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

Recommended by Heinrich Begehr 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 H Dp 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 D p. We prove that the completion H Dp of the Hurwitz space H Dp is uniformized by a non-nomal index p1 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 H Dp is a quotient of the upper half plane by the modular group ΓΓ2 ∩ Γ 0 p. Finally, we study the real forms of the Belyi projection H Dp → C and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.


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 International Journal of Mathematics and Mathematical Sciences spaces in positive characteristic.There are many recent works studying Hurwitz spaces by Fried, V ölklein, 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.
Let S 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: In Section 2, we present the uniformization of π : H D p → 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: We define ρ : Δ → PSL 2, p by In Section 3, we establish the relation between irregular p-fold dihedral coverings and elliptic curves.We show that the space H D p is isomorphic to the quotient of the hyperbolic plane by the modular group Γ 2 ∩ Γ 0 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 case p 3 establishing the relation between modular groups, Belyi curves, modular equations, and euclidean crystallographic groups.
Finally, in Section 4 we study the real forms for the Belyi function π : H D p → 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 r 1 , r 1 and r 2 , 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 We establish that the meromorphic function H D p → C admits two nonequivalent real forms: r 1 , r 1 and r 2 , 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 .

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 functions f 1 and f 2 are considered equivalent if there is an automorphism g : Let S 1 , f 1 and S 2 , f 2 be two pairs of Riemann surfaces S 1 and S 2 and meromorphic functions f 1 : S 1 → C and f 2 : S 2 → C with four branching points.We say that S 1 , f 1 and S 2 , f 2 are of the same topological type if there are homeomorphisms ϕ : S 1 → S 2 and ψ : C → C such that f 2 • ϕ ψ • f 1 and ψ 0 0, ψ 1 1 and ψ ∞ ∞.Let t 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 The group ω π 1 C − B f , O is called the monodromy group of the n-fold covering f.
Fixing ω, the variation of the point λ gives an 1-dimensional complex structure on the set of pairs S, f .We will denote the Hurwitz space of irregular p-fold dihedral branched coverings f : S → C whose branching set consists exactly of 0, 1, ∞ and a variable point λ ∈ C − {0, 1, ∞} by There is a covering π : Then H D p and π can be extended to a branched covering π : Belyi function.We will determine π and First of all we need to know the degree of π.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.

2.3
By conjugation in Σ p , we can assume that ω m 0 s.Now, either ω m 0 ω m 1 s or s ω m 0 / ω m 1 .If ω m 0 ω m 1 s, by an automorphism of D p , we can assume that ω m ∞ sr, and so ω m λ sr.If s ω m 0 / ω m 1 , again by an automorphism of the group D p , we can assume that ω m 1 sr.Now each value of ω m ∞ gives a class of monodromies.Then we have ω m 0 , ω m 1 , ω m ∞ s, sr, sr i , i 0, . . ., p.

2.4
Thus we have p 1 classes of monodromy representations.
We have found that the degree of π and π is p 1.
We can establish a bijection between monodromy classes and points of P 1 Z p .This bijection will be very useful in determining the monodromy representation of π:

2.5
Since the degree of π is p 1, the monodromy associated to the covering π is determined as follows.

2.9
Hence, the monodromy group of π : 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 * s 1 , . . ., s p s p 1 .
Summarizing, we can describe π : H D p → C as follows in Theorem 2.4.
Theorem 2.4.Let p be a prime integer, p > 3. Let Δ be a triangular Fuchsian group with signature 0; p, p, p and canonical presentation If φ : PSL 2, p → Σ p 1 is the natural map given by the geometrical action of PSL 2, p on P 1 Z p and δ φ • ρ, then

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A similar result it is obtained in 11 for some different types of Hurwitz spaces.In Figure 1 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 2.6.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 .
is a product of p−1 /2 transpositions for two branching points and a p-cycle for the remaining one.

The Hurwitz spaces H D p uniformized by modular groups
We establish first the relation between the irregular p-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 D p 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: Both vertical arrows are 2 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 The kernel of α 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 α j 1, 0 j, 0 ≤ j ≤ p − 1, If ℘ denotes the classical Weierstrass elliptic function, the arrows in 3.1 are obtained by for i ≤ j ≤ p − 1 and ℘ z; p, τ for i p. ℘ z; 1, τ is an even elliptic function for ker α j .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, G τ {z −→ ±z n mτ}.

3.5
We obtain corresponding subgroups for ker α j .Each mapping f may be visualized through appropriate fundamental regions for the group G τ and its non normal subgroups G ker α j .The theory of the automorphic function λ τ is classical and well known; we recall here the necessaries to fix the notations: The group Γ 2 is a normal subgroup of Γ of index 6 given by the kernel of the natural map from Γ to PSL 2, Z 2 Σ 3 .A fundamental region for Γ 2 that we will use is given in Figure 2.

