PRiME 2016: Purdue Research in Mathematics Experience

Research Projects in Algebraic Geometry

Research projects in Algebraic Geometry will be directed by Edray Goins, Associate Professor of Mathematics at Purdue University.

Dessin Explorers

Background

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.

Project #1: Examples of Bely? Maps for Elliptic Curves

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.

Project #2: Visualizing Dessins d'Enfants on the Torus

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.