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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) would be an article by Morishita, 0904.3399.

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 for number fields?

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

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3 Answers 3

I don't think there is any reason to think that one exists, especially because the analogy is not very tight. For example, if $X_K = \operatorname{Spec}(\mathcal{O}_K)$, there exist closed hyperbolic 3-manifolds $M$ such that the abelianization of the fundamental group is infinite. (In fact, one conjectures that all hyperbolic $M$ virtually (= after passing to a finite cover) have this property.) On the other hand, the abelianization of $\pi_1(X_K)$ is always finite, by class field theory. As has been remarked elsewhere, there are several non-trivial $K$ such that $pi_1(X_K)$ is trivial.

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It's not completely tight, but it works surpisingly well for some theorems, like Poincare conjecture -- see, e.g. my answer to the linked question. –  Ilya Nikokoshev Nov 4 '09 at 21:50
    
It doesn't give me the intuition, either, but the authors of 0904.3399 provide a different reformulation of Poincare, true for number fields, and they apparently have intuition in mind for it: "According to our analogy, the difficulty of the original Poincar ́e conjecture may be coming from that of the corresponding analytic method–gauge theory–in topology." –  Ilya Nikokoshev Nov 4 '09 at 22:23
    
@FC: makes sense, +1. Anyway, I agree I know nothing about what an analogue of Poincare for number fields could be. –  Ilya Nikokoshev Nov 10 '09 at 21:49
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McMullen's third lecture at the 2000 Washington DC AMS Colloquium is exactly addressing this question. See his slides at

http://www.math.harvard.edu/~ctm/expositions/home/text/talks/ams/dc00/html/index.html

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McMullen's lecture (which is based on his inspiring paper "From dynamics on surfaces to rational points on curves", available here : math.harvard.edu/~ctm/papers/home/text/papers/fermat/fermat.pdf) is about something entirely different. The theorem of Thurston he refers to is Thurston's classification of mapping classes (see here : en.wikipedia.org/wiki/Nielsen-Thurston_classification), not his geometrization conjecture. –  Andy Putman Nov 10 '09 at 4:51
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Continued. McMullen's purpose is to motivate some aspects of Faltings's proof of the Mordell conjecture using Bers's proof of Thurston's theorem. The last section contains a couple of paragraphs on the "analogy" between 3-manifolds and number theory, but that is incidental to the rest of the paper. –  Andy Putman Nov 10 '09 at 4:55
    
Well, I am probably misunderstanding things, but at least the last page seems directly related to the question asked, while the entire second half of the lecture is developing the asked-for analogy. –  Sam Nead Nov 10 '09 at 5:21
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I was surprised that there exist even arithmetic analogies to solitons (more) and Laumon's arithmetic version of an idea of Witten made a new proof of Weil II possible. What else may have arithmetic versions? Ricci flow?

On the Ricci flow-renormalization issue in the comments below, Urs Schreibers answer, an other expert yesterday: "The renormalization that is involved is not the same as in QFT, except for the fact that it can also be thought as realizing, in that geometric context, a subtraction of divergences that has the effect of keeping the flow solutions from blowing up. Whether there is in that context any role for algebraic structures of renormalization, such as Hopf algebras accounting for nested divergences, is a good question."

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I think Ricci flow is a name for a quite deep thing, so maybe -- just maybe -- it exists in arithmetic context. Physicists called it renormalization in my presence, btw. –  Ilya Nikokoshev Nov 15 '09 at 1:12
    
That's interesting! Manin apparently thinks renormalization is 'platonic', a universal idea, e.g. recently wrote on it's appl. in programming. Then arithmetic versions should exist. –  Thomas Riepe Nov 15 '09 at 8:41
    
hmmm,... do you refer to renormalization in QFT (what I thought at int the comment above), or some other use of the word? –  Thomas Riepe Nov 15 '09 at 21:33
    
I am not an expert on either of those things, but I thought those are two distinct but related phenomena. –  Ilya Nikokoshev Nov 16 '09 at 16:22
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