A theorem of Andre Weil and Gennadii Belyi asserts that, given a compact connected Riemann surface \( S \), we can choose the polynomial \(f(x,y) = \sum_{i,j} a_{ij} \, x^i \, y^j\) to have coefficients \(a_{ij} \in \mathbb C\) as algebraic numbers if and only if there exists a rational map \(\beta: S \to \mathbb P^1(\mathbb C)\) with at most critical values \(0, 1, \infty\). If such a map exists, we call it a Belyi Map.
For PRiME 2021, there will be three research projects.
For any positive integer \(N\), we have modular curves \(S = X_0(N)\), \(X_1(N)\), and \(X(N)\) which are known to be compact, connected Riemann surfaces. It is also known that we have Belyi maps \(\beta: S \to \mathbb P^1(\mathbb C)\) coming from sending an elliptic curve \(E: y^2 = x^3 + A \, x + B\) to its \(J\)-invariant \(J(E) = 4 \, A^3 / \bigl( 4 \, A^3 + 27 \, B^2 \bigr)\). This project will explicitly compute these Belyi maps when the modular curves have genus 0 (i.e., \(S \simeq S^2(\mathbb R)\) is the Riemann sphere) or genus 1 (i.e., \(S \simeq \mathbb T^2(\mathbb R)\) is the torus).
There are many examples of Belyi maps \(\beta: S \to \mathbb P^1(\mathbb C) \) associated to elliptic curves \( S = E(\mathbb C) \); several can be found online at LMFDB. Given such a Toroidal Belyi map of degree \(N\), we have the divisors
$$ \begin{aligned} \text{div}(\beta) & = \sum_{P \in B} e_P (P) - \sum_{P \in F} e_P(P) \\[5pt] \text{div}(\beta - 1) & = \sum_{P \in W} e_P (P) - \sum_{P \in F} e_P(P) \end{aligned} \qquad \text{where} \qquad \begin{aligned} B & = \beta^{-1} \bigl( \{ 0 \} \bigr) \\ W & = \beta^{-1} \bigl( \{ 1 \} \bigr) \\ F & = \beta^{-1} \bigl( \{ \infty \} \bigr). \end{aligned} $$
There exists a point \( P_0 \in E(\mathbb C) \) such that
$$[-N] P_0 = \bigoplus_{P \in B} [e_P] P = \bigoplus_{P \in W} [e_P] P = \bigoplus_{P \in F} [e_P] P. $$
Denote \( G = \left \{ P \oplus P_0 \ \bigl| \ P \in B \cup W \cup F \right \} \) as a translation of the critical points of the Belyi map. For example, the Toroidal Belyi map \( \beta: (x,y) \mapsto (y+1)/2 \) for the elliptic curve \( E: y^2 = x^3 + 1 \) has degree \( N = 3 \) and \( P_0 = O_E \), so that \( G = \bigl \{ (0,-1), \, (0,+1), \, O_E \bigr \} \simeq (\mathbb Z / 3 \mathbb Z) \). In this project, we will focus on this phenomenon, and will investigate when \( G \) is a subgroup of \( S = E(\mathbb C) \).
Say that \( \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)\) is a Dynamical Belyi map, that is, a Belyi map such that \( \beta \bigl( \{ 0, 1, \infty \} \bigr) \subseteq \{ 0, 1, \infty \} \); an example of such a map is \( \beta(z) = z^N \) for any positive integer \( N \). Given any Toroidal Belyi map \( \phi: E(\mathbb C) \to \mathbb P^1(\mathbb C) \), the composition \( \beta \circ \phi: E(\mathbb C) \to \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \) is also a Toroidal Belyi map. There is a group \( \text{Mon}(\beta) \), the monodromy group, which is contains information about the symmetries of a Belyi map \( \beta \). It is well-known that, for any Toroidal Belyi map \( \phi \), (1) there is always a surjective group homomorphism \( \text{Mon}(\beta \circ \phi) \twoheadrightarrow \text{Mon}(\beta) \), and (2) the monodromy group \( \text{Mon}(\beta \circ \phi) \) is contained in the wreath product \( \text{Mon}(\phi) \wr \text{Mon}(\beta) \). In this project, we study how the three groups \( \text{Mon}(\beta) \) and \( \text{Mon}(\beta \circ \phi) \) and \( \text{Mon}(\phi) \wr \text{Mon}(\beta) \) compare as we vary over Dynamical Belyi maps \( \beta \).
This material is based on work supported by the National Science Foundation under Grant No. DMS-2113782. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Contact edray.goins@pomona.edu for more information.