Computing Euclidean Belyi maps

Grothendieck in his Esquisse d’un Programme [5] described an action of the absolute Galois group $\operatorname{Gal}(\mathbb{Q}^{\textup{al}}\,|\,\mathbb{Q})$ of the rational numbers on the sets of Belyi maps and dessins d’enfants, linking combinatorics, topology, geometry, and arithmetic in a deep and surprising way. Computational aspects of this program [12] remain of significant interest, and there has been recent, fundamental progress using complex analytic techniques [6, 9, 10, 1]. A common thread underlying these approaches is to realize a Belyi map via uniformization as $\varphi\colon\Gamma\backslash{\mathcal{H}}\to\Delta\backslash{\mathcal{H}}$ where ${\mathcal{H}}$ is one of the three classical geometries (the sphere, the Euclidean plane, or the hyperbolic plane), and $\Gamma\leq\Delta$ is a finite-index subgroup of a triangle group. The case where ${\mathcal{H}}$ is spherical is truly classical, corresponding to certain triangulations of the Platonic solids. In the hyperbolic case, analytic methods can be employed to convert this geometric description into an algebraic one, using modular forms, finite element techniques, or conformal maps. What remains is the case of Euclidean triangle groups, those arising from the familiar regular triangular tessellations of the Euclidean plane. In this paper, we fill this gap: we compute Euclidean Belyi maps explicitly from maps of complex tori, forming a bridge between the classical and the general.

A Belyi map over $\mathbb{C}$ is a morphism $\varphi\colon X\to\mathbb{P}^{1}_{\mathbb{C}}$ of nice (projective, nonsingular, integral) curves over $\mathbb{C}$ that is unramified away from $\{0,1,\infty\}$. By the Riemann existence theorem, we may equivalently work with such a map of compact Riemann surfaces. Famously, Belyi [2, 3] proved that a curve $X$ over $\mathbb{C}$ can be defined over the algebraic numbers $\mathbb{Q}^{\textup{al}}$ if and only if $X$ admits a Belyi map.

Belyi maps admit a tidy combinatorial description, something we take as the input to our algorithm. A permutation triple of degree $d$ is a triple $(\sigma_{0},\sigma_{1},\sigma_{\infty})\in S_{d}^{3}$ of permutations on $d$ elements such that $\sigma_{\infty}\sigma_{1}\sigma_{0}=1$. A permutation triple is transitive if it generates a transitive subgroup of $S_{d}$. The monodromy around $0,1,\infty$ of a Belyi map of degree $d$ gives a permutation triple of degree $d$, giving a bijection between isomorphism classes of Belyi maps of degree $d$ and transitive permutation triples up to simultaneous conjugation. Lifting paths, one can compute (by numerical approximation) the permutation triple attached to a Belyi map; in this paper, we consider the harder, converse computational task.

Let $\sigma$ be a transitive permutation triple of degree $d$ and let $a,b,c$ be the orders of $\sigma_{0},\sigma_{1},\sigma_{\infty}$, respectively. By the theory of covering spaces, the permutation triple $\sigma$ defines a homomorphism $\pi\colon\Delta(a,b,c)\to S_{d}$ and thereby a subgroup $\Gamma\leq\Delta(a,b,c)$ of index $d$ (see section 2). The quotient $\Gamma\backslash{\mathcal{H}}$ can be given the natural structure of a Riemann surface $X(\Gamma)$, and the further quotient to $\Delta\backslash{\mathcal{H}}$ defines a Belyi map $\varphi\colon X(\Gamma)\to X(\Delta)\simeq\mathbb{P}^{1}_{\mathbb{C}}$. By the theorem of Belyi, the map $\varphi$ can be defined over the field of algebraic numbers $\mathbb{Q}^{\textup{al}}$.

We say that $\sigma$ (and its corresponding map $\varphi$) is Euclidean if $1/a+1/b+1/c=1$, in which case the attached triangle group $\Delta(a,b,c)$ is a group of symmetries of the Euclidean plane, whence $(a,b,c)=(3,3,3),(2,3,6),(2,4,4)$. Our main result provides an algorithmic way to compute algebraic equations for $\varphi$ given $\sigma$.

