in the punctures of order p, then F corresponds to the quadrilateral with angles2π/p, π/p,2π/p, π/p, a fundamental region for the Fuchsian groupΔinTheorem 2.4. The correspondence between generators is

A←→x2, C←→x3, B←→x1. 3.26

3.1. The casep3

The theory presented above is of course valid forp3 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

fx x

axb cxd

2

. 3.27

The reader may easily check that the function

fx x

2z−12x−3zz−2 32z−1x z−22

2

3.28

has double ramifications points at

3z z−2 2z−12,

z−2 2z−1

2

, −1

3

z−22 2z−1 ,

zz−2

2z−1 , 3.29

lying over the branching points 0,1,∞, λwith

λz3 z−2

1−2z 3.30 and simple points at 0,1,∞, λ∗with

λ∗z

z−2 1−2z

3

, 3.31

lying over the same branching points.

We recoverProposition 2.3since, for each value ofλ, there are four possible covers by 3.30. On the other hand, givenτinH, the 2 to 1 mapping fromT C/1, τtoC is given byy ℘u; 1, τand the mapping fromT∗ C/1,3τtoC is given by the corresponding functiony ℘u; 1,3τ. Given a pointu1inCmodulo1, τ, there are three preimages:u1, u2 1τ−u1andu3u1τmodulo1,3τ. Branching will happen whenyis branched but xis not. Thusyis a 3 to 1 rational function ofxwith simple points ate13τ, e23τ, e33τ,∞ lying overe1τ, e2τ, e3τ,∞. Normalizing these points we obtain the values 3.30and 3.31:

λλτ, λ∗λ3τ, 3.32

therefore,

z−2 2z−1

e2−e3 e2−e1

℘1/2τ/2−e1

℘1/2τ/2−e33τ. 3.33

We consider now a fundamental region for the groupΓ2∩Γ03of index 4 inΓ2in

1 2/3

1/2 1/3 0

−1/3 −1/2 −2/3 −1

Figure 3

If 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:

Aτ τ2 : 1←→10,

C3τ −τ

6τ−1 : 5←→6,

H1τ 5τ2

12τ5 : 4←→7, H2τ 7τ4

12τ7 : 3←→8,

H3τ 5τ4

6τ5 : 2←→9.

3.34

We observe that at the puncture at 0, λhas 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

H

Γ2∩Γ03−→ H

Γ2, 3.35

determined byλas a function ofzas in3.30. But the values ofλ3τyield also an interesting configuration. This function is automorphic with respect to the group

Γ∗aτb

cτd, ad−bc1, a, d∈Z, b∈2/3Z, c∈6Z

3.36

2/3

1/2

1/3−1/3

−1/2

−2/3 −1

∞

1

Figure 4

region forΓ2, as inFigure 2, pulls back to a fundamental region forΓ∗. We observe now that λ3τhas 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

H

Γ2∩Γ03−→ H

Γ∗ 3.37

determined byλ∗λ3τas a function ofzas in3.31. If we fill in the punctures ofFigure 3

we obtain the Euclidean crystallographic group0;3,3,3; as shown inFigure 4. We sumarize all this inTheorem 3.6.

Theorem 3.6. There is an isomorphism between the following spaces:

athe completionHD3of the Hurwitz space of irregular 3-fold dihedral coverings ofC;

bthe quotient spaceH/Γ2∩Γ03;

cthe quotient spaceC/GwhereGis the group generated bygu ρu,hu ρu 1−ρ, ρ −1i√3/2;

dthe curve with Belyi mapxz3z−2/1−2z; ethe algebraic modular curve

x4y4−256xy384x2yxy2−132x3yxy3−762x2y2

384x3_y2_x2_y3_−256x3_y3_0, 3.38

Proof. Onlyeremains to be proven. The algebraic equation is obtained by eliminating z 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

x1−2z3z2−z3, y1−2z z32−z 3.39

and lexicographic orderz > x > y.”