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In this figure the fundamental region is divided into twelve parts, each two adjacent parts being a fundamental region for Γ.The free generators for Γ 2 are where e 1 ℘ 1/2; 1, τ , e 2 ℘ 1/2 τ/2; 1, τ , e 3 ℘ τ/2; 1, τ .We also need to consider the following groups:

3.8
In order to explain why Γ 2 and not Γ is our main group it is necessary to review some basic facts of Teichm üller theory of Riemann surfaces.See 14 for complete details.
Let X be a fixed Riemann surface and f 1 : X → X 1 a quasiconformal homeomorphism.Two such maps f 1 , f 2 are considered equivalent if there is a conformal isomorphism g : is homotopic to the identity relative to the ideal boundary.Teichm üller space T X is the set of equivalence classes f 1 .
The set QC X of quasiconformal homeomorphisms of X acts on T X via If QC 0 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 /QC 0 X .
Proposition 3.1.The modular group of the four times punctured sphere is a semidirect product of Γ PSL 2, Z with Klein's group of order four.The group Γ 2 is isomorphic to the subgroup formed by the elements that give the identity on the punctures.
Proof.Let G 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
An element M in SL 2, Z acts on C as a linear mapping: M x, y ax by, cx dy 3.11 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 Σ 4 of permutations of {−1, 0, 1, ∞}.
To prove that the elements of Γ 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 Finally, Γ/Γ 2 is isomorphic to the group Σ 3 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 space H D p of irregular p-fold dihedral branched coverings of the sphere with four marked points is isomorphic to
Proof.Given τ in H, we consider the linear map , where τ r si, s > 0.

3.12
It sends the lattice 1, i to the lattice 1, τ in C and gives a quasiconformal homeomorphism from

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We define a left action of PSL 2, Z on f τ via which is given explicitely by Observe that M ∈ Γ 2 if and only if λ τ λ τ * .Now, M also acts on the right of the epimorphisms In particular, for α 0 in 3.3 , we have so that ker α 0 ker M α 0 if and only if c ≡ 0 mod p.
Given τ in H, we have a p-covering of the lattice 1, τ by α 0 • f −1 τ , 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 ≡ 0 mod p.The p 1 cosets of Γ 2 ∩ Γ 0 p in Γ 2 correspond to the p 1 homomorphisms α j and to the monodromy representations ω of Proposition 2.3: ker M α 0 α p if a ≡ 0.

3.17
An explicit set of coset representatives will be given next.
Lemma 3.3.Let ϕ : Γ 2 → PSL 2, Z p be the natural homomorphism that sends a matrix to its class modulo p:

3.18
Let P denote the subgroup of matrices a b 0 a −1 modulo p of order p p − 1 /2 and index p 1. Then Proof.We have to establish that ϕ is surjective.Since it is enough to prove that these two matrices generate PSL 2, Z p .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 i √ 3 /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, Z p is surjective, so is ϕ.The definitions of Γ p and Γ 0 p give ker ϕ and ϕ −1 P .
Observe that we have the following inclusions:

3.21
Proposition 3.4.One has the right coset decomposition where D C m B and 3 − 4 m ≡ 0 mod p.
Proof.We establish the decomposition

3.24
Therefore, if d / 0 mod p, we define k by c 2dk ≡ 0 mod p to obtain a matrix in P .If

3.25
Now, taking 4m − 3 ≡ 0 mod p, ϕ D Corollary 3.5.Let F be a fundamental region for Γ 2 as in Figure 2. Then When we compactify this region by filling in the punctures of order p, 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
This explains Figure 1 for p 5, where D F has been separated into two triangles for symmetry.

The case p 3
The theory presented above is of course valid for p 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 functions f : 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
The reader may easily check that the function lying over the branching points 0, 1, ∞, λ with and simple points at 0, 1, ∞, λ * with

3.31
lying over the same branching points.We recover Proposition 2.3 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 u 1 in C modulo 1, τ , there are three preimages: u 1 , u 2 1 τ −u 1 and u 3 u 1 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 e 1 3τ , e 2 3τ , e 3 3τ , ∞ lying over e 1 τ , e 2 τ , e 3 τ , ∞. Normalizing these points we obtain the values 3.30 and 3.31 :

3.33
We consider now a fundamental region for the group Γ 2 ∩ Γ 0 3 of index 4 in Γ 2 in 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: A τ τ 2 : 1 ←→ 10,

3.34
We observe that at the puncture at 0, 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 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 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. We sumarize all this in Theorem 3.6.

Theorem 3.6.
There is an isomorphism between the following spaces: Proof.Only e remains 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 and lexicographic order z > x > y." We obtain a map from H/Γ 2 ∩ Γ 0 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 z 8 x 3 /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.