The algorithm in Theorem 1.2.1 is specified in Algorithm 3.5.1. We implemented the algorithm in the computer algebra system Magma [4]: the running time is quite favorable. We computed a database of Euclidean Belyi maps with this implementation (see section 4) which we will upload to the LMFDB [7]. Our code is available as part of a Belyi maps package available online (https://github.com/michaelmusty/Belyi).

We now briefly indicate the idea behind the proof of Theorem 1.2.1. We first convert the permutation triple $\sigma$ into an explicit description of the group $\Gamma\leq\Delta$. Next, we write $\Gamma\simeq T(\Gamma)\rtimes R(\Gamma)$ as a semi-direct product, where $T(\Gamma)$ consists of the subgroup of translations in $\Gamma$ and $R(\Gamma)$ is generated by rotation around a particular point, which we can find explicitly. The quotients $E(\Gamma)\colonequals T(\Gamma)\backslash\mathbb{C}$ and $E(\Delta)\colonequals T(\Delta)\backslash\mathbb{C}$ define elliptic curves. We then have the following commutative diagram, which we call the master diagram:

To find the Belyi map $\varphi$, our strategy is to compute the other three maps in our diagram, filling in $\varphi$ by commutativity (“descending $\psi$ along $\alpha$”). The bottom map $\alpha$ depends only on $a,b,c$ and the choice of origin, giving six possibilities. The map $\psi$ is an isogeny of elliptic curves, which we compute from the inclusion of lattices implied by $T(\Gamma)\leq T(\Delta)$ by applying formulas of Vélu. The top map $\beta$ is computed by looking at the fixed field of $\mathbb{C}(E(\Gamma))$ under the finite subgroup of automorphisms corresponding to the rotations $R(\Gamma)$ (taking care to ensure these rotations act by automorphisms at the origin). The final step, to fill in $\varphi$ to make the diagram commute, is obtained via explicit substitution.

After reviewing background in section 2, we exhibit in section 3 the main algorithm (Algorithm 3.5.1) in pseudocode and then prove the main result (Theorem 1.2.1). In section 4 we describe an implementation in the Magma computer algebra system and then present some computed examples.

The authors would like to thank Sam Schiavone and Jeroen Sijsling for discussions. Voight was supported by a Simons Collaboration grant (550029).

First, a few basic facts and conventions. In this article, the symmetric group $S_{d}$ acts on the right on $\{1,\dots,d\}$, written in exponentiated form: e.g., if $\tau=(1\,2\,3)$ and $\mu=(2\,3)$ then $1^{\tau\mu}=(1^{\tau})^{\mu}=2^{\mu}=3$.

Recall that if $G$ is a group, a (finite) permutation representation of $G$ is a group homomorphism $\pi\colon G\to S_{d}$ for some $d\geq 1$, and we say that $\pi$ is transitive if its image is a transitive subgroup of $S_{d}$. A transitive permutation triple $\sigma$ defines a transitive permutation representation by $\pi(\delta_{s})=\sigma_{s}$ for $s=a,b,c$, and conversely.

Let $\pi\colon\Delta\to S_{d}$ be a transitive permutation representation. Let

be the preimage of the stabilizer of $1$ under $\pi$. (The stabilizer of $k\in\{1,\dots,d\}$ is conjugate to $\Gamma$ in $\Delta$.) Then $[\Delta:\Gamma]=d$. Conversely, given $\Gamma\leq\Delta$ of index $d$, the action of $\Delta$ on the cosets of $\Gamma$ gives a transitive permutation representation $\pi\colon\Delta\to S_{d}$, and this correspondence is bijective.

