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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 609425, 18 pages
http://dx.doi.org/10.1155/2008/609425
Research Article

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

1Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional de Educación a Distancia (UNED), Senda del rey, 9, 28040 Madrid, Spain
2Matematiska Institutionen, Linköpings Universitet, 581 83 Linköping, Sweden
3Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile

Received 9 June 2008; Accepted 13 October 2008

Academic Editor: Heinrich Begehr

Copyright © 2008 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.

Abstract

Hurwitz spaces are spaces of pairs where is a Riemann surface and a meromorphic function. In this work, we study -dimensional Hurwitz spaces of meromorphic -fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of transpositions and the monodromy group is the dihedral group . We prove that the completion of the Hurwitz space is uniformized by a non-nomal index subgroup of a triangular group with signature . We also establish the relation of the meromorphic covers with elliptic functions and show that is a quotient of the upper half plane by the modular group . Finally, we study the real forms of the Belyi projection and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.

1. Introduction

Hurwitz spaces are spaces of pairs where is a Riemann surface and 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 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.

Let be a Riemann surface and a meromorphic function branched on the set of points . Let be a prime integer, we define an irregular -fold dihedral covering as a meromorphic function having a monodromy:the monodromy group given by, is a product of transpositions, where is free homotopic to the boundary of a disc neighborhood of in .

Let be the Hurwitz space of irregular -fold dihedral branched coverings and let be the covering defined by . Then and can be extended, in the Deligne-Munford compactification, to a branched covering which is a Belyi function.

In Section 2, we present the uniformization of by a non-normal, index , subgroup of an hyperbolic (Euclidean for triangular group. Let be the triangular Fuchsian (Euclidean, for group acting on the hyperbolic plane with signature and canonical presentation:

We define by

If is the natural map given by the geometrical action of on and , then is isomorphic to and the orbifold covering is conformally equivalent to the covering . Therefore, the surface is a Riemann surface of genus which is the quotient of the underlying surface of a regular hypermap of type with automorphism group by the action of the stabilizer of infinity in (see [10]).

In Section 3, we establish the relation between irregular -fold dihedral coverings and elliptic curves. We show that the space is isomorphic to the quotient of the hyperbolic plane by the modular group . 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 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 . A real form for a meromorphic function is a reflection of and an anticonformal involution of such that is the lift by of . Two real forms and of a meromorphic function are conformally equivalent if there is an automorfism of and a lift of by to an automorphism of , such that and . We establish that the meromorphic function admits two nonequivalent real forms: and . The set of real points for the anticonformal involutions and is connected and nonseparating. Hence and are topologically conjugate (see [12]).

2. Hurwitz Spaces of Irregular Dihedral Coverings

Hurwitz spaces are spaces of pairs where is a Riemann surface and is a meromorphic function. We will consider the case when has four branching points . Definition 2.1. Two meromorphic functions and are considered equivalent if there is an automorphism satisfying

Let and be two pairs of Riemann surfaces and and meromorphic functions and with four branching points. We say that and are of the same topological type if there are homeomorphisms and such that and , and .

Let be a class of topologically equivalent meromorphic functions; denotes the set of topological classes of pairs with of topological type .

Given , the representative of a point in , we denote the branching set of by . Following [13], the pair is given by and the monodromy representation of the covering :

The group is called the monodromy group of the -fold covering .

Fixing , the variation of the point gives an -dimensional complex structure on the set of pairs .

Let be the branching set of and . Let be a disc in centered in and such that . A meridian of in based in is a path starting and finishing at and free homotopically equivalent to , where is positively oriented. We will denote the homotopy class in represented by a meridian of . Then we have the following presentation of :Definition 2.2. Define an irregular -fold dihedral covering as a covering having a monodromy , such that the monodromy group is a product of transpositions.

We will denote the Hurwitz space of irregular -fold dihedral branched coverings whose branching set consists exactly of and a variable point by .

There is a covering , defined by . Then and can be extended to a branched covering that is a Belyi function. We will determine and

First of all we need to know the degree of . The degree of is the number of different meromorphic functions of degree that are dihedral irregular coverings branched on four fixed points. In other words, we look for the number of irregular dihedral -fold coverings with monodromy representation as in Definition 2.2. Proposition 2.3. There are classes of monodromies of irregular -fold dihedral coverings.

