# Research Projects in Algebraic Geometry

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

## Background

Let $X$ be a compact, connected Riemann surface of genus $g_X$. It is well-known that $X$ is an algebraic variety, that is, $X \simeq \left \{ x \in \mathbb P^n(\mathbb C) \, \bigl| \, F_1(x) = F_2(x) = \cdots = F_m(x) = 0 \right \}$ in terms of a collection of homogeneous polynomials $F_i$ over $\mathbb C$ in $(n+1)$ variables. 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, a Riemann surface $Y \simeq \mathbb P^1(\mathbb C) = \mathbb C \cup \{ \infty \}$ if and only if $g_Y = 0$; then $\mathcal O_Y \simeq \mathbb C[z]$ consists of polynomials in one variable while $\mathcal K_Y \simeq \mathbb C(z)$ 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 $\gamma \mapsto \gamma \circ \beta$. The degree $N$ of such a map is the size of the group $\text{Gal} \bigl( \mathcal K_X / \beta^\ast \, \mathbb C(z) \bigr)$, that is, $N = | \mathcal K_X : \beta^\ast \, \mathbb C(z) |$.

For each $x \in X$, let $\mathcal O_{X, x}$ be the localization of $\mathcal O_X$ at the kernel of the evaluation map $\mathcal O_X \to \mathbb C$ defined by $f \mapsto f(x)$. We view $\mathcal O_{X,x}$ as a discrete valuation ring, so we have a normalized valuation $\text{ord}_x: \mathcal K_X^\times \twoheadrightarrow \mathbb Z$. For example, fix a rational map $\beta: X \to \mathbb P^1(\mathbb C)$. Denoting $\gamma(z) = z - \beta(x_0)$ for some $x_0 \in X$, we have $\big(\gamma \circ \beta)(x) = \beta(x) - \beta(x_0)$, so that the ramification index $e_\beta(x_0) = \text{ord}_{x_0}(\gamma \circ \beta)$ is a positive integer. We say that $z_0 \in \mathbb P^1(\mathbb C)$ is a critical value for $\beta$ if there exists $x_0 \in X$ such that $\beta(x_0) = z_0$ and $e_\beta(x_0) \geq 2$.

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 map $\beta: X \to \mathbb P^1(\mathbb C)$ with at most three critical values. Such a rational function is called a Belyĭ map; we may and do assume that these critical values are contained in the set $\{ 0, \, 1, \, \infty \}$. Say that $\beta: X \to \mathbb P^1(\mathbb C)$ is indeed a Belyĭ map of degree $N$. There are three related objects.

• Dessin d'Enfant. Following Grothendieck, we associate a bipartite graph $\Delta_\beta$ 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. For example, if $\beta: X \to \mathbb P^1(\mathbb C)$ is a Shabat polynomial, then its Dessin d'Enfant will be a tree embedded on the Riemann surface $X$.

• Degree Sequence. The Riemann-Hurwitz Genus formula asserts that the Belyĭ map has degree $N = \sum_{x \in B} e_\beta(x) = \sum_{x \in W} e_\beta(x) = \sum_{x \in F} e_\beta(x) = (2 \, g_X - 2) + |B| + |W| + |F|.$ The degree sequence is the multiset of multisets $\mathcal D_\beta = \bigl \{ \left \{ e_\beta(x) \, \bigl| \, x \in B \right \}, \ \left \{ e_\beta(x) \, \bigl| \, x \in W \right \}, \ \left \{ e_\beta(x) \, \bigl| \, x \in F \right \} \bigr \}$ as a collection of three partitions of $N$. Unfortunately a Belyĭ map is not uniquely determined by its degree sequence. Conversely, any graph $\Delta \subseteq X$ which can be embedded on $X$ without crossings corresponds to some degree sequence $\mathcal D$. The positive integer $e(x)$ is the number of edges adjacent to each vertex $x \in B \cup W$.

• Monodromy Group. A Belyĭ map is equivalent to an $N$-fold covering map $\beta: X^\circ \to Z$ from $X^\circ = \left \{ P \in X \, \bigl| \, \beta(P) \neq 0, \, 1, \, \infty \right \}$ to the punctured sphere $Z = \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \}$. For any base point $z_0 \in Z$, the fundamental group $\pi_1(Z, z_0) = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \, \bigl| \, \sigma_0 \ast \sigma_1 \ast \sigma_\infty \sim 1 \right \rangle$ is generated by the loops $\sigma_0$, $\sigma_1$, and $\sigma_\infty$ around $0$, $1$, and $\infty$, respectively. For each $x_i$ in the preimage $\beta^{-1}(z_0) = \bigl \{ x_1, \, x_2, \, \dots, \, x_N \bigr \}$, any closed loop $\sigma: [0,1] \to Z$ satisfying $\sigma(0) = \sigma(1) = z_0$ lifts back to a unique path $\widetilde{\sigma}: [0,1] \to X^\circ$ satisfying $\beta \circ \widetilde{\sigma} = \sigma$ and $\widetilde{\sigma}(0) = x_i$, so there exists a permutation $\tau \in S_N$ such that $\widetilde{\sigma}(1) = x_{\tau(i)}$. The image of the group homomorphism $\pi_1(Z, z_0) \to S_N$ which sends $\sigma \mapsto \tau$ is called the monodromy group; it is a subgroup of $S_N$. A Belyĭ map is uniquely determined up to Galois conjugacy by its monodromy group.

