This is going to be a fairly vague question but hopefully it will have concrete answers:

There is this recurrent phenomenon in the arithmetic of curves (possibly stacky,affine) where there is a "phase transition" associated with the euler characteristic being positive, zero or negative.

For instance, Mordell's conjecture (Falting's theorem) states that there are finitely many integral points if the euler characteristic is negative. In fact, this theorem is true not only for affine curves but also for stacky curves.

Another example is the failure of local-global principles for integral points for euler characteristic $\leq 0$.

Question:Is there some key set of features in the "hyperbolic" case that come up often in proofs?

Let me illustrate the kind of answer I am looking for with reference to the complex case. In this case, negative euler characteristic corresponds exactly to having the hyperbolic disc as the universal covering space. So a very common argument is that given a map to a hyperbolic curve, we can lift it in some appropriate sense to the universal cover and then by Liouville, this lift will often be constant.

Is there an analogous collection of arithmetic hyperbolic techniques to keep in mind?