Finding a model for $X_G$ with $G\subseteq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$

Question

I'm studying modular curves as part of my doctorate work and would like to understand how one gets from a subgroup $G$ of $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$ to an equation for $X_G$. By $X_G$ I mean a modular curve which parametrices elliptic curves whose Galois representation has image contained in $G$ (when restricted to $N$-torsion).

Is there a general algorithm which will produce an equation for $X_G$ given a list of generators of $G$? My guess from what Ive seen so far is no, but I want to find a good source to start approaching this area. I could only find information about specific groups ($\Gamma(N)$,$\Gamma_0(N)$, Cartan subgroups).

Answer 1

One more or less systematic method which you might use roughly goes as follows. Let $X_1\cong\mathbb{P}^1$ denote the $j$-line.

1. Describe the branched covering $f : X_G\rightarrow X_1$ combinatorially, via a triple of permutations $\sigma_0,\sigma_1,\sigma_\infty$ with $\sigma_0\sigma_1\sigma_\infty = 1$.
2. Find equations for the "Belyi map" $f$ (including equations for the curve $X_G$).

Part 1 is rather straightforward: Let $\tilde{G} := \langle G,-I\rangle$. Let $\varphi : \mathbb{Z}^2\rightarrow(\mathbb{Z}/n)^2$ be the reduction map, viewed as an element of $\text{Epi}(\mathbb{Z}^2,(\mathbb{Z}/n)^2)/\tilde{G}$, where $\tilde{G}$ acts on the codomain. The group $\text{PSL}(2,\mathbb{Z})$ acts on this set. Let $\mathcal{O}$ be the orbit of $\varphi$. Consider the matrices $$M_0 := \left(\begin{array}{ll} 1&1\\-1&0\end{array}\right)\qquad M_{1728} := \left(\begin{array}{ll} 0&-1\\1&0\end{array}\right)\qquad M_\infty := \left(\begin{array}{ll} 1&1\\0&1\end{array}\right)$$

The fundamental group $\Pi$ of $X_1^* := X_1 - \{0,1728,\infty\}$ is free of rank 2, generated by $x,y,z$ satisfying $xyz = 1$. Let $X_G^* := f^{-1}(X_1^*)$. Then the covering $f : X_G^*\rightarrow X_1^*$ corresponds, via the Galois correspondence, to the set $\mathcal{O}$ together with the $\Pi$ action where $x,y,z$ acts via $M_0,M_{1728},M_\infty$ respectively. In particular, the ramified points above $j = 0,1728,\infty$ correspond to the orbits of $x,y,z$ respectively, the ramification index being the orbit sizes. The permutations induced by $x,y,z$ are the desired $\sigma_0,\sigma_1,\sigma_\infty$.

Part 2 is much more subtle. My understanding is that there is an algorithm for this, but it is only really feasible for covers with relatively small degrees (e.g., less than 300). In practice there are a number of different techniques one can employ, some of which are described in this article by Sijsling and Voight.

Answer 2

Another approach is to directly access differentials on $X_{G}$ via modular forms. David Zywina has implemented this (see Sections 4 and 5 of his preprint here). Roughly speaking and when the genus of $X_{G}$ is $\geq 3$, one determines the action of $G$ on the space of weight $1$ Eisenstein series for $G$, computes products of these to produce weight $2$ forms, computes the subspace consisting of forms vanishing at the cusps, and then the quadratic relations between these forms to obtain the canonical embedding of $X_{G}$. (Variants of this idea are needed when the genus is lower and/or when $X_{G}$ is hyperelliptic.)

Some work has been done to implement these (and related ideas) and to run them at scale, and there is a hope to eventually have a database of equations for $X_{G}$ and maps $j : X_{G} \to \mathbb{P}^{1}$ in the LMFDB. Currently, a beta version of this database is available here. We hope to compute many more models, improve some models already computed, and do some careful checking before the database is officially released.