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davidac897 comments on "Why does representation theory encode all information about a modular form? (and some meta thoughts)" (21173)

davidac897 — Mon, 07 Jan 2013 03:20:42 -0800

Actually, I think I might wait a little while to post, until I know more. I think that then, even if I don't know the answer, I'll be better able to ask a question. I.e., if I understand how representation theory encodes modular forms, I'll be able to ask why, at least historically, one might have thought that representation theory wouldn't lose anything.

(And I'm going to learn more since I plan to attend/take a course on automorphic forms this semester...)

davidac897 comments on "Why does representation theory encode all information about a modular form? (and some meta thoughts)" (20667)

davidac897 — Fri, 07 Dec 2012 06:15:17 -0800

Okay great, I'll post it this weekend when I get a chance.

Marc Palm comments on "Why does representation theory encode all information about a modular form? (and some meta thoughts)" (20666)

Marc Palm — Fri, 07 Dec 2012 04:11:06 -0800

You have asked a similiar question here: http://mathoverflow.net/questions/35629/how-is-representation-theory-used-in-modular-automorphic-forms

But I think two years later, your question has become more concrete. So I think it's totally appropriate to ask an additional question. I'd love to see more big picture questions on MO again.

Mariano comments on "Why does representation theory encode all information about a modular form? (and some meta thoughts)" (20661)

Mariano — Thu, 06 Dec 2012 16:21:59 -0800

(We should somehow convince Matthew to come back!)

Will Jagy comments on "Why does representation theory encode all information about a modular form? (and some meta thoughts)" (20660)

Will Jagy — Thu, 06 Dec 2012 09:08:05 -0800

In summary, if you don't post the question on Main you are a wuss.

Also, you have those two years of experience of MO itself; that is, you could likely now write a really good version of your Fourier question.

As a final note, you probably want answers from Emerton and, if possible, Buzzard. For the former case, you might post here and then on MSE a week later.

Qiaochu Yuan comments on "Why does representation theory encode all information about a modular form? (and some meta thoughts)" (20658)

Qiaochu Yuan — Thu, 06 Dec 2012 00:39:45 -0800

I agree with François; this seems like a fine question.

François G. Dorais comments on "Why does representation theory encode all information about a modular form? (and some meta thoughts)" (20651)

François G. Dorais — Tue, 04 Dec 2012 16:33:40 -0800

I just gave a talk at The National Academies in Washington where I talked about the documentary value of MathOverflow. I argued that this is precisely the type of content that make MathOverflow worthwhile. The folklore of mathematics: things you should know if you want to work on something but can't learn from reading books or articles. Things that separate people "in the know" from the others...

So, yes, you should ask that question. It's clear that this knowledge is missing from the literature and needs to be out there somewhere!

See also this video of Anton speaking at the Open Science Summit 2011: http://fora.tv/2011/10/22/This_Year_in_Open_Science_Highlights_From_2011#chapter_07 Especially the bit starting at 1:30:55 with "Oh, this is a good one..."

davidac897 comments on "Why does representation theory encode all information about a modular form? (and some meta thoughts)" (20650)

davidac897 — Tue, 04 Dec 2012 14:16:45 -0800

Coming off of this question http://mathoverflow.net/questions/37021/is-fourier-analysis-a-special-case-of-representation-theory-or-an-analogue, I'm thinking of asking the following separate question:

(Aside: This leads into a more general question of mine about the use of representation theory as a generalization of modular forms. My question is the following: I understand that a classical Hecke eigenform (of some level N) can be viewed as an element of L2(GL2(Q) GL2(AQ) which corresponds to a subrepresentation. But what I don't get is why the representation tells you everything you would have wanted to know about the classical modular form. A representation is nothing more than a vector space with an action of a group! So how does this encode the information about the modular form?)

I'm thinking of this in part because Yemon Choi wrote in a very recent comment that he thinks it would be a good question. However, I'm already wondering whether I should have asked the original question in the first place (something I did more than 2 years ago, when I knew less, and when MO was much different, I believe). So that's one reason I'm not even sure about asking this new question.

I think this new question, while certainly appropriate for MO two years ago, fits into the following category. It is perfectly clear to someone who understands the representation-theoretic point of view of modular forms quite well. However, it might be a legitimate question (and this depends on what the nature of MO is) for the following reason: the answer to the question is NOT "just go read a book on automorphic representations." Why? Because the question speaks to a confusion that I believe a number of students first learning the subject face, one that will be hard to remedy by reading books on automorphic representations, which just give the definitions without explaining why we generalize them in the way we do.

Then again, shouldn't the answer just be "read Kudla's Chapter 7 of Introduction to the Langlands Program" (though he doesn't address my question explicitly, except arguably at the bottom of p.147).