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My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be compact (with respect to the $p$-adic topology)?

I more or less understand that if $G=SL_1(D)$ where $D$ is a division ring of dimension $n^2$ and of order $n$ in the Brauer group over $Q_p$ then $G(Q_p)$ is compact. I also understand that $Spin(q)$ when $q$ has more than $5$ variable cannot be compact over local non-archimedean fields.

Are there more examples? I think that one can classify all the examples but I don't manage to do it or find a reference for it.

Can someone outline a route to take in order to understand it thoroughly (for someone with basic understanding of algebraic groups and Galois cohomology)?

Thanks a lot!

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4 Answers 4

up vote 7 down vote accepted

Here are some remarks on the answers of Charles Matthews, Kevin Buzzard, and Victor Protsak. For justification of Kevin Buzzard's claim, see G. Prasad, "An elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits" from Bull. Soc. Math. France 110 (1982), pp. 197--202, for an incredibly elegant and short proof that over any henselian valued field $F$, a connected reductive $F$-group $G$ is $F$-anisotropic if and only if $G(F)$ is "bounded" (a property defined in terms of a choice of closed immersion of $G$ into an affine space over $F$, the choice of which doesn't matter; this is meaningful for any affine $F$-scheme of finite type and equivalent to compactness when $F$ is locally compact). Platanov-Rapinchuk has a universal assumption that all fields of characteristic 0 (except when they're finite), so unfortunately that reference is insufficient for uniform arguments over all non-archimedean local fields. I suppose (near?-)circularity (suggested by Victor Protsak) is a more serious issue. :)

There remains the matter of determining, for locally compact non-archimedean $F$ and connected reductive $F$-groups, precisely when the $F$-anisotropic case can actually occur. As Victor Protsak mentioned, via Bruhat-Tits theory one sees that for connected semisimple $F$-groups which are absolutely simple and simply connected over a non-archimedean local field $F$ (i.e., $G$ a simply connected $F$-form of a Chevalley group), such forms never exist away from type A, and in type A the $F$-anisotropic examples are precisely the $F$-groups of norm-1 units of central simple algebras over $F$. (Note the contrast with the case $F = \mathbb{R}$, for which there's always a "compact form" of any Chevalley type.)

Let me now briefly explain why this handles the general connected reductive case, by a standard kind of argument with central isogenies and separable Weil restriction. (This is explained also in the article [2] of Tits referenced in Charles Matthews' answer.) If $f:G' \rightarrow G$ is a (possibly inseparable) central $F$-isogeny between connected reductive $F$-groups then the preimage of an $F$-torus of $G$ is an $F$-torus of $G'$ (since maximal tori in $G'$ are their own functorial centralizers, so $\ker f$ is of multiplicative type, nothing funny happens when $f$ is not separable). Since $F$-anisotropcity of an $F$-torus is invariant under $F$-isogenies (as we see using the $F$-rational character group, or more direct arguments), it follows that $G$ is $F$-anisotropic if and only if $G'$ is. (This argument has the advantage of working over any field $F$, in contrast with a direct attack on the topology of rational points by using finiteness theorems for Galois cohomology of connected reductive groups.) Thus, by considering an arbitrary connected reductive $F$-group $G$ and letting $G'$ denote the product of its maximal central $F$-torus and the simply connected central cover of the derived group $D(G)$, we see that the problem comes down to the simply connected case.

But in the simply connected semisimple case, the general structure of connected semisimple groups over fields (in terms of central isogenous quotient of direct product of commuting simple "factors") implies that $G$ is uniquely a direct product of commuting $F$-simple connected semisimple $F$-groups, each of which is simply connected, so we may assume $G$ is $F$-simple. Then by an elementary result of Borel and Tits (6.21 in "Groupes reductif", IHES), $G = {\rm{Res}}_ {F'/F}(G')$ for a finite separable extension $F'/F$ and a connected semisimple $F'$-group $G'$ that is absolutely simple and simply connected. By the good behavior of Weil restriction with respect to the formation of the topological group of rational points, it follows that the equality ${\rm{Res}}_ {F'/F}(G')(F) = G'(F')$ of abstract groups is a homeomorphism, so we can replace $(G,F)$ with $(G',F')$ to reduce to the case when $G$ is also absolutely simple, the case addressed by Victor Protsak above. (A more algebraic argument with Galois descent relating maximal $F'$-tori in $G'$ and maximal $F$-tori in its Weil restriction to $F$ shows the equivalence of anisotropicity for $G'$ and its Weil restriction through the finite separable $F'/F$, where $F$ can be taken to be any field at all.)

