Timeline for Uniform proof of Hasse principle for algebraic groups?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 9, 2016 at 16:50 | comment | added | nfdc23 | @dhy: I prefer to leave it as a comment since I've never read the proof in full detail and someone who has more familiarity with those details might be able to provide a more erudite perspective later on. If in due time nothing else is forthcoming, feel free to copy my comments to make an official answer (the point stuff is meaningless to me). | |
| Dec 9, 2016 at 16:48 | comment | added | nfdc23 | @WillSawin: I agree that a case-heavy proof is not ideal, but nothing in mathematics is a coincidence. Maybe a uniform proof will be found when someone has a better idea for how to handle archimedean places. The vanishing of ${\rm{H}}^1(k,G)$ for non-archimedean local $k$ and simply connected semisimple $k$-groups $G$ was proved by Kneser by extensive case-checking (and only in char. 0), but later the uniform proof was found, so one can hope! That is part of the charm of semisimple groups: we can make progress by detailed study of special cases, and maybe later a uniform method can be found. | |
| Dec 9, 2016 at 14:10 | comment | added | Will Sawin | @nfdc23 This always bothers me a little. Is the fact that the Hasse principle holds in this generality a coincidence? | |
| Dec 9, 2016 at 13:10 | comment | added | dhy | @nfdc23: Thanks! If you write that as an answer, I'll accept it (so that this question no longer shows as unanswered.) | |
| Dec 9, 2016 at 0:23 | comment | added | nfdc23 | No. Even setting aside the E$_8$ headache, for the classical groups there's a lot of case-checking (e.g., the Hasse-Minkowski theorem on quadratic forms is used in the proof for spin groups, if I remember correctly). The book by Platanov and Rapinchuk gives the entire argument for number fields, and no "better" proof is known (as far as I'm aware). On the bright side, the vanishing of $H^1(k_v,G)$ for finite $v$ has a uniform (but deep) proof by combining work of Bruhat-Tits & Steinberg. That also highlights why the number field case is so much more thorny: real places! | |
| Dec 8, 2016 at 22:44 | history | asked | dhy | CC BY-SA 3.0 |