This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer.

As we all know, algebraic number theory is one of the oldest and most developed areas in math. New and spectacular results come very often, one can mention here the work of Wiles, or of several recent Fields medalists.

My point now: I might be wrong here, not being a specialist, but my impression is that all this new and spectacular work, and for some time already, rather consists in building new things, and not much in substantially simplifying the theory.

Question. Is there any big simplification in sight, in algebraic number theory? If so, in which precise direction?

I'm asking this question because I'm a bit puzzled by what is going on, when compared to other branches of math. Any healthy theory must alternate "building" and "simplifying" periods. By "healthy" here I mean something really viable, that cannot just die over the time from the lack of new and interesting results, but nor from over-complication either.

[Edited, Feb, 9, 2013, by A. Caicedo. Original question by Teo B.]

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    $\begingroup$ I don't see why anything should one day fit in a small book accessible to undergraduates. Try the Weil conjectures. $\endgroup$ – KConrad Feb 5 '13 at 22:40
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    $\begingroup$ Maybe the downvote is because the question "when will FLT proof fit in a short book" implicitly rules what to me seems like by far the most likely option, which is "proof of FLT will never fit into a short book". The current proof of FLT is still astronomically large. All proofs still currently use a lot of modern algebro-geometric machinery and deep work of Langlands on the trace formula (in fact the more modern proofs use far more Langlands -- they use base change as well as Jacquet-Langlands). The proof is getting longer! In terms of absolute length if you write out all details, at least. $\endgroup$ – user30035 Feb 6 '13 at 7:38
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    $\begingroup$ @Teo: history deals with the past. This here is futurology. And I have no idea what we are supposed to do here except speculate. In addition, your claims are highly subjective. What's next? "Do you think that there will be a proof of ABC fitting in a small booklet one day?" $\endgroup$ – Franz Lemmermeyer Feb 6 '13 at 8:52
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    $\begingroup$ There are theorems that can be briefly stated, whose proofs will never fit in a small book. This is because the function $$n \mapsto \sup_{\text{theorems } T \text{ of length }n} \inf_{\text{proofs } P \text{ of } T} \text{length}(P)$$ grows faster than any computable function. (Proof: If it were not true, one could prove theorems by checking all proofs of suitable length, and thereby solve the Halting Problem.) $\endgroup$ – S. Carnahan Feb 7 '13 at 5:57
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    $\begingroup$ I think quid has a good idea (particularly in view of the possibility that the question might be undeletable; I'm not sure): just rewrite the original question, removing parts that ask about the future, and probably also the stuff about all great theories fitting into a small book one day, and make it harmonize with the nice answer that appeared. In other words, just make the best of the situation from where it stands now. (My own problem with the question is that I don't know what 'simplification' should mean. Cf. Rota's The Phenomenology of Mathematical Beauty in his Indiscrete Thoughts.) $\endgroup$ – Todd Trimble Feb 8 '13 at 15:47

A major simplification in algebraic number theory occurred in the beginning of the 20th century when Hensel explicitly introduced his $\mathfrak{p}$-adic numbers. Compare the original cumbersome definition of the Hilbert symbol with the modern definition using local fields.

Another major simplification occurred in the 40s when Chevalley put all these local fields together into the ring of adèles. Compare the original cumbersome formulation of class field theory in terms of ray class groups with the modern formulation in terms of idèle class characters.

A third major simplification occurred in the 50s when Tate rederived results of Hecke and others about the analytic properties (analytic continuation and functional equations) of certain $\zeta$- and $L$-functions using Fourier analysis on the said adèles and idèles.

As far as Wiles's proof of Fermat's Last Theorem is concerned, I believe that many of the ingredients (such as modularity lifting theorems) have since been vastly simplified and generalised. You should ask the experts, or a more specific question here.

So the trend towards greater simplicity and generality will continue, but there is no reason to believe that the proof will one day fit into a small volume accessible to undergraduates. Is there such a proof of the Kronecker-Weber theorem, which is now more than a hundred years old ?

Addendum I didn't want to create the impression that simplifications have not been made in recent times. Consider for example the local-to-global principle for the existence of rational points. Various people (Lind, Reichardt, Selmer, Cassels, Swinnerton-Dyer, ...) found examples of the failure of this principle, but these examples could be understood from a unified and simplified perspective only after Manin (1970) introduced his obstruction based on the Brauer group.

Since then, Skorobogatov (1999) has found examples of the failure of the local-to-global principle which cannot be accounted for by the Manin obstruction. I get the impression that such examples are beginning to be understood from a general point of view only now, and that the theory of étale homotopy is being used to find a whole hierarchy of obstructions; see for example Homotopy Obstructions to Rational Points by Yonatan Harpaz and Tomer M. Schlank (http://arxiv.org/abs/1110.0164).

It might be objected that the introduction of such a high-level theory into what was initially an elementary question cannot be called a simplification. However, most mathematicians would concur that it is indeed a simplification when a general theory serves to illustrate various disparate phenomena. This simplification has not yet been fully worked out.


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