Timeline for What is the "reason" for modularity results?
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| Sep 13, 2011 at 21:18 | comment | added | Emerton | ... 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, | |
| Sep 13, 2011 at 21:14 | comment | added | Emerton | ... 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 ... | |
| Sep 13, 2011 at 21:11 | comment | added | Emerton | ... 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 ... | |
| Sep 13, 2011 at 21:09 | comment | added | Emerton | ... elliptic integrals and elliptic functions. The generalization to automorphic forms took place over a long period of time, and was placed in a representation theoretic context by Gelfand and his school (as far as I know): they shifted the focus from functions on $G/K$ satisfying an automorphy condition under the action of $\Gamma$ to functions on $\Gamma \backslash G$, which then admit a $G$-action. (Here $G$ is a real semisimple group, say.) From this point of view, the interpretation of Hecke operators in terms of an adelic group action is not so remote. But there are lots of ... | |
| Sep 13, 2011 at 21:04 | comment | added | Emerton | Dear James, The notion of automorphic representation (as an irreducible representation of an adelic group) is a generalization of the notion of Hecke eigenform. This aspect of the theory of automorphic forms (i.e. the theory of Hecke operators and their simultaneous eigenvectors) was initiated by Hecke, as a means of understanding and generalizing Mordell's proof of Ramanujan's conjectured multiplicative relations for the $\tau$ function. The notion of automorphic form itself arose as a generalization of the notion of modular form. The latter arose (as I said) out of the study of ... | |
| Sep 13, 2011 at 20:57 | comment | added | Laie | It's perhaps a little late in the game, but my interpretation of the question is a bit different from what's been discussed so far. It certainly makes sense to refer to a long history of development to explain why certain things are believed to be true at present. It seems to me though that even if one knows all the history, it's a valid question whether today, with hindsight spanning 100+ years, one can come up with a structural intuition that would lead from motivic L-functions to automorphic L-functions even if one had never heard of the latter. | |
| Sep 13, 2011 at 20:40 | comment | added | James D. Taylor | I am going very much off topic here, but was the original intent for defining automorphic representations? | |
| Sep 13, 2011 at 20:07 | comment | added | Emerton | 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 | |
| Sep 13, 2011 at 20:04 | comment | added | Emerton | ... who defined automorphic $L$-functions in general, and saw directly the relationship between his functoriality conjecture and Artin's conjecture, and more broadly saw that his $L$-functions were candidates to be Hasse--Weil (i.e. motivic) $L$-functions; but there was also the work and ideas of Taniyama and Shimura about modularity of elliptic curves, Shimura's work on (what are now called) Shimura varieties (which gave another, previously unknown, link between arithmetic and automorphic forms), Serre's ideas about $2$-dimensional Galois representations attached to modular forms, and so on. | |
| Sep 13, 2011 at 20:02 | comment | added | Emerton | Dear James, The notion of automorphic representation developed independently of any concern with modularity or class field theory. As I said, it has its own history, arising ultimately from the theory of elliptic integrals. The realization that algebraic number theory and automorphic forms were related by (what we now call) modularity was something that evolved slowly, over a long period of the twentieth century. Even when it became concretely articulated, in the 60s and 70s, there were several strands of development feeding into it: of course there is the work and ideas of Langlands, ... | |
| Sep 13, 2011 at 19:55 | comment | added | James D. Taylor | This seems to tell me people have a better insight than me about why one should expect either CFT or the modularity theorem. (I myself would certainly not be able to come up with the notion of an automorphic representation by myself!) | |
| Sep 13, 2011 at 19:55 | comment | added | James D. Taylor | Dear Matthew, this is more similar to what I'm asking. Indeed, as far as I can tell, no one knows "why" Langlands is true. But we do know why CFT is true (and I'm sure there are people here, you among them, who can answer that much better than I can), and we know why modularity for elliptic curves is true, and it seems that the notion of an automorphic representation grew out of an understanding (that I don't have) for how to extrapolate the notion of Dirichlet characters, and the notion of modular forms, in such that a way that the vague "reason" for modularity conditions holds. | |
| Sep 13, 2011 at 19:25 | comment | added | Emerton | ... integrals from the 1700s. Regards, Matthew | |
| Sep 13, 2011 at 19:24 | comment | added | Emerton | 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 ... | |
| Sep 13, 2011 at 17:41 | comment | added | James D. Taylor | *"one should gave same" should be "one should give some". Sorry, new laptop. I'm still adjusting to the new keyboard. | |
| Sep 13, 2011 at 17:36 | comment | added | James D. Taylor | I understand the statements. I'm just not sure why to believe them. Each motive has an associated L-function. But then one should gave same vague incentive to define automorphic functions, and to define their L-functions. One way to give the incentive, and that's the way I was exposed to, is to say: there is a lot of evidence that it's true, and it would be nice if it is. I want a more intuitive incentive. Why should there be a notion of an automorphic L-function, and why should it be related to the L-functions of motives? | |
| Sep 13, 2011 at 17:30 | history | answered | Emerton | CC BY-SA 3.0 |