Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:58:15Z http://mathoverflow.net/feeds/question/28361 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Menny 2010-06-16T08:26:53Z 2010-06-16T13:20:10Z <p>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)?</p> <p>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.</p> <p>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.</p> <p>Can someone outline a route to take in order to understand it thoroughly (for someone with basic understanding of algebraic groups and Galois cohomology)? </p> <p><strong>Thanks a lot!</strong></p> http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar/28364#28364 Answer by Charles Matthews for Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Charles Matthews 2010-06-16T09:00:47Z 2010-06-16T09:00:47Z <p>See <a href="http://eom.springer.de/a/a012530.htm" rel="nofollow">http://eom.springer.de/a/a012530.htm</a> and the concept of anisotropic group. As it says there, the classification involved is pretty much the classification of semisimple groups.</p> http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar/28365#28365 Answer by Kevin Buzzard for Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Kevin Buzzard 2010-06-16T09:03:46Z 2010-06-16T09:03:46Z <p>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 <code>$D^*$</code> are coming from fields $E/F$ embedding into $D$, so the maximal tori in $G$ are the norm 1 elements of <code>$E^*$</code> and these have no rational characters (the norm is the only rational character of <code>$E^*$</code> and you've just removed it).</p> http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar/28373#28373 Answer by Victor Protsak for Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Victor Protsak 2010-06-16T10:50:33Z 2010-06-16T10:50:33Z <p>As Charles said, for a semisimple group over a local field compact $\iff$ anisotropic.</p> <p>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.</p> http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar/28377#28377 Answer by Boyarsky for Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Boyarsky 2010-06-16T11:54:21Z 2010-06-16T13:20:10Z <p>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. :)</p> <p>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 <em>absolutely simple</em> and <em>simply connected</em> 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.) </p> <p>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. </p> <p>But in the <em>simply connected</em> 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 <em>absolutely</em> simple and simply connected. By the good behavior of Weil restriction with respect to the formation of the <em>topological</em> 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.) </p> <p>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 $F$.</p>