TITLE: Applications of Belyi's theorem
QUESTION [16 upvotes]: 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.

REPLY [10 votes]: 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