What is a “Dessin d’Enfant”?

For the past couple of years, I’ve been thinking about properties of Dessins d’Enfants. As my very first post, I’d like to give a rigorous definition. This is meant to be formal for the math folks out there who are comfortable with algebraic geometry.

Let  X be a compact, connected Riemann surface. It is well-known that  X is an algebraic variety, that is,  X \simeq \left \{ P \in \mathbb P^n(\mathbb C) \, \bigl| \, F_1(P) = F_2(P) = \cdots = F_m(P) = 0 \right \} in terms of a collection of homogeneous polynomials  F_i over  \mathbb C in  (n+1) variables  x_j . Denote  \mathcal O_X as the ring of regular functions on  X , that is, “polynomials”  f, g: X \to \mathbb P^1(\mathbb C) ; and denote  \mathcal K_X as its quotient field, that is, rational functions  f/g: X \to \mathbb P^1(\mathbb C) . For example, if  X \simeq \mathbb P^1(\mathbb C) = \mathbb C \cup \{ \infty \} , then  \mathcal O_X \simeq \mathbb C[z] consists of polynomials in one variable, while  \mathcal K_X \simeq \mathbb C(z) consists of rational functions in one variable. In particular, any rational map  \beta: X \to \mathbb P^1(\mathbb C) induces a map  \beta^\ast: \mathbb C(z) \to \mathcal K_X which sends  J \mapsto J \circ \beta . The degree of such a map is the size of the group  G = \text{Gal} \bigl( \mathcal K_X / \beta^\ast \, \mathbb C(z) \bigr) .

For each  P \in X , let  \mathcal O_P be the localization of  \mathcal O_X at the kernel of the evaluation map  \mathcal O_X \to \mathbb C defined by  f \mapsto f(P) . Let  \mathfrak m_P denote the maximal ideal of  \mathcal O_P ; we view this as the collection of rational maps  \beta \in \mathcal K_X which vanish at  P . As shown by Weil and Belyĭ, the Riemann surface  X can be defined in terms of homogeneous polynomials  F_i over an algebraic closure  \overline{\mathbb Q} if and only if there exists a rational function  \beta: X \to \mathbb P^1(\mathbb C) such that  \beta: \left \{ P \in X \, \bigl| \, \beta - \beta(P) \in {\mathfrak m_P}^2 \right \} \to \bigl \{ 0, \, 1, \, \infty \} . The difference  \beta - \beta(P) \in \mathfrak m_P for any  P \in X because the function vanishes at  P ; the condition  \beta - \beta(P) \in {\mathfrak m_P}^2 means the derivative of the function vanishes as well. A rational function as above where these critical values are at most  0 ,  1 , and  \infty is called a Belyĭ map.

Following Grothendieck, we associate a bipartite graph  \Delta_\beta to a Belyĭ map  \beta: X \to \mathbb P^1(\mathbb C) by denoting the “black” vertices as  B = \beta^{-1}(0) , “white” vertices as  W = \beta^{-1}(1) , midpoints of the faces as  F = \beta^{-1}(\infty) , and edges as  E = \beta^{-1}\bigl([0,1] \bigr) . This is a loopless, connected, bipartite graph, called a Dessin d’Enfant, which can be embedded on  X without crossings. The group  G = \text{Gal} \bigl( \mathcal K_X / \beta^\ast \, \mathbb C(z) \bigr) permutes the solutions  P to  \beta(P) = z , and hence acts on the dessin  \Delta_\beta . The hope is that in studying graphs  \Delta_\beta one can better understand quotients  G of the absolute Galois group  \text{Gal} \bigl( \overline{\mathbb Q}/\mathbb Q \bigr) .

We are motivated by the following question:

Given a loopless, connected, bipartite graph  \Gamma on a compact, connected Riemann surface  X , when is  \Gamma \simeq \Delta_\beta the Dessin d’Enfant of a Belyĭ map  \beta: X \to \mathbb P^1(\mathbb C) ?

Given such a loopless, connected, planar, bipartite graph  \Gamma , we wish to use properties of the symmetry group  G to construct a Belyĭ map  \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) such that  \Gamma arises as its Dessin d’Enfant.

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About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
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