# What is the “reason” for modularity results?

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!)

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Bauer's theorem says a Galois extension of Q is characterized by the prime numbers which split completely in it. (There is a similar statement for any Galois extension of a fixed number field.) This had been conjectured earlier by Kronecker, who thought of it as a number-theoretic boundary value theorem. It is natural to ask, once we have this theorem, if there is a concise way of describing such sets of primes. Class field theory does so for such sets corresponding to abelian extensions. See mathoverflow.net/questions/11747/… for non-abelian examples. –  KConrad Sep 13 '11 at 17:53
Dear James, It doesn't strike me as at all accurate to say that class field theory grew out of generalizations of Fourier analysis. Rather, it (largely) grew out of Kummer's work on the arithmetic of cyclotomic fields and higher reciprocity laws, and the earlier work of Gauss, Eisenstein (and maybe others) on which this was based, and the attempts of number theorists in the second half of the 19th century/early 20th century (Kronecker, Weber, Hilbert) to understand and systematize this work. Regards, Matthew –  Emerton Sep 13 '11 at 21:29
Please also look the first letter by Serre at smf.emath.fr/Publications/Gazette/2002/91/smf_gazette_91_55-57.pdf, esp. the bottom of the first page and the first column of the second one. –  KConrad Sep 13 '11 at 23:06
Maybe the sense in which class field theory is connected with a generalization of Fourier analysis is the role of Dirichlet characters (and cyclotomic fields) in explaining class field theory over Q via Dirichlet L-functions. Dirichlet characters on (Z/NZ)* are analogous to the functions e^{2pi*inx} on R which are central to ordinary Fourier analysis. [continued] –  KConrad Sep 13 '11 at 23:12
At the start of Chapter 8 of "From Fermat to Minkowski" there is quoted a letter from Jacobi where he writes "Dirichlet created a new part of mathematics, the application of those infinite series which Fourier has introduced in the theory of heat to the exploration of the properties of the prime numbers." This is an oblique reference to the analogue between classical Fourier series and the expansion of the characteristic function of a point in (Z/NZ)* into a finite linear combination of Dirichlet characters mod N. Do a Google search on Fermat Minkowski Dirichlet heat to see the book excerpt. –  KConrad Sep 13 '11 at 23:16

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.

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Regarding historic developpment the following question is somewhat related and perhaps of interest mathoverflow.net/questions/61959/taniyama-original-conjecture –  quid Sep 13 '11 at 19:43
Joel, you said: "Actually, what Weil's result proves is that those nice behavior is essentially the same thing as being modular." Would it be fair to say that one can deduce the definition of a cuspidal representation by having its L-function have analytic continuation and a functional equation is some way that I can't make precise but others (you?) can? –  James D. Taylor Sep 13 '11 at 19:58
James, automorphic $L$-functions for cusp forms on $GL_2$ are essentially Mellin transforms of cusp forms, so you could recover the cusp form by taking the inverse Mellin transform of the $L$-function (unless the cusp form is unramified everywhere you'd also need to worry about twists of the $L$-function). See Proposition 1.5.1 and Theorem 1.5.1 of Bump's book, or Theorem 7.3 of Iwaniec's Topics in Classical etc. For $GL_n$, inverting the integral requires the notion of an automorphic form on $GL_{n-1}$, so you could bootstrap it . . . –  B R Sep 14 '11 at 2:07

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.

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Dear James, I don't understand what you're asking for exactly. The notion of automorphic $L$-function was invented by Langlands in full generality, but it builds on an earlier tradition going back to Hecke (for the case $n = 2$) and Dirichlet (for the case $n = 1$). The notion of motivic $L$-functions also goes back a long way, and the two notions have been intertwined more-or-less continually throughout their development. As for the incentive to define automorphic forms, this is a subject with its own very long and detailed history, going back (at least) to the theory of elliptic ... –  Emerton Sep 13 '11 at 19:24
In summary, the history and motivations are complex and subtle. But nevertheless, the role of the converse theorems is key: e.g. if we look at Lafforgue's work, which settles the Langlands conjectures for $GL_n$ over function fields, they play a fundamental role there. Regards, Matthew –  Emerton Sep 13 '11 at 20:07
... other traditions feeding into the modern theory of automorphic forms, too. I believe that Maass was motivated to introduce his Maass forms in response to Hecke's theory relating Grossencharacters for imag. quad. fields to CM modular forms; Maass introduced automorphic forms that can play the same role for real quad. fields. I think that Selberg was motivated by Maass's papers to then study the spectrum of the Laplacian on modular curves, which led him to develop his trace formula, and, along the way, to effect the analytic continuation of Eisenstein series. It was generalizing this ... –  Emerton Sep 13 '11 at 21:11
... result to arbitrary groups that then led Langlands to discover general automorphic $L$-functions (see his book Euler products). From the beginning of the theory of modular forms, theta series (generating functions of quadratic forms) had played a key role, and Siegel's work on more general automorphic forms was aimed at, among other things, generalizing this theory. It was Tamagawa (I think) who saw how to phrase some of Siegel's main results in terms of properties of the adelic quotient $G(\mathbb Q)\backslash G(\mathbb A)$, and he (and then Weil in his book ... –  Emerton Sep 13 '11 at 21:14
... Adeles and algebraic groups) are thus responsible for introducing adelic groups into the subject (and for a reason not directly related to the theory of Hecke operators). To quote Langlands on the subject of automorphic forms: It is a deeper subject than I appreciated and, I begin to suspect, deeper than anyone yet appreciates. To see it whole is certainly a daunting, for the moment even impossible, task. Regards, –  Emerton Sep 13 '11 at 21:18

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.

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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.)

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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).

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Being a logician myself, I can’t recall anyone equating number theory with provability in Peano arithmetic (or any other formal theory of arithmetic for that matter). That sounds like a curious misconception. Could you be more specific? –  Emil Jeřábek Nov 15 '11 at 11:38
This link uses it in that kind of fashion: mathworld.wolfram.com/GoedelsFirstIncompletenessTheorem.html Of course the link there is to number theory as "the higher arithmetic" instead. While I can't prove that a logician wrote that text, it's what I meant. –  Charles Matthews Nov 16 '11 at 16:37
Well, neither Hofstadter nor Wolfram is a logician, actually. Anyway, you should read “a subset of” before any occurrence of “number theory” on that page (as is blatantly obvious from the Presburger arithmetic example: no one in their right mind would claim that all of number theory can be formulated in a system whose expressive power is limited to Boolean combinations of linear inequalities with integer coefficients and congruences). It is not an explanation of what number theory is in terms of a formal system, but vice versa. –  Emil Jeřábek Nov 18 '11 at 17:18
Shrug. You can treat my comment as a straw man if you insist. It doesn't mention "provability" at all. If you want a paraphrase of the whole thought, it would be that the hierarchy of results that matters to the mainstream tradition of number theory is no kind of logical hierarchy. –  Charles Matthews Nov 18 '11 at 19:39