## 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.