Applications of Belyi's theorem

Question

Belyi's theorem (1979) is stated as follows:

A smooth projective curve over $\mathbb C$ is defined over a number field if and only if there exists a finite morphism (of varieties over $\mathbb C$) $t:X\longrightarrow\mathbb P^1_\mathbb C$ with at most $3$ critical values.

I'd like to know if there are some important applications of this theorem.

Thanks in advance.

Answer 1

There is one fairly famous application, which says that the absolute Galois group of $\mathbb{Q}$ (a completely arithmetic object!) sits inside the outer automorphisms of $\pi_1(\mathbb{P}_\mathbb{C}^1-\{0,1,\infty\})$ (a completely topologically defined group).

In symbols, this says that there is an injective group map

$$\pi_1^{\text{et}}(\text{Spec}(\mathbb{Q}))=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\text{Out}(\pi_1^{\text{Top}}(\mathbb{P}^1_\mathbb{C}-\{0,1,\infty\}))$$

(Belyi's theorem is used to show that the map is injective). This is one of the first big theorems that motivated the study of what Grothendieck called dessin d'enfants. In fact, Belyi's theorem is an integral part of many aspects of the study of dessin d'enfants.

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Here is another, very surprising application of Belyi's theorem.

There is a famous theorem of Faltings (previously known as the Mordell conjecture) which states that if $C$ is a smooth curve over $\mathbb{Q}$ of genus greater than $1$, then the $\mathcal{O}_K$-points of $C$ are finite. This, of course, is super interesting. It implies, for example, that even if FLT were not true, there could only be finitely many solutions for each $n\geqslant 3$. But, it applies to so, so many more curves other than FLT. It furthers the trichotomy between genus $0$, $1$, and greater than $1$ curves.

The attempted proof of Mordell's conjecture (obviously by Falting's) was responsible for the development of many modern day theories. For example, Arakelov theory was developed largely to try and apply intersection theoretic techniques to prove Mordell.

There is another theorem which is famous, and for which I am sure you've encountered. The famous ABC conjecture with it's subtly simple statement. I'm also sure you've heard that recently Mochizuki (a big player in the field of anabelian geometry, something this question is very much related to!) has claimed to have proved that ABC conjecture.

One of the reasons that people care about the ABC conjecture is that it implies (by a theorem of Granville) the coveted Fermat's last theorem, for sufficiently large exponents. Less well known though is that ABC also implies the Mordell conjecture. This was proven by Noam Elkies, and fundamentally relies on Belyi's theorem. You can find the original paper here.

Answer 2

An equivalent statement of Belyi's Theorem is that for every algebraic curve $X$ defined over $\overline{\mathbb{Q}}$, there is a finite index subgroup $\Gamma$ of $\Gamma(1) = \operatorname{PSL}_2(\mathbb{Z})$ such that

$\Gamma \backslash \overline{\mathcal{H}} \cong X(\mathbb{C})$,

or in other words that the corresponding complex algebraic curve is uniformized by a finite index subgroup of the modular group. (The equivalence comes from the fact that $\Gamma(2)$ is a free group on two generators which acts freely on the upper halfplane and the correspondence between covering spaces and subgroups of the fundamental group.)

From the perspective of algebraic curve theory and modular curve theory, this is truly startling: somehow every "arithmetic" algebraic curve (i.e., with algebraic moduli) is a modular curve! However the catch is that the uniformizing subgroup $\Gamma$ will in general have to be very far from being a congruence subgroup, i.e., containing some principal congruence subgroup

$\Gamma(N) = \operatorname{Ker}(\operatorname{PSL}_2(\mathbb{Z}) \rightarrow \operatorname{PSL}_2(\mathbb{Z}/N\mathbb{Z}))$.

For instance every algebraic curve uniformized by a congruence subgroup of level $N$ can be defined, together with all its automorphisms, over $\mathbb{Q}(\zeta_N)$, and thus every algebraic curve uniformized by any congruence subgroup can be defined, together with all its automorphisms, over an abelian number field. So although the existence of noncongruence subgroups already follows from the group theory of the free group $\Gamma(2)$ -- e.g. that it has finite quotients with plenty of Jordan-Holder factors other than prime order groups and groups $\operatorname{PSL}_2(\mathbb{Z}/p\mathbb{Z})$ (e.g. the Monster simple group!) -- Belyi's Theorem shows that noncongruence subgroups not only exist but exist in enough abundance to let in all arithmetic curves.

(Warning: The above use of "arithmetic" is not the standard one in this subject. But I have come to dislike the standard use: it seems to warn against studying the arithmetic geometry of certain curves which I believe to be richly deserving of arithmetic study.)

Answer 3

Here are three other "applications":

1) Belyi's theorem (in a stronger form due to Lily Khadjavi) is used in the work of Couveignes-Edixhoven-Bruin on computational aspects of modular forms; see http://arxiv.org/abs/1403.6404 (Theorem 5.0.1) You can also use it to prove bounds on Faltings heights of covers of curves (Theorem 1.3.1 and Theorem 6.0.4)

2) Belyi's theorem can be used to establish a special case of "Szpiro's small points conjecture"; see http://arxiv.org/abs/1311.0043 .

The way these applications work is as follows. By Belyi's theorem, any curve over $\overline{\mathbb{Q}}$ has a well-defined Belyi degree. Lily Khadjavi proved that the Belyi degree of a curve $X$ can be explicitly (i.e., not just effectively as Belyi showed) bounded in terms of data associated to a rational function $\pi:X\to \mathbf P^1$. Now use Arakelov theory to relate the Faltings height of $X$ to the data associated to $\pi$.

3) Belyi's theorem is indispensable in the work of Bauer, Catanese et al. on the action of the absolute Galois group on the moduli of surfaces of general type; see for example http://arxiv.org/abs/1303.2248