Riemann surfaces: explicit algebraic equations

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 ?

• I think that you are asking how constructive the various isomorphisms are between the various versions of moduli space. I am going to guess that they are not constructive at all. (I used to be familiar with the proof of the Uniformization theorem: as I recall, it is a transcendental operation.) Perhaps somebody who knows the Kodaira Embedding theorem, and Chow's theorem, can enlighten us? – Sam Nead Dec 6 '09 at 13:03
• Perhaps an edit to the last sentence - giving a group via generators and relations is not the same as giving a representation into SL(2,R). What are you saying there? – Sam Nead Dec 6 '09 at 16:35

You are trying to relate the periods of the curve (which are analytic invariants), to algebraic invariants, so the most you can get is some power series. Suppose you get from Gamma to the period matrix of the Jacobian: tau, then:

In genus 1 - which you are not interested in - you have the j-invariant, which is a function of tau.

In genus 2 you have the Igusa invariants.

In genus 3 you don't have a formula (too complicated), but you have an algorithm: there are 28 tangents to the theta divisor at the 28 odd 2-torsion points, these are the 28 bitangents of the canonical curve, and you can (effectively) reconstruct the curve from the bitangents.

In higher genera you can start in a similar way: map the 4-torsion points to some grassmanian (which is an embedding of A_g plus some level: Grushevsky and Salvati-Mani), but I'm not aware of a reconstruction algorithm.

• "You are trying to relate the periods of the curve" - Is it easy to convert a representation of $\pi_1(S) \to SL(2, R)$ into a basis for the period lattice of the underlying complex manifold? – Sam Nead Dec 6 '09 at 16:45
• Yes. The pullback of the diagram (a little hard to draw on a comment) which sends both C^g, and the a curve to the Jacobian of the curve is the universal cover of the curve. See chapter 4 in Clemens's "A scrapbook of complex curve theory". – David Lehavi Dec 6 '09 at 18:00

A student of Bryan Birch (Steven Galbraith) wrote his thesis on computing equations defining modular curves, which of course are a very special case. This indicates that (a) the question is hard even in this case and (b) it might be a good place to start reading.

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.

Not an answer for individual curves, necessarily, but a related fact is that for large $g$, the moduli space of curves is not unirational. So that means that we can't do anything nice like write down a system of equation with $3g-3$ parameters such that a generic curve of genus $g$ is in that system for specified values. It doesn't quite answer the above question (which seems like it's the desire for a map from the appropriate space of groups to the moduli space of curves, or perhaps into the tricanonical embedding of the curve $\mathbb{H}/\Gamma$, and I don't know how to do that, though it seems like the lack of unirationality will just make this harder, if we have any chance at all.

For congruence subgroups of the modular group you can often express coordinates in terms of Weierstrass elliptic curves together with choices of torsion points.

Some references for the case of Shimura curves, which come from subgroups $\Gamma$ not commensurable with the modular group, include:

• Akira Kurihara, "On some examples of equations defining Shimura curves and the Mumford uniformization"
• Noam Elkies, "Shimura curve computations" (arXiv)