What is the "reason" for modularity results?

Question

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? .

I am aware of the statements of class field theory and the modularity theorem, as well as far-reaching generalizations that have to do with the conjectural Langlands group and motives. But on a basic level, I just don't understand why such statements should be true, other than that there is a lot of evidence that they are.

What is the philosophical impetus behind modularity results?

When I read about number theory, I can very easily understand the intuition behind ramification of primes (because the intuition is geometric), but as soon as we start talking about splitting of primes, and are therefore in the realm of modularity results, I lose all intuition of why things should be true (even though I can read and understand the results as an undergraduate can -- agreeing line by line).

An example of an answer for CFT can be the following thing I've heard, but was somewhat unsatisfied with because I didn't fully understand it: that it grew out of generalizations of Fourier analysis. (if you also think of this as the reason it's true, and can expatiate -- do!)

Answer 1

Serre relates that early in the sixties, at the IAS, Shimura asked him: "is it true that the L-function of every elliptic curve over $\mathbb{Q}$ is modular" and he replied "why would it be so?". Serre goes on explaining that a question like that is of little value (not even worth of being called a conjecture) if not motivated by either strong evidence or philosophical reasons. It is even an important point in his argument that the name of Weil really belongs in the name of the Shimura-Tanyama-Weil conjectue, as Weil provided both (according to Serre): (1) the observation that there was no Elliptic curve over $\mathbb{Q}$ of small conductor, related to the absence of cuspidal modular forms of weight 2 of small level (with the same precise sense of "small"), and (2) as mentioned by Emerton, Weil's converse theorem, that if the L-function of an elliptic curve say, with enough of its twists, satisfy the basic behavior we have come to expect from all kind of L-functions sine Riemann (analytic continuation and functional equation) then they are modular. Actually, what Weil's result proves is that those nice behavior is essentially the same thing as being modular.

I said the above because you're in the same state of mind than Serre was (in the early sixties). But he became satisfied with the conjecture in the early seventies, after Weil's work on it. Right now I am not sure we have made so much progress in understanding why philosophically those higher reciprocity law should be true. We have a lot of evidence, provided by the huge numbers of particular cases and analog problems (e.g. the function field case solved by Lafforgue) we have solved. But the main philosophical reasons we believe such things should hold are the same they were then, namely the ones Emerton has recalled, Weil's converse theorems.

We might get a deeper understanding of why those things hold when we eventually prove everything we want in the Langlands program. For some (cf. some philosophical texts by Michael Harris on his webpage), this is even the main interest in proving things in mathematics. But even that is perhaps too optimistic.

Answer 2

The $L$-functions of motives have (conjecturally) the same analytic properties as do the $L$-functions of automorhpic representations. Converse theorems suggest that these $L$-functions are then necessarily automorphic $L$-functions.

Answer 3

I don't think it is too much an overstatement to say that nobody has any idea why the most general conceivable form of the modularity conjectures-say a combination of Langlands program and the Fontaine-Mazur conjecture-should be true. As in the case of conjectures on special values of $L$-function, the most one could probably say is that their inner consistency is absolutely impressive so that in some sense, they feel too good to be not true.

That said, not all is lost, I think, in your quest to get a philosophical understanding of this topic, especially if you set yourself a more modest goal at first. Because why things should be true is probably inherently subjective, I will only offer my personal experience with modularity results for $\operatorname{GL}_2$. I think that the first significative experience I had towards a modicum of understanding of the deep reasons why these should be true was to realize how utterly surprising they were. The more I understood about abstract universal deformation rings and the less I could see why they should be Hecke algebras. The Taylor-Wiles method, I still don't claim any deep or philosophical understanding of, but this is mostly because I never read closely enough the literature. Some papers from Kisin, for instance, do explain that there seems to be a trade-off between how singular a deformation ring can be and the local behaviour of the Galois representation at p. The next big step for me was to read carefully Taylor's paper on potential modularity. This paper makes it very clear that modularity results are very amenable to bootstrapping: prove one, and you may get a lot for free. So to recap: modularity results should be true because (in certain settings), one can reduce them to much simpler modularity results and then get rid of the singularities of the universal deformation ring (provided you have what you need to do so).

Not very philosophical perhaps but since the name of Weil has appeared in the answers by Emerton and Joël, let me conclude by quoting his magnificent (if slightly depressing) words on finding philosophical understanding of mathematical theories.

Rien n’est plus fécond, tous les mathématiciens le savent, que ces obscures analogies, ces troubles reflets d’une théorie à une autre, ces furtives caresses, ces brouilleries inexplicables ; rien aussi ne donne plus de plaisir au chercheur. Un jour vient où l’illusion se dissipe ; le pressentiment se change en certitude ; les théories jumelles révèlent leur source commune avant de disparaître ; comme l’enseigne la Gita, on atteint à la connaissance et à l’indifférence en même temps. La métaphysique est devenue mathématique, prête à former la matière d’un traité dont la beauté froide ne saurait plus nous émouvoir.

Answer 4

No one knows. Or at least, no one knows why we know.

I do not mean this flippantly. If you study the proofs of e.g. local and global class field theory (global especially), they use over and over again all kinds of tricks in the yoga of group cohomology to reduce everything down to understanding cases we can do by hand, like cyclotomic extensions of $\mathbf{Q}$ and Kummer extensions more generally. But I think this style of proof is very far from a satisfying "why", and I have heard the same opinion from other people (Tate, Rosen). The most satisfying proof in class field theory for me is the Lubin-Tate construction of totally ramified extensions of local fields, precisely because you can make canonical choices and it's reasonably explicit.

Likewise, the Taylor-Wiles method, while an extremely beautiful and powerful idea, is ultimately unsatisfying (to me) as a reason for why Hecke algebras should match deformation rings. If you read the "context-free version" in Section 2 of Diamond's paper "The Taylor-Wiles method and multiplicity one", you'll notice that a subsequence of (quotients of) the auxiliary modules $H_n$ is chosen using compactness. Compactness! Roughly speaking, this corresponds to controlling the relation between deformation rings and Hecke rings at some fixed level by smooshing together a bunch of modular forms at sporadic higher levels, which are chosen in some gratuitously noncanonical way.

I agree with Emerton that converse theorems provide an extremely persuasive reason for believing modularity results. But I also believe that the ultimately "correct" method of proof has not surfaced yet, and who knows how many decades or centuries until it does?

(Let me stress that these are simply my opinions, and nothing more. But they are not unconsidered.)

Answer 5

You'll find logicians who'll explain to you that "number theory" is some collection of theorems to do with Peano arithmetic. Almost exactly wrong, if designed to bring on an existential crisis in number theorists (is no one theorem more significant than another?) The counter-attack begins with the assertion that there are major planks of number theory, even if G. H. Hardy's conception of their "depth" isn't really tenable. But this is more a matter of "faith" than anything else. If people believe that there is a complete, detailed theory of Hasse-Weil L-functions to be had eventually, rather than there being inexplicable "junk" in that theory, I think they are usually appealing to some sort of traditional thinking, rather than the existence of an ultimate top-down theory (though there is a minority view, cf. comments of Weil in the introduction to Basis Number Theory).