We refer to Magnus [8, §II.4] for classical background on Euclidean triangle groups; we briefly summarize some classical facts. Let $T^{*}$ be a triangle in the Euclidean plane $\mathbb{C}$ with angles $\pi/a$, $\pi/b$, and $\pi/c$ at the vertices $v_{a}$, $v_{b}$, and $v_{c}$ labeled clockwise, with $a,b,c\in\mathbb{Z}_{\geq 2}$. Then in fact there are only three possibilities, namely

corresponding to the solutions to $1/a+1/b+1/c=1$; the corresponding tessellations of the Euclidean plane by triangles are sketched in Figure 2.2, with alternating triangles colored white and black.

The group generated by the reflections in the sides of $T^{*}$ generates a discrete group of isometries acting properly on $\mathbb{C}$, with fundamental domain $T^{*}$. The further subgroup of orientation-preserving isometries $\Delta(a,b,c)$ has index $2$, described as follows. For $s\in\{a,b,c\}$, let $\delta_{s}$ be the counterclockwise rotation about $v_{s}$ by an angle of $2\pi/s$.

Visibly from Figure 2.2 we have $\Delta(3,3,3)\trianglelefteq\Delta(2,3,6)$ with index $2$ (halving a fundamental triangle), and $T(\Delta(3,3,3))=T(\Delta(2,3,6))$. Attached to each translation subgroup is the orbit of $0$

which defines a lattice $\Lambda_{\Delta}\simeq\mathbb{Z}^{2}$. We write

In this section, we describe fundamental domains for the groups under consideration. A fundamental domain for the action of $\Delta$ is obtained from any pair of one shaded triangle and one unshaded triangle which we may take to share an edge. This gives a region where all the interior points are distinct under the identification $\Delta\circlearrowright\mathbb{C}$. Furthermore, we can divide the four sides of the quadrilateral into two pairs of consecutive sides identified under the quotient by $\Delta$ as in Figure 2.3, so that $X(\Delta)$ has genus 0.

$$\begin{gathered}\includegraphics[scale={.2}]{./pictures/333Dregion.png}\includegraphics[scale={.2}]{./pictures/244Dregion.png}\includegraphics[scale={.2}]{./pictures/236Dregion.png}\\[4.0pt] \text{Figure \ref{fig:DeltaFundoms}: Fundamental domains for $\Delta$, like colored edges and vertices are identified}\end{gathered}$$

Since $T(\Delta)$ is generated by two noncollinear translations, we can take as its fundamental domain the parallelogram determined from two sides sharing a vertex at the origin. Opposite edges are identified while consecutive edges are distinct, so the fundamental region is equivalent to a torus (genus 1). Similar statements hold for $T(\Gamma)$.

$$\begin{gathered}\includegraphics[scale={.32}]{./pictures/333TDPar.png}\includegraphics[scale={.30}]{./pictures/244TDPar.png}\includegraphics[scale={.24}]{./pictures/236TDPar.png}\\[4.0pt] \text{Figure \ref{fig:FundomsTD}: Fundamental domains for $T(\Delta)$, like colors identified}\end{gathered}$$

Finally, a fundamental domain for $\Gamma$ is constructed in the usual manner: we choose coset representatives $\Delta=\bigsqcup_{i=1}^{d}\gamma_{i}\Gamma$, and then for $D(\Delta)$ the fundamental domain for $\Delta$ we have the fundamental domain $D(\Gamma)=\bigcup_{i=1}^{d}\gamma_{i}D(\Delta)$.

We now consider the genus of the surface $X(\Gamma)\colonequals\Gamma\backslash\mathbb{C}$. Given a permutation $\tau\in S_{d}$, let $k(\tau)$ be the number of disjoint cycles in $\tau$ and define its excess as $e(\tau)\colonequals d-k(\tau)$. Then by the Riemann–Hurwitz formula, the genus of $X(\Gamma)$ is equal to [12, (1.5)]

Let

be the subgroup of translations in $\Gamma$; then $T(\Gamma)\trianglelefteq\Gamma$, as $T(\Delta)\trianglelefteq\Delta$ by Proposition 2.2.2. Writing $E(\Gamma)\colonequals T(\Gamma)\backslash\mathbb{C}$ and similarly for $\Delta$, the containments of these four groups give quotient maps which fit into the diagram (1.3.1).

