I suppose this is an extremely general question, so I apologize, and perhaps it should be deleted. On the other hand it's an awesome question.

Is it clear exactly how much (assumedly algebraic) number theory can written down diagrammatically, and if so, has there been any effort to write such problems in the category of spectra (whichever category you like) and solve problems there? It seems that some problems may become easier to solve, if only because there in some sense "more" spectra to work with than there are regular algebraic objects (i.e. he have the Eilenberg-MacLane spectrum for whatever ring or field of whatever, but we also have things that don't come from any algebraic object).

I have heard about Rognes' work on Galois extensions in this sense, and that there are lots of connections to things like Morava K-theory (and the associated spectra), and plan on at least attempting to pursue such ideas.


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    $\begingroup$ Can I get a comment explaining the down vote? $\endgroup$ Oct 28 '11 at 21:49
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    $\begingroup$ Sorry, I was trying to think of how to say this diplomatically. I downvoted the question because it is way too broad and speculative. It has the flavor of "Here are two big fields of mathematics. Are there any connections between them?". Such questions are a little too b.s'y for my taste. $\endgroup$
    – Adam Smith
    Oct 28 '11 at 22:01
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    $\begingroup$ (though just to give you something in return, there do exist deep connections between algebraic topology and arithmetic algebraic geometry; see, for instance, the work of Voevodsky on Milnor's conjecture and the Bloch-Kato conjecture), $\endgroup$
    – Adam Smith
    Oct 28 '11 at 22:03
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    $\begingroup$ Fair enough. I'm not really intending to ask "Are there any connections here?" though I've been known to ask such questions. To put a label on it to avoid it just being a "discussion question" I would say it's primarily a reference request. I.e., who (if anyone) is doing this? Obviously Galois theory can be framed thusly, is that the end of the story? Maybe that's still too broad, but it is genuinely a question I'd like some answers to. $\endgroup$ Oct 28 '11 at 22:04
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    $\begingroup$ this IS an awesome question. $\endgroup$
    – Jacob Bell
    Nov 13 '11 at 0:27

This is probably not the answer you are looking for but, since there has been no other answers, it may be of some use. Recently, I have enjoyed finding out how wonderful and mysterious the relationship between (algebraic) number theory and low-dimensional topology is through the links between Galois groups and homotopy/fundamental groups. In particular the framing of properties of prime numbers in class field theory and arithmetic geometry having duality correlations with knots, links and three-dimensional manifolds! This subject of Arithmetic Topology was hinted time ago by several people, like D. Mumford and B. Mazur, which has very recently got its first MARVELOUS book:

Some of the first explorations on this were:

The whole point is the study of the fundamental groups of topology but from a number-theoretic perspective, that is, working with the (étale) algebraic fundamental group of varieties and schemes as originally thought by A. Grothendieck in his anabelian geometry. For this and as a prerequisite to Morishita's book, along with standard arithmetic geometry, one should get the specific background on the relationship of Galois gropus and fundamental groups of schemes, provided by:

These new dualities between apparently so different worlds may open the door to a deep understanding of the mysterious arithmetic-topologic relationship.


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