Conclusion: for non-archimedean local $F$, the $F$-anisotropic connected reductive $F$-groups are precisely the central quotients of products of an $F$-anisotropic torus and groups of norm-1 units of central division algebras over finite separable extensions of $F$.

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Very thorough explanation, thank you! By the way, Margulis showed me an elegant short proof of the equivalence between anisotropic and bounded (I hadn't been able to find one in Platonov-Rapinchuk and so asked him), I am not sure if it's the same one you've mentioned. Just a small personal request: can you, please, use my first name (or my ID, i.e. first+last name) in the future? –  Victor Protsak Jun 16 '10 at 13:15
    
Victor, I have updated how I use names in the answer. –  Boyarsky Jun 16 '10 at 17:05
    
Thanks a lot! For the clear explanation and references! I'll probably post some more questions when I'll try to digest it... This is my first time around and I feel that I won the lottery! –  Menny Jun 16 '10 at 18:58

I'm no expert, but I think the theorem is that (for $G$ reductive over a local field $F$) $G(F)$ is compact iff $G$ is $F$-anisotropic, that is, every $F$-torus in $G$ (or equivalently every maximal $F$-torus) has the property that its only $F$-character is the trivial character. This will be somewhere in Platonov-Rapinchuk but I don't have it to hand so can't give a more precise reference :-( . So, for example, in your $SL_1(D)$ example (presumably this means the norm 1 elements of $D$?) the maximal tori in $D^*$ are coming from fields $E/F$ embedding into $D$, so the maximal tori in $G$ are the norm 1 elements of $E^*$ and these have no rational characters (the norm is the only rational character of $E^*$ and you've just removed it).

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PS aah wonderful, my hesitant answer is vindicated by the man who taught me what a Cauchy sequence was :-) Hey Charles! Remember me? Thought not. –  Kevin Buzzard Jun 16 '10 at 9:06
    
Name familiar, face escapes me? That would be a long time ago. –  Charles Matthews Jun 16 '10 at 9:12
    
Indeed! grin –  Kevin Buzzard Jun 16 '10 at 9:18

See http://eom.springer.de/a/a012530.htm and the concept of anisotropic group. As it says there, the classification involved is pretty much the classification of semisimple groups.

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The overview (in English) by Tits of the classification in the 1965 Boulder conference proceedings is freely available at the AMS online books page: e-math.ams.org/publications/online-books/pspum9-index But as Victor observes, it can be troublesome to find complete and accessible details for your specific purpose without tackling the Bruhat-Tits papers, etc. –  Jim Humphreys Jun 16 '10 at 11:54
    
There is also a later survey by Tits in the Corvallis volume (PSPM 33), also available on the AMS website. –  Victor Protsak Jun 16 '10 at 13:17

As Charles said, for a semisimple group over a local field compact $\iff$ anisotropic.

I confirm that Chapter 6 of Platonov-Rapinchuk contains the proof of (a) the vanishing of the Galois cohomology $H^1(K, G)$ for a connected simply connected semisimple group $G$ over a local field $K$ and (b) Classification theorem: if $G$ is anisotropic simple connected simply connected then $G=SL_1(D).$ However, the proofs are neither self-contained nor transparent and rely on an argument close to circular. The original proofs due to Kneser (Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern. II. Math. Z. 89 1965 250--272, MR0188219) also use case-by-case arguments. A uniform proof follows from Bruhat-Tits theory, but if I understand it right, it comes closer to the end of their series of papers.

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