Rational Exceptional Belyi Coverings

In Chapter 2, we start with some preliminaries about ramified coverings and then give the definition of Belyi coverings. We give examples of Shabat polynomials [Sha94]. We also state the famous Belyi’s Theorem saying that there exists a Belyi function on a compact surface that can be defined over a number field as an algebraic curve [Bel80]. Later, we introduce dessin d’enfants as some connected bipartite maps. They have a topological structure too as mentioned in [GáD11] and [SS03]. The theory of dessins was first introduced by Grothendieck [Gro03]. Dessins and Belyi coverings are in one-to-one correspondence [Zvo]. We also mention the action of the absolute Galois group on dessins as in [LZ04], [GáD11] annd [CG94]. We conclude this chapter by giving examples from [SV90], [BZ96] and [Zvo].

Next we deal with how to count these coverings in Chapter 3. We define the Eisenstein number of a covering [LS04]. Dessin of a polynomial covering is a bicolored plane tree. They were first studied by G. Shabat [Sha94]. Counting these trees is a combinatorial problem solved by Tutte [Tut64]. We then give a formula for counting coverings with given ramification schemes. This formula includes both Eisentein number and irreducible characters of symmetric groups [KK98]. Preliminaries from representation theory can be found in [Kür15].

In Chapter 4; we focus on Belyi coverings that are uniquely determined by either ramification schemes or by respective dessin d’enfants. We call them exceptional Belyi coverings. We characterize them with their ramification indices in fibers 0,1, and $\infty$. We give the well-known examples that are cyclic, dihedral and Chebyshev coverings. Cyclic coverings, dihedral coverings and the coverings of regular polyhedra ([CG94], [MZ00]) are known as Klein’s coverings. We found a number of rational exceptional coverings which include Klein’s coverings and Chebyshev polynomials. We also found new infinite series. We give a classification of exceptional polynomial coverings by Adrianov ([Adr09]). Then we state the formula to count the exceptional Belyi coverings. Lastly, we talk about the field of definition, which is the smallest field in which both the Riemann surface, or equivalently the corresponding algebraic curve, and the coefficients of the covering are defined. We state the field of definition of a rational exceptional Belyi covering with the help of theorems in [Ser73] and [Ser97]. We conclude with our Maple algorithm finding all coverings with a given genus and degree. A table of all rational exceptional Belyi coverings up to degree 6 with ramification schemes, Belyi functions and dessins d’enfants and some of the ones that are of degree 7 is given.

Now let $f:S\rightarrow\mathbb{P}^{1}$ be a covering of the Riemann sphere ramified at $k$ points $s_{1},\ldots,s_{k}$ with degree $n\,.$ The fundamental group is $\left\langle\gamma_{1},\gamma_{2},\ldots,\gamma_{k}\mid\gamma_{1}\gamma_{2}\ldots\gamma_{k}=id\right\rangle$ and the monodromy group, as a subgroup of $S_{n}$ is $\left\langle g_{1},g_{2},\ldots,g_{k}\right|g_{1}g_{2}\ldots g_{k}=$ $id\rangle\,.$ For each fiber $f^{-1}\left(s_{i}\right)$, let the cycle structure of $g_{i}$ be denoted by $\lambda_{i}\,.$ $\lambda_{i}$ ’s are the ramification indices and the expression $\left[\lambda_{1}\right]\left[\lambda_{2}\right]\ldots\left[\lambda_{k}\right]$ is called the ramification scheme of the covering.

A dessin is a connected bipartite map with a topological structure. When the underlying surface is clear, we simply express a dessin as $\mathcal{D}$. The genus of $(X,\mathcal{D})$ is simply the genus of the topological surface $X$.

Now suppose that the edges of the dessin are numbered from the set $\Omega=$ $\{1,2,3,\ldots\}$. Each edge joins a black vertex to a white vertex, and incident with every black vertex, we have some of these edges. Using the anticlockwise orientation of the surface, we get a cyclic permutation of these edges. Thus if we have $b$ black vertices, we have a permutation $\sigma_{0}$ that is a product of $b$ disjoint cycles. Similarly, if we have $w$ white vertices then we get a permutation $\sigma_{1}$ consisting of $w$ disjoint cycles. The permutation $\sigma_{2}:=\left(\sigma_{0}\sigma_{1}\right)^{-1}$ describes the edges going around a face, each cycle of length $u$ corresponds to a $2u$-gonal face.

Let $k\subset\overline{\mathbb{Q}}$ be a number field. Every automorphism $k$ may be extended to an automorphism of $\overline{\mathbb{Q}}$. Subgroups of $\Gamma$ of finite index are in one-to-one correspondence with finite extensions of $\mathbb{Q}$ inside $\overline{\mathbb{Q}}$.