We again have a lattice

with $\Lambda_{\Gamma}\leq\Lambda_{\Delta}$ a subgroup of finite index. When no confusion can arise, we will identify translation maps by the corresponding lattice element. We define

In the following algorithm, we compute a convenient basis for $T(\Gamma)$.

In this section, we study rotations in $\Gamma$. Restricting the exact sequence (2.2.4) we obtain

where $R(\Gamma)\colonequals\rho(\Gamma)\leq\mathbb{Z}/c\mathbb{Z}$. Evidently, $R(\Gamma)$ is a cyclic group with order dividing $c$.

In Proposition 2.2.2(c) we split the exact sequence using $\delta_{c}$. Indeed, the analogous sequence for $\Gamma$ above is again split, but not necessarily by a power of $\delta_{c}$: instead, $R(\Gamma)$ is generated by a rotation about some vertex (an element in the $\Delta$ orbit of $v_{a}$, $v_{b}$, or $v_{c}$), as follows.

With Lemma 2.5.3, we can be more precise about the possible vertices of maximal rotation.

Because a permutation triple which is simultaneously conjugate to $\sigma$ gives an isomorphic Belyi map (with differently labelled sheets), we may suppose without loss of generality that one of $v_{a},v_{b},v_{c}$ is a vertex of maximal rotation: after simultaneous conjugation, we just insist that $1$ belongs to a cycle as in Corollary 2.5.5. This “preprocessing” step is given as follows.

From here forward, we may suppose without loss of generality that this “preprocessing” step has been applied.

We now see the exact circumstances when $g(X(\Gamma))=1$.

We begin with the bottom map $\alpha\colon E(\Delta)\to X(\Delta)\simeq\mathbb{P}^{1}$, which depends only on $\Delta$ (with the choice of the origin at $v_{c}$). From Proposition 2.2.2(c), the map $\alpha$ is the quotient by a cyclic group of rotations of order $c$ at a vertex $v_{c}$, which we may take as the origin of the elliptic curve $E(\Delta)$. Accordingly, these rotations act by automorphisms of the elliptic curve $E(\Delta)$, and so their equations are well-known [14, §II.2] (see also Lemma 3.3.2 below). Define the elliptic curves

over $\mathbb{Q}$, the automorphisms

and the quotient maps

We recall the lattices defined in (2.2.10). After homothety, the Weierstrass map $z\mapsto(\wp(z),\wp^{\prime}(z)/2)$ gives an analytic isomorphism from the complex elliptic curve $\mathbb{C}/\Lambda_{\square}$ to $E_{\square}(\mathbb{C})$. Moreover, the rotation $\delta_{4}$ acts by $(x,y)\mapsto(-x,iy)$ (gently abusing notation), and the quotient map $E(\Delta)\to X(\Delta)$ is given by $\alpha_{4}$ in these coordinates. Similar statements hold for the two $\hexagon$ cases.

We now turn to the left map $\psi$ in (1.3.1), an isogeny $E(\Gamma)\to E(\Delta)$.

We first show how to work explicitly with torsion on $E(\Delta)$ using exact arithmetic. To handle the three cases uniformly, let $j=i$ or $j=\zeta_{6}$, so that $\Lambda=\Lambda_{\Delta}=\mathbb{Z}[j]$, let $K\colonequals\mathbb{Q}(j)\subseteq\mathbb{C}$, and let $E=E_{\square}$ or $E=E_{\hexagon}$.

For an integer $N\geq 1$, the torsion group

is a cyclic $\mathbb{Z}[j]$-module; we use the symbol $P\in E[N]$ to denote a generator of $E[N]$ as a $\mathbb{Z}[j]$-module.

Next, we recall section 2.4, where we defined $N\colonequals[T(\Delta):T(\Gamma)]$ and computed in Algorithm 2.4.4 a basis for $\Lambda_{\Gamma}$. Since $N\Lambda_{\Delta}\subseteq\Lambda_{\Gamma}$, we have an isogeny

dual to our desired isogeny $\psi$. From this setup, we compute an equation for $\psi$ using Vélu’s formulas, as in the following algorithm.