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Let $G$ be a simply connected semi-simple linear algebraic group over a global field $k$. The Hasse principle for algebraic groups states that the map $$H^1(k,G)\rightarrow\prod_vH^1(k_v,G)$$ is injective, where the $H^1(k,-)$ denotes Galois cohomology and the product is taken over all valuations.

For number fields, this was proven for $G$ with no factors of type $E_8$ by Kneser and Harder in the 1960s. It took 20 more years for the $E_8$ case to be settled, by Chernousov. On the other hand, I believe there is a uniform proof for all $G$ in the function field case (maybe due to Harder?)

My question is: Do we now have a uniform (i.e. not case-by-case) proof over number fields?

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    $\begingroup$ 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! $\endgroup$
    – nfdc23
    Commented Dec 9, 2016 at 0:23
  • $\begingroup$ @nfdc23: Thanks! If you write that as an answer, I'll accept it (so that this question no longer shows as unanswered.) $\endgroup$
    – dhy
    Commented Dec 9, 2016 at 13:10
  • $\begingroup$ @nfdc23 This always bothers me a little. Is the fact that the Hasse principle holds in this generality a coincidence? $\endgroup$
    – Will Sawin
    Commented Dec 9, 2016 at 14:10
  • $\begingroup$ @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. $\endgroup$
    – nfdc23
    Commented Dec 9, 2016 at 16:48
  • $\begingroup$ @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). $\endgroup$
    – nfdc23
    Commented Dec 9, 2016 at 16:50

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