Proof. A monodromy is given by (up to conjugacy in ). Let
By conjugation in , we can assume that .
Now, either or .
If , by an automorphism of , we can assume that , and so .
If , again by an automorphism of the group , we can assume that . Now each value of gives a class of monodromies. Then we have
Thus we have classes of monodromy representations.

We have found that the degree of and is .

We can establish a bijection between monodromy classes and points of . This bijection will be very useful in determining the monodromy representation of :

Since the degree of is , the monodromy associated to the covering is determined as follows.

The meridian in is represented by a closed path around , with base point , together with the marking . If we start with a monodromy for , then at the monodromy transforms in a new monodromy . The monodromy is precisely , where is the isomorphism of induced by the braid acting on . We say that the effect of on the monodromies is given by the braid .

In the same way, the effect on the monodromies of the meridian is given by the braid and the effect of the meridian by .

The value of (resp., , ) is given by the transformation of the monodromies when moves along (resp., , ). Since acts on the meridians by we obtain that the monodromy is defined by the following action on the monodromies of the meromorphic functions: and .

The bijection between monodromies and points of yields us

Hence, the monodromy group of is , (see [10]). The function is a -fold covering with three branching points: (a Belyi function). The preimage of each branching point contains a ramification point of local degree and a pseudoramification point of local degree one. In terms of the monodromy : .

Summarizing, we can describe as follows in Theorem 2.4.

Theorem 2.4. Let be a prime integer, . Let be a triangular Fuchsian group with signature and canonical presentation
Define by
If is the natural map given by the geometrical action of on and , then
(1) is uniformized by , that is, is isomorphic to ,(2)the orbifold covering is analytically equivalent to the covering .

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 and its subgroup for . In Section 3, we obtain a fundamental region for all . Remark 2.5. The signature of the Fuchsian group is . See Section 3.

609425.fig.001
Figure 1
Remark 2.6. For , there is a completely analogous description using the Euclidean crystallographic group (the group in crystallographic notation) instead of .Remark 2.7. Let us consider the regular covering , the Riemann surface is the underlying surface to a regular hypermap of type with automorphism group . Then is the quotient of by a subgroup of isomorphic to the semidirect product of with (the stabilizer of infinity [10]).Remark 2.8. The points in are of two types.
Points of where is a local homeomorphism: there are noded Riemann surfaces consisting in Riemann spheres joined by nodes.
Singular poins of corresponding to meromorphic functions of degree having three branching points and monodromy , such that the monodromy group is is a product of transpositions for two branching points and a -cycle for the remaining one.

3. The Hurwitz Spaces Uniformized by Modular Groups

We establish first the relation between the irregular -fold dihedral coverings of and elliptic curves. As before, let be a rational function of degree with branching points at given by a monodromy representation as in Definition 2.2. The Galois covering, given by the kernel of the monodromy, is a torus where acts by a translation of order and the elliptic involution. The quotient of by the translation group is again a torus and the natural projection gives the following conmutative diagram: eq3.1(3.1)

Both vertical arrows are to maps. The horizontal arrows are to maps. On the other hand we may start with a torus , an elliptic involution , and the 2 to 1 projection with branching points . We obtain different coverings as follows. Let , Im , be the group of translations on so that . Consider the group epimorphisms

The kernel of defines a subgroup of index and the quotient of by this subgroup defines the torus ; there are homomorphisms with different kernels given by

If denotes the classical Weierstrass elliptic function, the arrows in (3.1) are obtained by eq3.3(3.4)for and for . is an even elliptic function for . Thus is a rational function of giving us an explicit formula for . It may be worthwhile noticing that, for each , we get a discrete group acting on , depending analytically on and uniformizing an orbifold of genus and four conic points of order , that is, an Euclidean crystallographic group with signature namely, We obtain corresponding subgroups for . Each mapping may be visualized through appropriate fundamental regions for the group and its (non normal) subgroups .

The theory of the automorphic function is classical and well known; we recall here the necessaries to fix the notations:

(i), the modular group acting on the upper half plane ;(ii) in .