We obtain a map fromH/Γ2∩Γ03into this modular curve. Since the quotient is of genus 0, this is the modular curve and the map is onto. Given nowx, ywe may determine zup to an eight root of unity byz8 _x3_{/y. For genericzthese 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

LetΔbe the triangular Fuchsian group with signature0;p, p, pand canonical presentation

x1, x2, x3:x1px p 2 x

p

3 1;x1x2x31

. 4.1

We defineρ:Δ→PSL2, pby

ρx1

1 −2 2 −3

, ρx2 1 2 0 1

, ρx3

1 0

−2 1

. 4.2

Lemma 4.1. For each primep > 3 there are non-Euclidean crystallographic (NEC) groupsΛ1and

Λ2, such thatΛ₁ Δ Λ₂ with signatures

sΛ10;;−;p, p, p, sΛ20;;p;p. 4.3

There are two epimorphisms:

ρ1:Λ1→PGL2, p, ρ2:Λ2→PGL2, p 4.4 such thatρ1|_Δρ2|_Δρ.

Proof. By 15 see also 16, we obtain the existence of the groupsΛ1 and Λ2 such that Λ

1 Δ Λ2. Let

c0, c1, c2:c20c21 c221;

c0c1 p

c1c2 p

c2c0 p

1 4.5

be a canonical presentation for the NEC groupΛ1and let

x, c0, c1, e:c20c12xp1;xe

c0c1 p

1;c0ec1e−1

4.6

be a canonical presentation for the NEC groupΛ2. Then we defineρ1:Λ1→PGL2, pas

ρ1 c0 1 0 2 −1

, ρ1

c1

1 −2 0 −1

, ρ1 c2 1 0 0 −1

, 4.7

and we defineρ2 :Λ2→PGL2, pby

ρ2x

1 2 0 1

, ρ2 c0 0 1 1 0

. 4.8

Remark 4.2. For p 3, the two extensions of 0;3,3,3 or p3 are the classical plane Euclidean crystallographic groupsp3m1 andp31m.

An anticonformal involutionr ofC conjugated to the complex conjugationz→z is called a reflection of C.

Definition 4.3real form of a meromorphic function; see17. LetSbe a Riemann surface andf :S→C a meromorphic function. A real form forf :S→C is a reflectionrofCand an anticonformal involutionσofSsuch thatσis the lift ofrbyf.

Definition 4.4equivalence of real forms of a meromorphic function. Two real formsr1, σ1 and r2, σ2of a meromorphic funcionf : S→C are biconformally equivalent if there are automorfismsαofCandαofS, such that

α◦ff◦α, r1α◦r2◦α−1, σ1α◦σ2◦α−1_.

4.9

Proposition 4.5. The meromorphic functionHDp→H/Δ C_{admits two nonequivalent real forms.}

Proof. With the same notation as in Theorem 2.4andLemma 4.1. Letφ : PGL2, p→Σp1 be the natural representation given by the geometrical action of PGL2, p onP1_{Zp. Let} δiφ◦ρi,i1,2.

The orbifold coverings

HDp H

δ−1_Stab1 −→

H

δ−_i1Stab1, i1,2, 4.10

provide us the existence of two anticonformal involutionsσ1andσ2inHDp, defining the two required real forms.

The fact that the signatures ofΛ1andΛ2are 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

functionHDp→H/Δ_{in the above proposition is connected and nonseparating.}

Proof. 1The 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 of δ−1

i Stab1. We will use the technics in18,19.

Following18 see also19, we construct the Schreier graph given by the action of the canonical generators ofΛ1byρion the cosets ofφ−1Stab1≤PGL2, p. Each connected component of this graph corresponds to a period cycle insδ−1

i Stab1. For each reflection cjinΛiwe have thatρicjhas two fixed points inP1Zpand then the permutationδicjleft invariant two indices, so each reflection gives rise to two vertices of the graph:vj1, vj2. Since all periods inΛi are prime integers, then we connect the verticesvjk with vj1k by an edge and we have thatδ−1

i Stab1has one or two cycles. Finally, the hyperbolic generator with axis in the fixed point set of the reflection inδ_i−1Stab1is sent byδito an element of order two:

1 −1

2 −1 forδ1

or

0 −1

1 0 forδ2

So the verticesv31 andv02 forδ1andv11 andv02 forδ2are joined by an edge. Hence the graph is connected and there is only one period cycle insδ−1

i Stab1. Therefore, the set of real points of each real form is connected.

