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 Bely?, 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 Bely? map.

Following Grothendieck, we associate a bipartite graph $\Delta_\beta$ to a Bely? 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 $\text{Gal} \bigl( \overline{\mathbb Q}/\mathbb Q \bigr)$.

There will be two main projects.

**Draw Dessins d'Enfants on Elliptic Curves:**We will focus on elliptic curves $X$ defined as the collection of $(x_1:x_2:x_0) \in \mathbb P^2(\mathbb C)$ such that $x_2^2 \, x_0 = 4 \, x_1^3 + b_2 \, x_1^2 \, x_0 + 2 \, b_4 \, x_1 \, x_0^2 + b_6 \, x_0^3$. Elliptic logarithms induce an embedding $X \hookrightarrow \mathbb R^3$ in the form \[ \begin{matrix} {X} & \simeq & {\mathbb C / \mathbb Z[\omega_1, \omega_2]} & \simeq & {\mathbb T^2(\mathbb R) = \left \{ (u,v,w) \in \mathbb R^3 \, \biggl| \, \bigl( \sqrt{u^2 + v^2} - R \bigr)^2 + w^2 = r^2 \right \}} \\[10pt] {P} & \mapsto & \log_X(P) = \theta \, \omega_1 + \phi \, \omega_2 & \mapsto & (u,v,w) = \biggl( \bigl(R + r \, \cos \theta \bigr) \, \cos \phi, \ \bigl(R + r \, \cos \theta \bigr) \, \sin \phi, \ r \, \sin \theta \biggr) \end{matrix} \] in terms of the integrals \[ \begin{aligned} \log_X(x:y:1) & = \dfrac {\sqrt{y^2}}{y} \cdot \int_x^{\infty} \dfrac {dz}{\sqrt{4 \, z^3 + b_2 \, z^2 + 2 \, b_4 \, z + b_6}}, \\[5pt] \omega_1 & = \dfrac {1}{\pi} \int_{e_1}^{e_3} \dfrac {dz}{\sqrt{4 \, z^3 + b_2 \, z^2 + 2 \, b_4 \, z + b_6}}, \\[5pt] \omega_2 & = \dfrac {1}{\pi} \int_{e_2}^{e_3} \dfrac {dz}{\sqrt{4 \, z^3 + b_2 \, z^2 + 2 \, b_4 \, z + b_6}}; \end{aligned} \] where $e_1$, $e_2$, and $e_3$ are roots of the 2-division polynomial $4 \, z^3 + b_2 \, z^2 + 2 \, b_4 \, z + b_6 = 0$. We seek to write code in Mathematica and Sage which will visualize Dessins d'Enfant $\Delta_\beta \subseteq X$ coming from Bely? maps $\beta: X \to \mathbb P^1(\mathbb C)$ on elliptic curves.**Find More Examples on the Torus:**We would like to draw cubic symmetric graphs on the torus $\mathbb T^2(\mathbb R)$. Examples would be the Tetrahedral graph $K_4$, the Utility graph $K_{3,3}$, the Petersen graph, and the Pappus graph. The idea would be to realize each of these graphs $\Gamma$ as Dessins d'Enfant $\Delta_\beta \subseteq X$ coming from Bely? maps $\beta: X \to \mathbb P^1(\mathbb C)$ on elliptic curves $X \simeq \mathbb T^2(\mathbb R)$. For example, it has been claimed that the Utility graphs $K_{3,3}$ may be realized as a Dessin d'Enfant coming from the Bely? map $\beta: (x_1 : x_2 : x_0) \mapsto (x_1 / x_0)^3$ on the elliptic curve $X: x_1^3 + x_2^3 + x_0^3 = 0$. We plan to visualize this Dessin d'Enfant in 3-dimensions.

A Bely? map $\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ is a rational function with at most three critical values; we may assume these values are $\{ 0, \, 1, \, \infty \}$. Replacing $\mathbb P^1$ with an elliptic curve $E: \ y^2 = x^3 + A \, x + B$, there is a similar definition of a Bely? map $\beta: E(\mathbb C) \to \mathbb P^1(\mathbb C)$.

The complete bipartite graph $K_{m,n}$ is a graph with $|V| = m + n$ vertices and $|E| = m \, n$ edges, where each of the $m$ ``black'' vertices is connected by an edge to each of the $n$ ``white'' vertices. In 1964, Ringel showed that such a graph $K_{m,n} \hookrightarrow X$ can be embedded in a compact connected Riemann surface $X$ of genus $g$ without edge crossings, where \begin{equation*} |V| - |E| + |F| = 2 - 2 \, g \qquad \text{in terms of} \qquad g = \left \lfloor \dfrac {(m-2) \, (n-2) + 3}{4} \right \rfloor. \end{equation*}

As a consequence, the only complete bipartite graphs which can be embedded on the sphere $X = \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$ without edge crossings are $K_{1,n}$ and $K_{2,n}$, and the only complete bipartite graphs which can be embedded on the torus $X = E(\mathbb C) \simeq \mathbb T^2(\mathbb R)$ without edge crossings are $K_{3,3}$, $K_{3,4}$, $K_{3,5}$, $K_{3,6}$, and $K_{4,4}$. In a 2009 paper by Coste et. al., the authors introduce a rational function $\beta: E(\mathbb C) \to \mathbb P^1(\mathbb C)$ which inequivalent to the one defined by $\beta(x,y) = 216 \, x^3 / (y + 36)^3$ for the elliptic curve $E: \ y^2 = x^3 - 432$. We verified that $\beta$ is a Bely? map having critical values $\bigl \{ (0:1), \, (1:1), \, (1:0) \bigr \}$ whose Dessin d'Enfant is $K_{3,3}$. Riemann's Existence Theorem asserts that each of the complete bipartite graphs $K_{3,3}$, $K_{3,4}$, $K_{3,5}$, $K_{3,6}$, and $K_{4,4}$ can be realized as a Dessin d'Enfant for some Bely? map $\beta: E(\mathbb C) \to \mathbb P^1(\mathbb C)$. We seek to find explicit Bely? maps $\beta$ for each of these complete bipartite graphs.

A Bely? map $\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ is a rational function with at most three critical values; we may assume these values are $\{ 0, \, 1, \, \infty \}$. A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: $\beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$. Replacing $\mathbb P^1$ with an elliptic curve $E$, there is a similar definition of a Bely? map $\beta: E(\mathbb C) \to \mathbb P^1(\mathbb C)$. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: $\beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R)$.

In this project, we use the open source Sage to write code which takes an elliptic curve $E$ and a Bely? map $\beta$ to return a Dessin d'Enfant on the torus -- both in two and three dimensions. Following a 2013 paper by Cremona and Thongjunthug we make the elliptic logarithm $E(\mathbb C) \simeq \mathbb C / \Lambda$ explicit using a modification of the arithmetic-geometric mean, then compose with a canonical one-to-one correspondences $\mathbb C / \Lambda \simeq \mathbb T^2(\mathbb R)$. Using this, we focus on several examples of Bely? maps which appear on Elkies' Harvard web page Elliptic Curves in Nature.