The group is a normal subgroup of of index given by the kernel of the natural map from to . A fundamental region for that we will use is given in Figure 2.

609425.fig.002
Figure 2

In this figure the fundamental region is divided into twelve parts, each two adjacent parts being a fundamental region for . The free generators for are with . fixes . We have the relation .

The function is the universal covering map from to with a group of covering automorphisms , that is, , , . In terms of elliptic functions, where .

We also need to consider the following groups:

In order to explain why (and not ) is our main group it is necessary to review some basic facts of Teichmüller theory of Riemann surfaces. See [14] for complete details.

Let be a fixed Riemann surface and a quasiconformal homeomorphism. Two such maps , are considered equivalent if there is a conformal isomorphism such that is homotopic to the identity relative to the ideal boundary. Teichmüller space is the set of equivalence classes .

The set of quasiconformal homeomorphisms of acts on via

If is the normal subgroup consisting of those maps homotopic to the identity relative to the ideal boundary, then the modular group is . Proposition 3.1. The modular group of the four times punctured sphere is a semidirect product of with Klein's group of order four.
The group is isomorphic to the subgroup formed by the elements that give the identity on the punctures.

Proof. Let be the group of transformations generated by , , . acts properly discontinously on with quotient surface isomorphic to the Riemann sphere with the set deleted. An explicit isomorphism is given by the restriction to of the elliptic function
An element in acts on as a linear mapping: porducing an element of the modular group. Observe that and provide the same action on . The homeomorphism induced by on permutes in general the three points . Together with elements of Klein's group of order four such as , they fully generate the modular group and induce the group of permutations of .
To prove that the elements of fix the punctures, it is enough to check this for the generators and given above. Now, acts as the linear map that sends the pair and to and , thus it sends to itself, to and to . In the same manner, , , and .
Finally, is isomorphic to the group of permutations of so that if an element of fixes them, it belongs to .

Theorem 3.2. The Hurwitz space of irregular -fold dihedral branched coverings of the sphere with four marked points is isomorphic to .

Proof. Given in , we consider the linear map It sends the lattice to the lattice in and gives a quasiconformal homeomorphism from to .
We define a left action of on via which is given explicitely by
Observe that if and only if .
Now, also acts on the right of the epimorphisms via . In particular, for in (3.3), we have so that if and only if .
Given in , we have a -covering of the lattice by , therefore a covering of . Two such coverings will be equivalent in the sense of Definition 2.1 if and only if . The cosets of in correspond to the homomorphisms and to the monodromy representations of Proposition 2.3:

An explicit set of coset representatives will be given next. Lemma 3.3. Let be the natural homomorphism that sends a matrix to its class modulo :
Let denote the subgroup of matrices modulo of order and index . Then

Proof. We have to establish that is surjective. Since it is enough to prove that these two matrices generate . Consider , in . Then has order three and fixes , whereas has order two and fixes . It is well known that generate . Since the natural homomorphism is surjective, so is . The definitions of and give and .

Observe that we have the following inclusions:

Proposition 3.4. One has the right coset decomposition where and .

Proof. We establish the decomposition
Now
Therefore, if , we define by to obtain a matrix in . If , then
Now, taking , , as required. ().

Corollary 3.5. Let be a fundamental region for as in Figure 2. Then is a fundamental region for in .

When we compactify this region by filling in the punctures of order , then corresponds to the quadrilateral with angles , a fundamental region for the Fuchsian group in Theorem 2.4. The correspondence between generators is

This explains Figure 1 for , where has been separated into two triangles for symmetry.

3.1. The Case

The theory presented above is of course valid for 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 of degree three with four branching points. If such a function has simple points at and double ramification points, then it must be of the form

The reader may easily check that the function has double ramifications points at lying over the branching points with and simple points at with lying over the same branching points.

We recover Proposition 2.3 since, for each value of , there are four possible covers by (3.30). On the other hand, given in , the 2 to 1 mapping from to is given by and the mapping from to is given by the corresponding function . Given a point in modulo , there are three preimages: , and modulo . Branching will happen when is branched but is not. Thus is a 3 to 1 rational function of with simple points at lying over . Normalizing these points we obtain the values (3.30) and (3.31): therefore,

We consider now a fundamental region for the group of index in in Figure 3.