### Project #1: Visualizing Monodromy Groups

Let's make the monodromy group computation more explicit. We can consider a loops $\sigma_0, \, \sigma_1: \ [0,1]: \mathbb P^1(\mathbb C)$ with $\sigma_0(0) = \sigma_1(1) = z_0$ and centered around 0 and 1, respectively, rather explicitly as the paths $\sigma_0(t) = z_0 \, e^{2 \pi \sqrt{-1} t}$ and $\sigma_1(t) = 1 + (z_0 - 1) \, e^{2 \pi \sqrt{-1} t}$. If we have a Belyĭ map $\beta(z) = p(z) / q(z)$ as a rational function of degree $N = \deg(\beta) = \max \{ \deg(p), \, \deg(q) \}$, then paths $\widetilde{\sigma}_0, \, \widetilde{\sigma}_1: \ [0,1] \to \mathbb P^1(\mathbb C)$ satisfying $\beta \circ \widetilde{\sigma}_0 = \sigma_0$ and $\beta \circ \widetilde{\sigma}_1 = \sigma_1$ correspond to unique solutions to the differential equations $\dfrac {d \widetilde{\sigma}_0}{dt} = 2 \pi \sqrt{-1} \, \dfrac {p(\widetilde{\sigma}_0) \, q(\widetilde{\sigma}_0)}{p'(\widetilde{\sigma}_0) \, q(\widetilde{\sigma}_0) - p(\widetilde{\sigma}_0) \, q'(\widetilde{\sigma}_0)} \qquad \text{and} \qquad \dfrac {d \widetilde{\sigma}_1}{dt} = 2 \pi \sqrt{-1} \, \dfrac {\bigl[ p(\widetilde{\sigma}_1) - q(\widetilde{\sigma}_1) \bigr] \, q(\widetilde{\sigma}_0)}{p'(\widetilde{\sigma}_1) \, q(\widetilde{\sigma}_1) - p(\widetilde{\sigma}_1) \, q'(\widetilde{\sigma}_1)}$ subject to the initial conditions $\widetilde{\sigma}_0(0) = \widetilde{\sigma}_1(0) = x_k$ for any point $x_k$ in the inverse image $\beta^{-1}(z_0) = \bigl \{ x_1, \, x_2, \, \dots, \, x_N \bigr \}$. (We can choose $z_0$ small enough so that each $x_k \in \mathbb A^1(\mathbb C)$ is a point in the plane.) This gives a collection of $N$ paths on the Riemann sphere $X = \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$. An animation of this can be found here:

Animation featuring Monodromy Action (120 MB)

The goal for the summer is to generalize this to other Riemann surfaces, in particular for the torus. 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 movie on the torus -- both in two and three dimensions. Following a 2013 paper by Cremona and Thongjunthug we make the elliptic logarithm $X = E(\mathbb C) \simeq \mathbb C / \Lambda$ 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 Elliptic Curves in Nature.

### Project #2: Computing Monodromy Groups

Say that $X = E(\mathbb C) \simeq \mathbb C / \Lambda \simeq \mathbb T^2(\mathbb R)$ for some elliptic curve $E$. A Belyĭ map $\beta: X \to \mathbb P^1(\mathbb C)$ of degree $N$ yields an extension $\mathcal K_X / \beta^\ast \, \mathbb C(z)$ of degree $N$. In general, this extension is not Galois, but there is a subfield $\mathcal K_\beta \subseteq \beta^\ast \, \mathbb C(z) \subseteq \mathcal K_X$ such that $\mathcal K_X / \mathcal K_\beta$ is indeed Galois. The monodromy group $\text{Mon}(\beta) = \text{Gal} \bigl( \mathcal K_X / \mathcal K_\beta \bigr)$ may be viewed as its Galois group, it is a transitive subgroup of $S_N$. We may compose with an isogeny $\phi: E' \to E$ to yield another Belyĭ map $\beta \circ \phi: E'(\mathbb C) \to X \to \mathbb P^1(\mathbb C)$, and hence another monodromy group $\text{Mon}(\beta \circ \phi) = \text{Gal} \bigl( \mathcal K_X / \mathcal K_{\beta \circ \phi} \bigr)$.

In this project, we seek to understand the relationship between $\text{Mon}(\beta)$ and $\text{Mon}(\beta \circ \phi)$. We will write code in Sage and Mathematica which will allow us to compute various examples, then formulate a conjecture. We will build on the past two summers (PRiME 2015 and PRiME 2016) who formulated conjectures based on a limited number of examples.