The Department of Mathematics at Purdue University offered an 8-week residential program to conduct research in pure mathematics from June 15, 2015 through August 7, 2015. The program was sponsored by the National Science Foundation (NSF) as well as generous gifts from Ruth and Joel Spira (BS '48, Physics) and Andris "Andy" Zoltners (MS '69, Mathematics).

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 \(\text{Gal} \bigl( \overline{\mathbb Q}/\mathbb Q \bigr)\).

There were 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 $$\begin{matrix} {X} & \simeq & {\mathbb C / \mathbb Z[\omega_1, \omega_2]} & \simeq & {\mathbb T^2(\mathbb R) = \left \{ (u,v,w) \in \mathbb R^3 \, \biggl| \, \bigl( \sqrt{u^2 + v^2} - R \bigr)^2 + w^2 = r^2 \right \}} \\[10pt] {P} & \mapsto & {\begin{aligned} & \log_X(P) \\[5pt] & \ = \theta \, \omega_1 + \phi \, \omega_2 \end{aligned}} & \mapsto & {\begin{aligned} (&u,v,w) \\[5pt] & = \biggl( \bigl(R + r \, \cos \theta \bigr) \, \cos \phi, \ \bigl(R + r \, \cos \theta \bigr) \, \sin \phi, \ r \, \sin \theta \biggr) \end{aligned}} \end{matrix}$$ in terms of the integrals $$\begin{aligned} \log_X(x:y:1) & = \dfrac {\sqrt{y^2}}{y} \cdot \int_x^{\infty} \dfrac {dz}{\sqrt{4 \, z^3 + b_2 \, z^2 + 2 \, b_4 \, z + b_6}}, \\[5pt] \omega_1 & = \dfrac {1}{\pi} \int_{e_1}^{e_3} \dfrac {dz}{\sqrt{4 \, z^3 + b_2 \, z^2 + 2 \, b_4 \, z + b_6}}, \\[5pt] \omega_2 & = \dfrac {1}{\pi} \int_{e_2}^{e_3} \dfrac {dz}{\sqrt{4 \, z^3 + b_2 \, z^2 + 2 \, b_4 \, z + b_6}}; \end{aligned}$$ here \(e_1\), \(e_2\), 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 \texttt{Mathematica} and \texttt{Sage} which will visualize Dessins d'Enfant \(\Delta_\beta \subseteq X\) coming from Belyĭ 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 Belyĭ maps \(\beta: X \to \mathbb P^1(\mathbb C)\) on elliptic curves \(X \simeq \mathbb T^2(\mathbb R)\). For example, a 2009 paper by Coste, Jones, Streit, and Wolfart claims that the Utility graphs \(K_{3,3}\) may be realized as a \textit{Dessin d'Enfant} coming from the Belyĭ 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 \textit{Dessin} in 3-dimensions.

- Sofia Sorokina Lyrintzis (Indiana University -- Purdue University at Ft. Wayne)
- Bronz McDaniels (Purdue University)
- Yesid Sanchez Arias (Purdue University)
- Danny Sweeney (Purdue University)
- Hongshan Li (Graduate Student in Mathematics, Purdue University)
- Dr. Edray Goins (Associate Professor of Mathematics, Purdue University)

The first aspect of this project seeks to determine infinite families of Belyĭ maps for elliptic curves. We have shown that given any elliptic curve \(E\) there exist infinitely many Belyĭ maps of degree 2; they are in the form \(\beta(x,y) = (a \, x + b) / (c \, x + d)\). We have also shown that any elliptic curve has at least one Belyĭ map of degree 3 with critical values \(\{ 0, \, 1, \, \infty \}\). After placing the curve in Hessian normal form \(y^2 + a_1 \, x \, y + a_3 \, y = x^3\), the Belyĭ map is in the form $$\beta(x,y) = \dfrac {\bigl( 2 \, a_1^3 - 27 \, a_3 + 2 \, \sqrt{a_1^6 - 27 \, a_1^3 \, a_3} \bigr) \, y - 27 \, a_3^2}{\bigl( 2 \, a_1^3 - 27 \, a_3 - 2 \, \sqrt{a_1^6 - 27 \, a_1^3 \, a_3} \bigr) \, y - 27 \, a_3^2} \qquad \text{for \(a_1 \neq 0\).}$$

The second aspect of this project seeks to determine of Belyĭ maps for elliptic curves corresponding to the complete bipartite graphs. A 2009 paper by Coste, Jones, Streit, and Wolfart claims that the Utility graph \(K_{3,3}\) may be realized as a Dessin d'Enfant coming from the Belyĭ map \(\beta: (x_1 : x_2 : x_0) \mapsto (x_1 / x_0)^3\) on the elliptic curve \(E: x_1^3 + x_2^3 + x_0^3 = 0\). It is known that the complete bipartite graphs which can be embedded on the torus (and not on the sphere) are \(K_{3,3}\), \(K_{3,4}\), \(K_{3,5}\), \(K_{3,6}\), and \(K_{4,4}\). Riemann's Existence Theorem asserts that each of these can be visualized as a Dessin d'Enfant. In this project, we seek to find explicit corresponding Belyĭ maps.

- Slides from the Final Presentation at Purdue University on August 6, 2015
- Video from the Final Presentation at Purdue University on August 6, 2015

- Leonardo Azopardo (Purdue University)
- Maxim Millan (Purdue University)
- Sarah Thomaz (Purdue University)
- Avi Steiner (Graduate Student in Mathematics, Purdue University)
- Dr. Edray Goins (Associate Professor of Mathematics, Purdue University)

In this project, we use the open source **Sage** to write code which takes an elliptic curve \(E\) and a Belyĭ 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 Belyĭ maps which appear on Elkies' Harvard web page entitled ``Elliptic Curves in Nature.''