All Questions
11 questions
10
votes
1
answer
719
views
what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?
Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
- 10.7k
9
votes
1
answer
553
views
What matrix groups can be embedded in $Sp_4$?
In a joint paper with Yifan Yang we constructed an "exotic" embedding
of $SL_2(\mathbb R)$ in $Sp_4(\mathbb R)$ (in fact, of $PSL_2(\mathbb R)$ in $PSp_4(\mathbb R)$),
namely,
$$
\iota\colon\begin{...
- 13.5k
9
votes
0
answers
99
views
Derived subgroups of 2-adic congruence subgroups of $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}$Let $p$ be a prime, and let $\Gamma_r$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^r\mathbb{Z}_p)$.
Explicit formulas with formal ...
- 7,527
7
votes
1
answer
270
views
Is $\Gamma(p) := \text{Ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{F}_p)$ a "standard" subgroup?
Let $\Gamma(p) := \text{ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{Z}_p/p))$.
Viewing $SL_2(\mathbb{Z}_p)$ as an analytic group, is there a formal group law $F$ in three variables, defined over $...
- 7,527
7
votes
1
answer
456
views
Can Galois conjugates of lattices in SL(2,R) be discrete?
Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
- 454
7
votes
0
answers
172
views
Subgroups of $\mathbb{Z}^{n}$ with rotational symmetries
Schmidt (https://projecteuclid.org/euclid.dmj/1077377618) showed that the number of $m$-dimensional subgroups of $\mathbb{Z}^{n}$ of covolume $\leq X$
is
$$c_{1}\left(m,n\right)X^{n}+O\left(X^{n-\...
- 273
4
votes
1
answer
212
views
Subgroups of $Sp_{2g}$ giving rise to Shimura data
Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...
user70064
3
votes
0
answers
160
views
Orbit representatives for the action of the maximal compact subgroup
Let $F$ be a non-Archimedean local field and $O$ be the ring of integers in $F$. Take $G=GL(2,F)$ and $K=GL(2,O)$. Consider the action of $K$ on $G$ by conjugation. Is it possible to explicitly write ...
- 185
2
votes
1
answer
185
views
Fields of definition of parabolically induced representations of $\mathrm{SL}(2,q)$
Let $\alpha_0$ be the unique non-trivial character satisfying $\alpha_0^2=1$ of the split torus $\mathrm{T} \subset \mathrm{SL}(2,q)$ and denote by $\mathrm{R}(\alpha_0)$ the character of $\mathrm{SL}(...
- 381
2
votes
0
answers
74
views
generalization of the discreteness of Hecke groups to general reductive groups
Consider the subgroup $G_{\lambda}$ of $SL_2(\mathbb R)$ generated by $N_{\lambda} = \begin{bmatrix} 1 & \lambda \\ 0 & 1 \end{bmatrix}$ and $S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{...
- 6,622
1
vote
1
answer
106
views
Actions of torsionfree discrete subgroups on hermitian symmetric domains
Let $D$ be a bounded hermitian symmetric domain with automorphism group $G(\mathbb R)$. In the example I have in mind, $D$ is Siegel upper half-space of degree $g$ and $G(\mathbb R) = \mathrm{Sp}(2g,\...
- 13