Timeline for Understand the $p$-adic local Langlands correspondence with examples
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 22, 2022 at 16:34 | vote | accept | Marsault Chabat | ||
| Dec 22, 2022 at 15:44 | comment | added | David Loeffler | Some people are never satisfied! "How do I know whether I should consider classical cohomology or completed cohomology" -- how do you know when you should use a screwdriver or a hammer? You use the tool that's appropriate to the problem you're trying to solve, that's how. (PS: You might want to look up the meaning of the word "cuspidal" in the context of smooth representation theory; it does not mean what you seem to think it means.) | |
| Dec 22, 2022 at 10:53 | comment | added | Marsault Chabat | (Sorry my comments are a bit long...) If I see things correctly then these two representations are $Hom(\rho,H)$ and $Hom(\rho,\tilde{H})$ (I think the second one as been constructed by Emerton) so how do you deal with that? And what is this second representation? I think that's why I was asking for examples, because in everyday life how do I know whether I should consider classical cohomology or completed cohomology (which is, you might know, something that gets happened in my research) | |
| Dec 22, 2022 at 10:44 | comment | added | Marsault Chabat | Thanks for this great answer David. Something I'm struggling with is local-global compatibility, in fact with your answer I know what's my problem (so thanks again for always helping on MO, it's really helpful). Here is what I am not comfortable with (I stay in the $\mathbf{Gl}_{2}$ case). Given a global Galois representation, we can associate (under assumption) a cuspidal representation. But know that there are two possibilities for the case "l=p", we can locally associate two representations, and therefore (with restricted tensor product) two global representations. | |
| Dec 20, 2022 at 21:19 | history | answered | David Loeffler | CC BY-SA 4.0 |