3
votes
1answer
190 views

What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field. Here, I think the word "compact" is used ...
1
vote
1answer
128 views

Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field), $g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...
3
votes
2answers
257 views

isogeny and congruence subgroup

Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$. By ...
1
vote
0answers
154 views

Compactifications of group schemes

Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...
1
vote
1answer
134 views

arithmetic group over function fields and its fundamental domain

Let $G$ be a semi-simple algebraic group defined over a global function field $K$. Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take ...
2
votes
1answer
177 views

S-arithmetic subgroup question

I've been reading a proof concerning S-arithmetic subgroups of algebraic groups and I'm having trouble determining why the following step should be true. First, the setup: Let $G$ be a connected ...
2
votes
2answers
436 views

Classification of quasi-split unitary groups

Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...
2
votes
1answer
311 views

Group of connected components of the global Néron-Raynaud model of a torus

Let $K = \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$ defined over a finite field of constants $\mathbb{F}_q$. Let $T$ be a $K$-torus. We choose one ...
4
votes
2answers
327 views

density in SU(2,1)

Let $K=Q(\sqrt{-3})$ , is $SU(2,1)(K)$ dense in $SU(2,1)(C)$ for the complex topology?
1
vote
0answers
224 views

Why do twists of an algebraic group over k correspond to k-torsors over G

Let $G$ be an algebraic group over a field $k$. Let $k^s$ be the separable closure of $k$. I can't seem to figure out why isomorphism classes of twists of $G$ correspond to $k$-torsors over $G$. ...
2
votes
1answer
209 views

Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively?

Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively? see also the link:mathoverflow.net/questions/36762,
30
votes
1answer
859 views

Is the group of integer points on a finite-type group scheme over Z finitely presented?

Let $G$ be a group scheme of finite type over $\mathbf{Z}$. Must $G(\mathbf{Z})$ be finitely presented? (The question is inspired by a not yet successful attempt to answer a question of Brian ...