2The real points are nonseparating

For p ≡ 1 mod 4, there are square roots of−1 in Zp by Wilson’s theorem. For p ≡ 3 mod 4, there arep−1/2-roots of−1 inZpsinceZp∗is cyclic. Letqbe such a root of−1. Consider the following element of PGL2,Zp:

1 0 0 q

. 4.12

The above element in PGL2,Zp−PSL2,Zphas order 4 orp−1 and fixes1 : 0 ∈ P1_{Zp, then there are orientation reversing transformations inδ}−1

i Stab1of order at least 4. Hence there is a − sign in sδ_i−1Stab1 and the real parts of the two real forms are nonseparating.

Remark 4.7. With the notation in the previous theorem, the complete signatures of sδ−1

i Stab1, forp >3 are _p−3

2 ;−;−;p, p, p

forδ1,

_p−3

2 ;−;p;p

forδ2. 4.13

Remark that there is only one exceptional case,p 3 when the two real parts are connected but separatingin this casep−3/20, the signatures are

i 0;;−;3,3,3 the Euclidean crystallographic groupp3m1, ii 0;;3;3 the Euclidean crystallographic groupp31m.

Acknowledgment

This research was partially suported by Fondecyt 1050904.

References

1 A. Clebsch, “Zur Theorie der Riemann’schen Fl¨ache,” Mathematische Annalen, vol. 6, no. 2, pp. 216– 230, 1873.

2 D. Eisenbud, N. Elkies, J. Harris, and R. Speiser, “On the Hurwitz scheme and its monodromy,” Compositio Mathematica, vol. 77, no. 1, pp. 95–117, 1991.

3 A. Hurwitz, “Uber Riemann’sche Fl¨achen mit gegebenen Verzweigungspunkten,” Mathematische Annalen, vol. 39, no. 1, pp. 1–60, 1891.

4 W. Fulton, “Hurwitz schemes and irreducibility of moduli of algebraic curves,” Annals of Mathematics, vol. 90, no. 3, pp. 542–575, 1969.

5 I. I. Bouw, “Reduction of the Hurwitz space of metacyclic covers,” Duke Mathematical Journal, vol. 121, no. 1, pp. 75–111, 2004.

6 I. I. Bouw and S. Wewers, “Reduction of covers and Hurwitz spaces,” Journal f ¨ur die reine und angewandte Mathematik, vol. 574, pp. 1–49, 2004.

7 B. Dubrovin, “Geometry of 2D topological field theories,” in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), vol. 1620 of Lecture Notes in Mathematics, pp. 120–348, Springer, Berlin, Germany, 1996.

9 G. V. Belyi, “On Galois extensions of a maximal cyclotomic field,” Mathematics of the USSR-Izvestiya, vol. 14, no. 2, pp. 247–256, 1980.

10 B. Huppert, Endliche Gruppen. I, vol. 134 of Die Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1967.

11 G. Berger, “Fake congruence subgroups and the Hurwitz monodromy group,” Journal of Mathematical Sciences, vol. 6, no. 3, pp. 559–574, 1999.

12 E. Bujalance and D. Singerman, “The symmetry type of a Riemann surface,” Proceedings of the London Mathematical Society, vol. 51, no. 3, pp. 501–519, 1985.

13 H. Seifert and W. Threfall, A Textbook of Topology, Academic Press, New York, NY, USA, 1980. 14 C. J. Earle, Teichm ¨uller Spaces as Complex Manifolds, Lecture Notes, University of Warwick, Warwick,

UK, 1993.

15 E. Bujalance, “Normal N.E.C. signatures,” Illinois Journal of Mathematics, vol. 26, no. 3, pp. 519–530, 1982.

16 E. Bujalance, J. J. Etayo, J. M. Gamboa, and G. Gromadzki, Automorphism Groups of Compact Bordered Klein Surfaces. A Combinatorial Approach, vol. 1439 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1990.

17 S. M. Natanzon, Moduli of Riemann Surfaces, Real Algebraic Curves, and Their Superanalogs, vol. 225 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 2004. 18 A. H. M. Hoare, “Subgroups of N.E.C. groups and finite permutation groups,” The Quarterly Journal

of Mathematics, vol. 41, no. 161, pp. 45–59, 1990.