A New Approach to Modularity

Posted on February 28, 2024 by woit

I was assuming that Peter Scholze’s Emmy Noether lectures at the IAS would be the big news about advances in the Langlands program this coming week, but an anonymous correspondent just sent me this link. Tomorrow Andrew Wiles will be giving a talk in Oxford on “A New Approach to Modularity”, with abstract:

In the 1960’s Langlands proposed a generalisation of Class Field Theory. I will review this and describe a new approach using the trace formua as well as some analytic arguments reminiscent of those used in the classical case. In more concrete terms the problem is to prove general modularity theorems, and I will explain the progress I have made on this problem.

I’m curious to hear from anyone who knows what this is about or can report tomorrow after the talk. Wiles does have a certain track record of unveiling unexpected huge progress in a talk like this…

Update: Well, the talk should be over now. Sure someone who was there can let us know what happened?

Update: There’s a tweet with some photos.


Update: The first of Scholze’s talks is available at the IAS youtube channel.

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4 Responses to A New Approach to Modularity

  1. GG says:
    February 29, 2024 at 3:26 pm

    I was at the seminar but I am reluctant to comment as I am not a number theorist. To my understanding, Wiles showed that proving the 2006 Michel-Venketash mixing conjecture would imply modularity. I can provide the photographs of the whiteboard if interested.

  2. Peter Woit says:
    February 29, 2024 at 6:11 pm

    GG,
    Thanks! That’s consistent with third-hand reports I’ve heard from another source, that this is about new cases of modularity, but conditional on cases of the mixing conjecture. Better I not try and provide a garbled fourth-hand version, I hope someone else who was there can provide more details, ideally a set of notes. Not clear to me if my hosting photographs of a talk would be a good idea if the speaker/sponsoring institution don’t approve.

  3. David Roberts says:
    February 29, 2024 at 7:31 pm

    FWIW, the comment on the first photo in the Twitter post saying “proved by” is I think referring to the Annals paper Joint equidistribution of CM points by Ilya Khayutin. This paper proves a particular special case of the Mixing Conjecture over the rationals.

  4. Z Y says:
    March 4, 2024 at 1:35 pm

    Isn’t the mixing conjecture already proved if we assume GRH? That would make for a very interesting turn of events

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