Timeline for Examples of non-split algebraic groups
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2015 at 20:03 | answer | added | Jay Taylor | timeline score: 1 | |
| Feb 26, 2015 at 7:55 | answer | added | Kimball | timeline score: 5 | |
| Feb 14, 2014 at 14:03 | comment | added | Abhishek Parab | @user76758 - I was sloppy but I thought there is no confusion, since Humphreys' comment said minimally but I edited the question. Could you suggest a reference where I can find about unit groups of CSAs that you mention? (PS: $E\cap E'$ is ill-posed too :) ) | |
| Feb 14, 2014 at 13:54 | history | edited | Abhishek Parab | CC BY-SA 3.0 |
made more precise
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| Feb 14, 2014 at 13:42 | comment | added | Keerthi Madapusi | @user76758-Thanks for the corrections! | |
| Feb 14, 2014 at 5:27 | history | made wiki | Post Made Community Wiki by Ben Webster♦ | ||
| S Feb 14, 2014 at 4:50 | history | suggested | user76758 | CC BY-SA 3.0 |
The hypotheses and notations for $G$ in both questions were imprecise and have now been clarified.
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| Feb 14, 2014 at 4:42 | comment | added | user76758 | The first question is ill-posed: if $E, E'$ are finite (separable) extensions of $F$ and a connected semisimple $F$-group $G$ splits over $E$ and $E'$ then it generally doesn't split over $E \cap E'$. Unit groups of central simple algebras provide lots of examples over number fields (since the global splitting over a finite extension is controlled by local splitting over a finite set of places). So speaking of "the smallest such extension" doesn't quite make sense (aside from special cases like tori). | |
| Feb 14, 2014 at 4:21 | review | Suggested edits | |||
| S Feb 14, 2014 at 4:50 | |||||
| Feb 14, 2014 at 4:18 | comment | added | user76758 | @KeerthiMadapusiPera: Without a rational point, "connected" is not "geometrically connected" for schemes over a field. If $K$ is a nontrivial finite extension of $\mathbf{Q}$ then ${\rm{Spec}}(K)$ is a smooth connected $\mathbf{Q}$-scheme with no smooth affine integral model (over $\mathbf{Z}[1/N]$ for sufficiently divisible $N$, say) having all but finitely many fibers connected. Also, without reductivity the special and generic fibers of a smooth affine group with connected fibers over a dvr may have Borels of different dimensions (Bruhat-Tits group schemes!), so Lang's theorem isn't enough. | |
| Feb 14, 2014 at 2:20 | comment | added | Keerthi Madapusi | On the other hand, one doesn't really need reductiveness for this. Lang's theorem applies to any smooth connected algebraic group over a finite field and Hensel's lemma only needs smoothness. So all you really need is that $G$ has smooth connected fibers for almost all primes $p$. This is a much easier assertion and is true for smooth connected $\mathbb{Q}$-varieties in general. | |
| Feb 14, 2014 at 2:19 | comment | added | Keerthi Madapusi | Abhishek: Okay, I wasn't sure if it was a typo or not. As for the statement on reductiveness, one can assume that $G$ has a smooth, affine model over $\mathbb{Z}[1/n]$ for some suitable integer $n$. Now apply Corollary 2.6 from Exp. XIX of SGA 3. | |
| Feb 14, 2014 at 1:50 | history | edited | Abhishek Parab | CC BY-SA 3.0 |
added 4 characters in body
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| Feb 14, 2014 at 1:49 | comment | added | Abhishek Parab | Keerthi: Sorry for the error, I really meant $v \not\in S$. Also could you give a reference for the statement "For almost all p, G has a smooth reductive model over Z_p". Thanks. | |
| Feb 14, 2014 at 1:38 | comment | added | Keerthi Madapusi | If $S$ has an even number of elements, then an example is the group of units in a quaternion algebra $D$ that is non-split exactly at the places in $S$. | |
| Feb 14, 2014 at 1:36 | comment | added | Keerthi Madapusi | So the correct formulation of (2) is: Given a finite set of primes $S$, find an example of a group $G$ such that $S$ consists precisely of the primes at which $G$ is not quasi-split. | |
| Feb 14, 2014 at 1:35 | comment | added | Keerthi Madapusi | (2) is basically impossible: Any connected reductive group over $\mathbb{Q}$ will be quasi-split over almost all primes: The variety $\mathcal{B}$ of Borel sub-groups of any such group $G$ is smooth and projective. For almost all finite primes $p$, $G$ has a smooth reductive model over $\mathbb{Z}_p$. By Lang's theorem, combined with Hensel's lemma, $\mathcal{B}(\mathbb{Q}_p)$ is non-empty, for almost all $p$. | |
| Feb 14, 2014 at 1:15 | comment | added | Abhishek Parab | Thanks Prof. Humphreys, I edited the question - yes, in both cases. | |
| Feb 14, 2014 at 1:14 | history | edited | Abhishek Parab | CC BY-SA 3.0 |
added hypotheses based on Jim Humphreys' comment.
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| Feb 14, 2014 at 0:58 | comment | added | Jim Humphreys | Two comments: 1) It would help to specify whether you are looking for semisimple groups or other types. 2) In the first question, I suspect you want $E$ to be a minimal splitting field over $F$ (?) | |
| Feb 14, 2014 at 0:35 | comment | added | Keenan Kidwell | Once source of many examples is Weil restriction of scalars. For example, take a non-trivial finite Galois extension $E/F$ and a split torus $T$ over $E$. Then $\mathrm{Res}_{E/F}(T)$ is a non-split torus over $F$ that is split by $E$ (it's non-split because its character lattice has a non-trivial Galois action). | |
| Feb 14, 2014 at 0:28 | history | asked | Abhishek Parab | CC BY-SA 3.0 |