Edray H Goins

AGEP PRiME

AGEP PRiME 2013

Background: High-Brow Version

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 \textit{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 \textit{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) $.

In the proposed projects below, 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 \textit{Dessin d'Enfant} of a Belyĭ map $ \beta: X \to \mathbb P^1(\mathbb C) $?

Background: Low-Brow Version

Prerequisites

Sample Problems

Given a loopless, connected, planar, bipartite graph $ \Gamma $, 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.