All Questions
9 questions
10
votes
1
answer
886
views
Definition of an arithmetic subgroup of an algebraic group
I'm struggling with the definition of an arithmetic subgroup of an algebraic group defined over $\mathbb{Q}$.
In Wikipedia you can read:
If $\mathrm G$ is an algebraic subgroup of $\mathrm{GL}_n(\...
- 563
2
votes
1
answer
288
views
Stabilizer in $G(\mathbb{Z})$ of point in fundamental domain $G(\mathbb{Z}) \backslash G(\mathbb{R}) / K$
Let $G$ be a semisimple group (the cases of primary interest to me are where $G$ is a special linear group or a special orthogonal group), let $K$ be a maximal compact subgroup of $G(\mathbb{R})$, and ...
5
votes
3
answers
448
views
Variation of centraliser in $\operatorname{GL}(n,\mathbb{Z})$
$\DeclareMathOperator\GL{GL}$Let $n$ be a positive integer $\geq 2$. The setting is that $K \in \GL(n,\mathbb{Z})$, and people are interested in understanding the centralizer:
$$
C(K)=\{ B \in \GL(n,\...
- 145
4
votes
1
answer
185
views
On the orthogonal group of a lattice on a quadratic space over dyadic local field
Let $F$ be a local field with valuation ring $R$. $V$ is a n dimensional non-singular quadratic space over $F$ with bilinear form $B$ and quadratic map $Q$.
As usual, $O(V)$ denotes the orthogonal ...
user125345
3
votes
0
answers
175
views
$\text{PGL}_n(\mathbf{Q}_p)$ and the Congruence Subgroup Property
Suppose $\Gamma$ is a torsion-free lattice of $\text{PGL}_n(\mathbf{Q}_p)$ for $n\geq 3$. Then I know that by the Margulis arithmeticity theorem, $\Gamma$ must be arithmetic. My question is does $\...
user72293
4
votes
1
answer
274
views
Subgroups of $Sp(2n,\mathbb{R})$ between $Sp(2n,\mathbb{Z})$ and some arithmetic group
The fantastic answers to my previous question Subgroups of $SL_2(\mathbb R)$ which contain $SL_2(\mathbb Z)$ as a finite index subgroup led me to the following question.
Let $O_K$ be the ring of ...
- 195
4
votes
1
answer
355
views
Volume of arithmetic quotients of symmetric spaces
Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...
- 685
10
votes
3
answers
581
views
Is there a generalization of the "characteristic polynomial" to other split/quasi-split algebraic groups?
Let $G = GL_n$ over a field $F$, and let $\gamma \in G(F)$ be a semisimple element. The characteristic polynomial $c_\gamma(t)$ of $\gamma$ encodes a fair bit of information about $\gamma$. ...
- 1,453
11
votes
2
answers
3k
views
Why are $S$-arithmetic groups interesting?
Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$.
Define the $S$-numbers $K_S$ to be the direct product $\prod_{s \in S} K_s$ where $K_s$ denotes the completion ...
- 349