TITLE: Riemann surfaces: explicit algebraic equations
QUESTION [17 upvotes]: Suppose $\Gamma$ is a nice discrete subgroup of $SL(2,\mathbb{R})$ such that the genus of the Riemann surface $\mathbb{H}/\Gamma$ is larger than 1. We know that this Riemann surface is also an algebraic curve over $\mathbb{C}$ defined by a bunch of polynomials. Is there any explicit/canonical way of going back and forth between the group $\Gamma$ and a set of polynomials defining the curve? For example, if the group is given in terms of generators and relations, is there any algorithm for obtaining a set of polynomials defining the associated curve?
Somehow I couldn't add a comment so let me write it here.
Thanks for the responses. It will take me some time to digest the suggestions and look up the references provided. Please ignore the last comment about generators and relation. Let me ask a more specific question just to clarify what I wanted. What I am wondering is whether something akin to what happens for genus one curve also happens for higher genus. Recall that if $L=\mathbb{Z}+\tau \mathbb{Z}$, then we can write an equation for the elliptic curve $\mathbb{C}/L$ with the Eisenstein series $G_4(\tau)$ and $G_6(\tau)$ as coefficients. Can we do (or hope to do) something similar if we replace $\mathbb{C}$ by $\mathbb{H}$ and the lattice by a discrete group ?

REPLY [6 votes]: I'm not sure what you mean by "if the group is given by generators and relations" -- you need the group as a subgroup of $\mathrm{PSL}_2(\mathbb{R})$, not just as an abstract group.
If your group is commensurable with $\mathbb{SL}_2(\mathrm{Z})$ -- let's say for simplicity is a finite-index subgroup of the congruence subgroup $\Gamma(2)$ -- then your Riemann surface is a finite cover of the modular curve (or orbicurve, or stack) $X(2)$, which is (again, ignoring stacky issues) $\mathbb{P}^1$ with three points removed.  And your Riemann surface is an finite etale cover of $\mathbb{P}^1$ - three points, and you can easily recover the monodromy action of $\pi_1(\mathbb{P}^1 -$ three points$)$ on the sheets of this cover from the group you have in your hand.
In other words, the data you have is that of a Belyi curve.  Unfortunately it's quite hard to write down algebraic equations for Belyi curves in general, though there's a reasonably large literature working out many examples.  The best-understood cases are those where $\mathcal{H}/\Gamma$ has genus 0 ("dessins d'enfant") which is exactly what you're not interested in! 
This paper by Couveignes and Granboulan should be relevant:
http://www.di.ens.fr/~granboul/recherche/publications/abs-1994-CoGr.html
G.V. Shabat also has extensive tables of explicit equations for dessins d'enfants, but I don't think they're online.