[Math] Applications of Belyi’s theorem


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

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

Thanks in advance.

Best Answer

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


(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.

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.

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