This is really a two-part question, but I would be happy to get an answer for either bit.
By a hyperbolic curve as defined by e.g. Szamuely in Galois Groups and Fundamental Groups (p.137) I mean an open subcurve $U$ of an integral proper normal curve $X$ over a field $K$ such that $2g-2+n>0$ where
- $g$ is the genus of the base change $X_\bar{K}$
- $n$ is the number of closed points in $X_\bar{K}\backslash U_\bar{K}$.
Many theorems/conjectures in anabelian geometry involve hyperbolic curves – for example, the section conjecture states that the rational points $U(K)$ on a hyperbolic curve $U$ correspond bijectively to (conjugacy classes of) sections of $\rho$ in the exact sequence of étale fundamental groups
$1\rightarrow \pi_1 (U_\bar{K}) \rightarrow \pi_1 (U)\xrightarrow{\rho}G_K \rightarrow 1$
where $G_K$ is the absolute Galois group of $K$.
Now, amongst all hyperbolic curves I have seen many particular references to the curve $U = \mathbb{P}_{\mathbb{Q}}^1 \backslash \left\{0,1,\infty\right\}$. Indeed, in his Notes on Etale Cohomology (p.30), J.S. Milne says that $\pi_1 (U)$ is arguably the most interesting object in mathematics. I know that in part this is because it ought to give us insight into the absolute Galois group of the rationals, but the particular choice of removed points seems quite mysterious to me.
So my questions are:
- Where does the interest in hyperbolic curves come from?
- Why is $= \mathbb{P}_{\mathbb{Q}}^1 \backslash \left\{0,1,\infty\right\}$ of particular interest?
Best Answer
The main theorem to mention is:
Theorem. (Belyi) Let $X$ be a smooth projective curve over $\mathbb C$. Then $X$ is defined over $\bar {\mathbb Q}$ if and only there exists a map $X \to \mathbb P^1_{\mathbb C}$ ramified over at most three points.
Remark. The 'only if' part is due to Belyi in 1979, and is basically an algorithm. The 'if' part was known before that (I believe it was proven by Weil in 1956). See this paper for a modern account of the 'if' part, and some background.
Note that three points on $\mathbb P^1$ are in general position, i.e. for any ordered set $(x,y,z)$ of distinct [closed] points, there exists an automorphism of $\mathbb P^1$ mapping $(x,y,z)$ to $(0,1,\infty)$. This is just linear algebra: we can choose coordinates so that $x$, $y$, and $z$ correspond to $[0:1]$, $[1:1]$, and $[1:0]$ respectively. Thus, if $X$ is a curve defined over a number field $K$, then we get a finite étale map $$U \to \mathbb P^1_K\setminus\{0,1,\infty\}$$ of some open $U \subseteq X$. This suggests that the study of $\pi_1^{\operatorname{\acute et}}(\mathbb P^1_{\mathbb Q} \setminus\{0,1,\infty\})$ could be interesting for number theoretic purposes.
Indeed, the 'fibration' $$\begin{array}{ccc}\mathbb P^1_{\bar{\mathbb Q}}\setminus\{0,1,\infty\} & \to & \mathbb P^1_{\mathbb Q} \setminus\{0,1,\infty\} \\ & & \downarrow \\ & & \operatorname{Spec} \mathbb Q \end{array}$$ induces a short exact sequence $$1 \to \pi_1^{\operatorname{\acute et}}(\mathbb P^1_{\bar{\mathbb Q}}\setminus\{0,1,\infty\}) \to \pi_1^{\operatorname{\acute et}}(\mathbb P^1_{\mathbb Q}\setminus\{0,1,\infty\}) \to \pi_1^{\operatorname{\acute et}}(\operatorname{Spec} \mathbb Q) \to 1.$$ The first group is the free profinite group $\hat F_2$ on two generators, since a change of algebraically closed field does not alter the fundamental group (and over the complexes we get the profinite completion of the topological fundamental group). The last group, of course, is just $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$. Thus, this in turn defines a map $$\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q) \to \operatorname{Aut}(\hat F_2) \twoheadrightarrow \operatorname{Out}(\hat F_2).$$
Corollary (of Belyi's theorem). The map $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q) \to \operatorname{Out}(\hat F_2)$ is injective.
See Szamuely's Galois groups and fundamental groups, Theorem 4.7.7. This means that we can view $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ as a subgroup of an object coming from topology. If we could understand what the image of the above injection is, we would solve all of number theory. This is the motivation for Grothendieck's study of dessins d'enfants.
There are also applications to the inverse Galois problem; see for example [loc. cit.], section 4.8.
See this post for some further applications and alternative perspectives on Belyi's theorem.