## 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/

#### 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of mine so I think I'd better lie low :-) But they are talking a lot about the paper with the assumption of a lot of unpublished work in it (all of which is of course almost certainly fine, and true, and provable -- and also unpublished).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: May 09 2021 at 10:11 UTC