It's been known for a while that primes in number fields can be thought of, from an algebraic point of view, to be similar to knots in 3-manifolds. A good reference (thanks to [this question][1]) would be an article by Morishita, [0904.3399][2]. 

There are therefore many good analogues of operations, such as covers, or objects, like zeta-functions, that are defined purely algebraically. For example, a *linking number* of two knots has an easy algebraic definition as the image of one knot in the homology of the complement to the other which is analogous to *residue symbol* in number theory.

However, the operations of taking connected sum and cutting/gluing along a subsurface don't appear immediately to have an analogue in number fields. If you know how to make sense of "gluing" two schemes $\operatorname{Spec} \mathcal{O}_K$ and $\operatorname{Spec} \mathcal{O}_L$ along the "common element $x \in K, L$, by all means, please tell us!

Either way, here's my question:

> What could be an analogue of the [Thurston geometrization program][3] for number fields?

(may be this analogue will not be using gluing-like operations after all?)


  [1]: http://mathoverflow.net/questions/4075/questions-about-analogy-between-spec-z-and-3-manifolds
  [2]: http://arxiv.org/abs/0904.3399
  [3]: http://en.wikipedia.org/wiki/Geometrization_conjecture