Real forms of the Belyi map H D p → C
Let Δ be the triangular Fuchsian group with signature 0; p, p, p and canonical presentation We define ρ : Δ → PSL 2, p by

4.3
There are two epimorphisms: Proof.By 15 see also 16 , we obtain the existence of the groups Λ 1 and Λ 2 such that Then we define ρ 1 : Λ 1 → PGL 2, p as and we define ρ 2 : Λ 2 → PGL 2, p by

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Remark 4.2.For p 3, the two extensions of 0; 3, 3, 3 or p3 are the classical plane Euclidean crystallographic groups p3m1 and p31m.
An anticonformal involution r of C conjugated to the complex conjugation z → z is called a reflection of C.
Proof.With the same notation as in Theorem 2.4 and Lemma 4.1.Let φ : PGL 2, p → Σ p 1 be the natural representation given by the geometrical action of PGL 2, p on P The orbifold coverings provide us the existence of two anticonformal involutions σ 1 and σ 2 in H D p , defining the two required real forms.The fact that the signatures of Λ 1 and Λ 2 are different implies that the two defined real forms are not equivalent.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 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 Λ 1 by ρ i on the cosets of φ −1 Stab 1 ≤ PGL 2, p .Each connected component of this graph corresponds to a period cycle in s δ −1  i Stab 1 .For each reflection c j in Λ i we have that ρ i c j has two fixed points in P 1 Z p and then the permutation δ i c j left invariant two indices, so each reflection gives rise to two vertices of the graph: v j1 , v j2 .Since all periods in Λ i are prime integers, then we connect the vertices v jk with v j 1k by an edge and we have that δ −1  i Stab 1 has one or two cycles.Finally, the hyperbolic generator with axis in the fixed point set of the reflection in δ −1 i Stab 1 is sent by δ i to an element of order two: 1 −1 2 −1 for δ 1 or 0 −1 1 0 for δ 2 .

4.11
So the vertices v 31 and v 02 for δ 1 and v 11 and v 02 for δ 2 are joined by an edge.Hence the graph is connected and there is only one period cycle in s δ −1  i Stab 1 .Therefore, the set of real points of each real form is connected.
2 The real points are nonseparating For p ≡ 1 mod 4, there are square roots of −1 in Z p by Wilson's theorem.For p ≡ 3 mod 4, there are p − 1 /2 -roots of −1 in Z p since Z p * is cyclic.Let q be such a root of −1.
Consider the following element of PGL 2, Z p : 1 0 0 q .4.12 The above element in PGL 2, Z p − PSL 2, Z p has order 4 or p − 1 and fixes 1 : 0 ∈ P 1 Z p , then there are orientation reversing transformations in δ −1 i Stab 1 of order at least 4. Hence there is a − sign in s δ −1  i Stab 1 and the real parts of the two real forms are nonseparating.

4.13
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 .

such that 1. 1 the monodromy group ω π 1 C
− B f , O D p , given by b ∈ B f , ω m b is a product of p − 1 /2 transpositions, where m b is free homotopic to the boundary of a disc neighborhood of b in C − B f − {b} .Let H D p be the Hurwitz space of irregular p-fold dihedral branched coverings and let π : H D p → C − {0, 1, ∞} be the covering defined by f : C → C → λ ∈ B f − {0, 1, ∞}.Then H D p and π can be extended, in the Deligne-Munford compactification, to a branched covering π : H D p → C which is a Belyi function.

2 International 2 . 2 .
Let B f be the branching set of f and b ∈ B f .Let D b be a disc in C centered in b and such that D b ∩ 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 ∂D b , where ∂D b is positively oriented.We will denote m b the homotopy class in π 1 C − B f , O represented by a meridian of b.Then we have the following presentation of π 1 C − B f , O : m b , b ∈ B f : b∈B f m b 1 .2.Journal of Mathematics and Mathematical Sciences Definition Define an irregular p-fold dihedral covering as a covering having a monodromy ω : π 1 C − B f , O → Σ p , such that the monodromy group ω π 1 C − B f , O D p , ω m b is a product of p − 1 /2 transpositions.

Remark 2 . 7 .
Let us consider the regular covering R H/ ker δ → 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 H D p is the quotient of R by a subgroup of PSL 2, p isomorphic to the semidirect product of C p with C p−1 the stabilizer of infinity 10 .

Remark 2 . 8 .
The points in H D p − H D p are of two types.Points of H D p where π : H D p → C is a local homeomorphism: there are noded Riemann surfaces consisting in p 1 Riemann spheres joined by p nodes.Singular poins of π : H D p → C corresponding to meromorphic functions f : C → C of degree p having three branching points and monodromy ω :

Figure 2 i
Figure 2

1; c 0 c 1 p c 1 c 2 p 2 0 c 2 1 x p 1; xe c 0 c 1 p 1 ;
presentation for the NEC group Λ 1 and let x, c 0 , c 1 , e : c c 0 ec 1 e −1 4.6 be a canonical presentation for the NEC group Λ 2 .

Definition 4 . 3
real form of a meromorphic function; see 17 .Let S 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 r 1 , σ 1 and r 2 , σ 2 of a meromorphic funcion f : S → C are biconformally equivalent if there are automorfisms α of C and α of S, such that

Proposition 4 . 6 .
Let p > 3. The set of real points for each of the two real forms of the meromorphic function H D p → H/Δ in the above proposition is connected and nonseparating.Proof. 1 The set of real points is connected.