Curves admitting Belyi maps are defined over $\overline{\mathbb{Q}}$.

Question

Belyi's theorem states that a complex algebraic curve $X$ admits a model over $\overline{\mathbb{Q}}$ if and only if it admits a map to $\mathbb{P}^1$ which is ramified over at most three points.

In fact, Belyi proves that if the curve X admits such a model, then there is a map to $\mathbb{P}^1$. The converse, which was previously known, follows from the existence of the étale fundamental group.

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I have to admit that I don't see why the converse is true just from knowing basic facts about the étale fundamental group, and the references I've found all approach it obliquely, with a summary more or less like what I said in the two paragraphs above.

Would someone mind writing the argument out carefully?

Answer 1

Let $U$ be a variety over $\overline{\mathbb{Q}}$, where variety means finite type separated integral scheme over $\overline{\mathbb{Q}}$. Then the category of finite 'etale covers of $U$ is equivalent to the category of finite 'etale covers of $U_{\mathbb{C}}$. This is stated in Szamuely's book "Galois groups and fundamental groups" (but not proven). In any case, it implies the "converse of Belyi's theorem" by taking $U = \mathbb{P}^1\setminus \{0,1,\infty\}$.

The way you prove this equivalence of categories is essentially by using the "rigidity" of finite etale covers of bounded degree. Much like Richard D. James explains in the comments, the basic idea is to look at the set of isomorphism classes of $\mathrm{Aut}(\mathbb{C})$-conjugates of a given finite etale morphism $V\to U_{\mathbb{C}}$. Any such conjugate $V^{\sigma}\to U^{\sigma}_{\mathbb{C}} = U_{\mathbb{C}}$ (with $\sigma \in\mathrm{Aut}(\mathbb{C})$) is finite etale of the same degree as $V\to U_{\mathbb{C}}$. By the finite generation of $\pi_1(U_{\mathbb{C}})$ (proven in SGA7) this implies that the set of conjugates is finite. From this is follows (not immediately, though) that $V\to U_{\mathbb{C}}$ can be defined over $\overline{\mathbb{Q}}$.

It's the same proof as Richard D. James explained, except that I invoke a more general finite generation result for arbitrary varieties.