609425.fig.003
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:

We observe that at the puncture at has a triple value and a simple value at the puncture at . It has a simple at and a triple at and a simple pole at and a triple pole at . This gives us a Belyi map determined by as a function of as in (3.30). But the values of yield also an interesting configuration. This function is automorphic with respect to the group with . Indeed, this group is conjugated to via the transformation and to . Multiplying by sends the fundamental region in Figure 3 to a region bounded by arcs at , , , , , , . The fundamental region for , as in Figure 2, pulls back to a fundamental region for . We observe now that has a simple value at the puncture at and a triple value at . Similar configurations are obtained at the other punctures; we have then a Belyi map determined by as a function of as in (3.31). If we fill in the punctures of Figure 3 we obtain the Euclidean crystallographic group ; as shown in Figure 4.

609425.fig.004
Figure 4

We sumarize all this in Theorem 3.6. Theorem 3.6. There is an isomorphism between the following spaces:
(a)the completion of the Hurwitz space of irregular -fold dihedral coverings of ;(b)the quotient space ;(c)the quotient space where is the group generated by , , ;(d)the curve with Belyi map ;(e)the algebraic modular curve where , .

Proof. Only (e) remains to be proven. The algebraic equation is obtained by eliminating 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 .”
We obtain a map from into this modular curve. Since the quotient is of genus , this is the modular curve and the map is onto. Given now we may determine up to an eight root of unity by . For generic these give eight different values of . But a generic one to one rational function from surfaces of genus is one to one, proving the equivalence.

4. Real Forms of The Belyi Map

Let be the triangular Fuchsian group with signature and canonical presentation

We define by Lemma 4.1. For each prime there are non-Euclidean crystallographic (NEC) groups and , such that with signatures There are two epimorphisms: such that .

Proof. By [15] (see also [16]), we obtain the existence of the groups and such that .
Let be a canonical presentation for the NEC group and let be a canonical presentation for the NEC group .
Then we define as and we define by
It is clear that .

Remark 4.2. For , the two extensions of (or are the classical plane Euclidean crystallographic groups and .

An anticonformal involution of conjugated to the complex conjugation is called a reflection of. Definition 4.3 (real form of a meromorphic function; see [17]). Let be a Riemann surface and a meromorphic function. A real form for is a reflection of and an anticonformal involution of such that is the lift of by .Definition 4.4 (equivalence of real forms of a meromorphic function). Two real forms and of a meromorphic funcion are biconformally equivalent if there are automorfisms of and of , such that Proposition 4.5. The meromorphic function admits two nonequivalent real forms.

Proof. With the same notation as in Theorem 2.4 and Lemma 4.1. Let be the natural representation given by the geometrical action of on . Let , .
The orbifold coverings provide us the existence of two anticonformal involutions and in , defining the two required real forms.
The fact that the signatures of and are different implies that the two defined real forms are not equivalent.

Proposition 4.6. Let . The set of real points for each of the two real forms of the meromorphic function 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 of . 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 by on the cosets of . Each connected component of this graph corresponds to a period cycle in . For each reflection in we have that has two fixed points in and then the permutation left invariant two indices, so each reflection gives rise to two vertices of the graph: . Since all periods in are prime integers, then we connect the vertices with by an edge and we have that has one or two cycles. Finally, the hyperbolic generator with axis in the fixed point set of the reflection in is sent by to an element of order two:
So the vertices and for and and for are joined by an edge. Hence the graph is connected and there is only one period cycle in . Therefore, the set of real points of each real form is connected.
(2) The real points are nonseparating
For , there are square roots of in by Wilson's theorem. For , there are -roots of in since is cyclic. Let be such a root of . Consider the following element of :
The above element in has order or and fixes , then there are orientation reversing transformations in of order at least . Hence there is a sign in 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 , for are
Remark that there is only one exceptional case, when the two real parts are connected but separating (in this case ), the signatures are
(i) (the Euclidean crystallographic group ),(ii) (the Euclidean crystallographic group ).

Acknowledgment

This research was partially suported by Fondecyt 1050904.

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