Zulip Chat Archive

Stream: maths

Topic: quanta on langlands

Johan Commelin (Apr 08 2020 at 14:49):

If people want a nice introduction to the maths that Kevin is always complaining about: https://www.quantamagazine.org/amazing-math-bridge-extended-beyond-fermats-last-theorem-20200406/

Patrick Massot (Apr 08 2020 at 15:02):

Now, who will add comments pointing out none of this is actually proved?

Kevin Buzzard (Apr 08 2020 at 15:33):

Essentially every mathematician in that article is a friend or colleague of mine so I think I'd better lie low :-) But they are talking a lot about the paper with the assumption of a lot of unpublished work in it (all of which is of course almost certainly fine, and true, and provable -- and also unpublished).

Andrew Ashworth (Apr 08 2020 at 15:48):

I think I will stick to the quanta magazine article in order to understand this. The first result after I looked up some terminology in the article was a paper with this abstract: "the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. " whew

Kevin Buzzard (Apr 08 2020 at 15:56):

That's a pretty important result. It's not hard for Shimura curves but the general case is much deeper.

Reid Barton (Apr 08 2020 at 16:38):

Kevin are you familiar at all with this "topological automorphic forms" stuff?

Reid Barton (Apr 08 2020 at 16:43):

oh hmm, I think I tried looking at this 10-author paper when it came out to see whether it was useful for TAF and it seemed Hard

Kevin Buzzard (Apr 08 2020 at 17:19):

I know epsilon about topological modular forms from e.g. conversations with @Neil Strickland about 20 years ago :-)

Reid Barton (Apr 08 2020 at 20:22):

So the computation of the homotopy groups of tmf starts with some etale cohomology $H^i(\overline{\mathcal{M}_\mathrm{ell}}, \omega^{\otimes j})$ . There's supposed to be a "higher chromatic level" version of this story where you start from a Shimura variety. Is this the same kind of cohomology that the paper is talking about?

Kevin Buzzard (Apr 08 2020 at 20:25):

The unitary Shimura varieties are moduli spaces for certain abelian varieties, and they're compact. The sheaf $\omega$ on $\overline{\mathcal{M}_{\mathrm{ell}}}$ is usually a coherent sheaf. Caraiani--Scholze are considering etale cohomology with constant coefficients $\mathbb{Z}/p\mathbb{Z}$ .

Patrick Massot (Apr 08 2020 at 20:56):

I don't know why but the style of this sequence of two messages brought me a flash of second world war coded messages on BBC for French people that were still fighting. I don't know if you see what I mean.

Patrick Massot (Apr 08 2020 at 20:57):

I think it's the combination of completely cryptic meaning, the fact it's not clear Kevin is really answering Reid, and the sequence of short sentences.

Patrick Massot (Apr 08 2020 at 20:59):

https://www.warhistoryonline.com/world-war-ii/the-bbcs-messages.html

Reid Barton (Apr 08 2020 at 21:56):

Kevin Buzzard said:

Caraiani--Scholze are considering etale cohomology with constant coefficients $\mathbb{Z}/p\mathbb{Z}$ .

OK that was my question, thanks.

Reid Barton (Apr 08 2020 at 21:57):

Haha, now I see it's good that I confirmed this for the onlookers :upside_down:

Last updated: Dec 20 2023 at 11:08 UTC