We conclude this section with the following theorem describing another facet of the action of $\Gamma$ :

Tutte found the Eisenstein number of planar trees with $n$ edges and given degrees $d_{i}^{\bullet}$ and $d_{j}^{\circ}$ of black and white vertices. Clearly, $\sum_{i}d_{i}^{\bullet}=\sum_{j}d_{j}^{\circ}=n$, the number of edges of $T$, or what is the same, the degree of the respective covering. The degrees $d^{\bullet}=\left(d_{1}^{\bullet},d_{2}^{\bullet},\ldots\right)$ and $d^{\circ}=\left(d_{1}^{\circ},d_{2}^{\circ},\ldots\right)$ give two partitions of $n$. In practice it is more convenient to deal with partitions $q^{\bullet}=\left(q_{1}^{\bullet},q_{2}^{\bullet},\ldots\right)$ and $q^{\circ}=\left(q_{1}^{\circ},q_{2}^{\circ},\ldots\right)$ where $q_{i}^{\bullet}$ and $q_{i}^{\circ}$ is the number of black and white vertices of degree $i$. Observe that

We’ll often use the last two sums and introduce for them special notations

In this notations the (slightly modified) Tutte result may be stated as follows:

Topologically, coverings $\pi:X\rightarrow Y$ of degree $n$ unramified outside $k$ points $y_{i}\in$ $Y$ are classified by conjugacy classes of homomorphisms $\pi_{1}\rightarrow S_{n}$ of the fundamental group $\pi_{1}=\pi_{1}\left(Y\backslash\left\{y_{1},y_{2},\ldots,y_{k}\right\}\right)$, which is known to be defined by the unique relation

where $g$ is the genus of $Y$ and the brackets denote the commutator $[f,h]=fhf^{-1}h^{-1}$. Thus the coverings of Riemann sphere $\pi:X\rightarrow\mathbb{P}^{1}$ of given degree $n$ and ramification indices are parametrized by solutions of the equation

up to conjugacy, where cycle lengths of the conjugacy class $C_{i}\subset S_{n}$ are equal to ramification indices of points in fibers $\pi^{-1}\left(y_{i}\right)$.

The following theorem gives the number of solutions of the equation above for an arbitrary group $G$ in terms of irreducible characters:

Now if we turn to Belyi coverings $\beta$ with 3 respective conjugacy classes of monodromy permutations for ramification points 0,1 and $\infty$, the formula above will be as follows:

More specifically, this formula for polynomial coverings of degree $n$ with ramification scheme $n,d^{\bullet},d^{\circ}$ will be:

or equivalently,

where $z\left(d^{\bullet}\right)$ is the order of centralizer of a permutation with cycle structure $d^{\bullet}$.

There are some examples illustrating these exceptional coverings: Klein studied the first three examples by classifying $G\subset\mathbb{P}GL(2,\mathbb{C})$ as cyclic, dihedral, cubic etc. where $G=\operatorname{Aut}\left(\mathbb{P}^{1},\beta\right)$ in the natural projection $\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}/G$.

We used Maple to calculate the respective Belyi function along with the real graph and dessin when the degree is 9. The Belyi function for corresponding to the ramification scheme $[3,6][2,2,2,3][1,2,2,2,2]$ is

This function has a pole at $\frac{5}{3}$ $\Bigg{(}$In general, the pole is at $\frac{n^{2}+d^{2}}{n^{2}-d^{2}}\Bigg{)}\,.$

Dessin d’enfants corresponding to polynomial coverings are trees.

Recall the formulas from Chapter 3:

where $\chi\left(c_{i}\right)$ are the irreducible characters of permutations $c_{i}$ in the ramification scheme and $\left|C_{i}\right|$ are the size of conjugacy classes with representatives $c_{i}$. Also, we stated Tutte formula for bicolored trees(corresponding to polynomial coverings):

where $q_{i}^{\bullet},q_{i}^{\circ}$ is the number of black and white vertices of degree $i$ respectively and $\sigma^{\bullet}=\sum_{i}q_{i}^{\bullet},\sigma^{\circ}=\sum_{j}q_{j}^{\circ}$.

The sums on the left-hand side in two formulas above simply reduces to the inverse of an integer when the case is exceptional Belyi coverings: By definition, these Belyi coverings are unique, so the Eisenstein number will be $\frac{1}{|\operatorname{Aut}\beta|}$ and $\frac{1}{|\operatorname{Aut}T|}$.

We developed a Maple algorithm finding all exceptional Belyi coverings with a given genus and degree. This allows us to classify them up to degree 15. This code could be found on the next page. We were able to calculate the respective Belyi functions up to degree 6 completely in addition to some of degree 7. The ramification schemes, Belyi functions and dessins d’enfants can be seen in the table attached. More sample Maple codes based on “fundamental identities” to find Belyi functions can be found in [Kür15]. Some rational coverings come from modular curves; e.g. R6.11 in the table. This is the only one we found whose field of definition is different than $\mathbb{Q};$ it is $\mathbb{Q}(\sqrt{10})\,.$

See pages - of maple.pdf

See pages - of table.pdf