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Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:

1) $X$ is defined over $\overline{\mathbb{Q}};$

2) There exists a meromorphic function $\phi: X\to\mathbb{P}^1\mathbb{C} $ ramified at most at $0,1,$ and $\infty$;

3) $X$ is isomorphic to $\Gamma \backslash \mathbb{H}$ (compactified at cusps) for a finite index subgroup $[\mathrm{PSL}_2(\mathbb{Z}), \Gamma]<\infty.$

The remaining question is: $$\boxed{\text{ Is there a way to treat singularities in this or a similar framework? }}$$

The following of my original questions have been answered:

Can this be generalized to arbitrary projective nonsingular varities of higher dimensions? (I discussed this with one professor here in Goettingen. That seems to be ongoing research. Please see also comment of David Roberts.)

What compactification do they mean here? $\Gamma \backslash \mathbb{H} \cup \mathbb{Q}$! (see the answer of Robin Chapman)

What is a nice reference for the proof of Belyi's theorem? (see answer of YBL and Koeck + the comments of Emerton)

How does the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ enter the picture? (see comment of Ariyan and answer of YBL)

Where can I find nice examples where these computations have been done explicitly? (see answer of Andy Putnam and JSE)

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By other global fields do you mean function fields? –  Emerton Nov 16 '10 at 17:49
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2) is wrong. The map should go the other way and there should be only 3 branch points. –  Felipe Voloch Nov 16 '10 at 19:15
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For a number field $K, {\bar K} = {\bar \mathbb{Q}}$, so the statement is the same as for $\mathbb{Q}$. –  Felipe Voloch Nov 16 '10 at 19:20
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Look at Chapter 4.7 of Szamuely's book: Galois Groups and Fundamental Groups. In that book you should look at Theorem 4.7.6. That's Belyi's theorem. Corollary 4.7.7 "explains" how the absolute Galois group comes into play. Remark 4.7.9 says that the significance of Corollary 4.7.7 is that it embeds the absolute Galois group in the outer automorphism group of something "topological". That's the beginning of Grothendieck's theory of dessins d'enfants (=children's drawings). –  Ari Nov 16 '10 at 20:41
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You can find Szamuely's book here renyi.hu/~szamuely/fg.pdf . I'm not sure if it's ok to post the link like that but the book is really worth checking out. –  Ari Nov 16 '10 at 20:43

5 Answers 5

up vote 7 down vote accepted

The compactification is the usual one coming up in the theory of modular forms, with the cusps being orbits of $\Gamma$ on $\mathbb{Q}\cup\{\infty\}$.

As for the proof, I like this paper by Bernhard Koeck.

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Okay, that was also what I suspected! Thanks for the reference, although it just threats 1<=>2. – Marc 0 secs ago –  plusepsilon.de Nov 16 '10 at 17:17
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The equivalence of (2) and (3) follows from the fact that the upper half-plane mod $\Gamma_1(4)$ (for example) is equal to $\mathbb P^1 \setminus \{0,1,\infty\}$, and hence giving a finite degree cover of $\mathbb P^1$ unramified outside $0,1,\infty$ is the same as giving a finite index subgroup of $\Gamma_1(4)$. –  Emerton Nov 16 '10 at 17:48
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The preceding comment should actually explains not the equivalence of (2) and (3), but rather the equivalence of (3) and (2'), where (2') is the statement that $X$ can be written as a cover of $\mathbb P^1$ unramified outside 3 points (which one can take to be $0,1,\infty$). The equivalence of (1) and (2') is what I usually think of as being Belyi's theorem. –  Emerton Nov 16 '10 at 17:53

2) should read: There exists a non constant function $X \to \mathbb{P}^1$ unramified outside $\{0,1,\infty\}$.

The Galois action is the obvious one: given a curve $X$ defined over $\overline{\mathbb{Q}}$, $G_{\mathbb{Q}}$ acts on the coefficients of the equations defining $X$. Belyi's theorem tells you that every $\overline{\mathbb{Q}}$-curve appears as a finite étale covering of $\mathbb{P}^1_{\mathbb{C}}-\{0,1,\infty\}$ and each covering is naturally defined over $\overline{\mathbb{Q}}$. So the obvious action on equations with coefficients in $\overline{\mathbb{Q}}$ translates into an action on the (profinite) fundamental group of $\pi_1(\mathbb{P}^1_{\mathbb{C}}-\{0,1,\infty\})$ (free profinite group on 2 generators). One checks that the action on elliptic curves induces the obvious action on their $j$-invariant so this action is faithful. This means we can study the whole Galois group $G_\overline{\mathbb{Q}}$ through its action $\pi_1(\mathbb{P}^1_{\mathbb{C}}-\{0,1,\infty\})$. This is what Grothendieck-Teichmuller theory and Dessins d'enfants do.

For the proof, a simple google search gives this article. For more detailed answers, you should look at the books "Geometric Galois actions" edited by Schneps and Lochak. They provide a good introduction to the numerous aspects of Grothendieck's "Esquisse d'un programme".

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There is a nice generalization of Belyi's theorem in positive characteristic, proved by M. Saïdi in his paper Revêtements modérés et groupe fondamental de graphe de groupes. (Compositio Math. 107 (1997), no. 3), Théorème 5.6:

Let $C$ be a smooth projective curve defined over a field $K$ of characteristic $p>2$. The following conditions are equivalent:

  • The curve $C$ can be defined over $\bar{\mathbf F}_p$,
  • There exists a finite cover $C\to\mathbf P^1$ tamely ramified above $\infty,0$ and $1$ (and étale elsewhere).

The proof, very short and elegant, relies on a result of Fulton on the existence of covers with only double ramification. Unfortunately, the argument does not apply in characterisctic $2$, for which the question is still open (some recent progress in this situation can be found in S. Schröer's paper Curves with only triple ramification, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7).

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Can you post a reference for proof in English please? –  zroslav Jun 6 '11 at 17:45
    
Wushi Goldring wrote an expository paper focussed on Belyi's theorem and its generalizations. It will appear on Serge Lang memorial volume and it contains Saïdi's proof (Theorem 4.6). You can find the preprint at math.harvard.edu/~wushi –  Leonardo Jun 27 '11 at 0:55

Chapter 2 of Lando and Zvonkin's lovely book "Graphs on Surfaces and Their Applications" contains a nice, down-to-earth exposition of the proof of Belyi's theorem together with a large number of explicit examples.

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As for explicit computations, Jean-Marc Couveignes and G.B. Shabat did a bunch of these. Couveignes gives a nice discussion of the problem here.

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Indeed, Shabat polynomials are the (I believe) most understood case of this theorem –  David Roberts Nov 18 '10 at 4:08

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