The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
0answers
43 views

symmetric theta structures and arithmetic subgroups

A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface. ...
8
votes
1answer
249 views

Reference for the fact that $SL_n(O_K)$ surjects onto $SL_n(O_K/I)$ for any ideal I

Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map ...
11
votes
2answers
357 views

Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free

My question stems from Misha's answer of a MathOverflow question. Misha supplied the following question in his answer: Open question: Does there exist a finitely generated Zariski-dense ...
3
votes
3answers
485 views

Non existence of cyclic infinite linear algebraic groups

Let $G$ be a linear algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of ...
11
votes
1answer
240 views

What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?

While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical ...
6
votes
2answers
338 views

Pre-images of unipotent elements in $\operatorname{SL}_{n}(A)$

The starting point of this question is the (presumably) well-known theorem (the proof I know is from Abelian $\ell$-adic representations and elliptic curves from J-P.Serre in which it is a lemma for ...
5
votes
1answer
201 views

Is SL_n of an order in a number ring finite-index in SL_n of the number ring?

Let $\mathcal{O}$ be the ring of integers in an algebraic number field and let $R \subset \mathcal{O}$ be an order. For instance, we might have $\mathcal{O} = \{\text{$x+i y$ $|$ $x,y \in ...
2
votes
0answers
134 views

Discussion of specific arithmetic triangle groups?

Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of ...
5
votes
0answers
187 views

Generating congruence subgroups of SL_n over totally imaginary number rings

Fix some $n \geq 3$. Let $k$ be an algebraic number field with ring of integers $\mathcal{O}$ and let $\alpha$ be an ideal of $\mathcal{O}$. Define $\text{SL}_n(\mathcal{O},\alpha)$ to be the ...
6
votes
1answer
211 views

Characterisation of Q-rank 1

I'm looking for a reference and/or the original source for the following fact: An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...
2
votes
1answer
60 views

Action of GL(2,O_k) on 1d subspaces of (O_k)^2

Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$. Let $M$ be a rank $1$ projective module over $\mathcal{O}_k$ (in other words, $M$ is a projective module such that $k ...
1
vote
0answers
156 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 ...
2
votes
1answer
141 views

Conjugacy of torsion subgroups in Gl(n, Z) for small n [duplicate]

Have the conjugacy classes of the torsion subgroups of Gl(n, Z) been determined for small n (say, n<=6)? In general, can much be said about the torsion subgroup?
7
votes
1answer
182 views

Is the group of integer points of ${\rm SO}(n,1)$ maximal?

That is, is it true that there does not exist a lattice in $G = {\rm SO}(n,1)$ which contains the group of integer points of $G$ as a proper subgroup (obviously then of finite index)? if such a ...
9
votes
3answers
530 views

Generators for a certain congruence subgroup of SL(n,Z)

I'm looking for a reference (or quick proof) of the following fact. Fix some $n \geq 3$ and some $\ell \geq 2$. Set $\Gamma_n(\ell) = \text{ker}(\text{SL}_n(\mathbb{Z}) \rightarrow ...
0
votes
1answer
213 views

Siegel set in SO(n,1) modulo integer points?

I wonder what is known about a fundamental region for SO($n,1$) modulo its integer points? is there only one cusp? and if one writes a Siegel set in the form of $K A_\tau N_c$, where $N_c$ is compact ...
2
votes
1answer
188 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 ...
7
votes
0answers
242 views

subgroups of higher rank lattices

This is related to the question $G=\langle a\rangle H$ for subgroup $H$ raised a few days ago. Suppose $\Gamma $ is a higher rank lattice (for example, $SL_3({\mathbb Z})$). As Misha says in his ...
5
votes
1answer
244 views

2 generated arithmetic groups

Suppose $G({\mathbb Z})$ is a higher rank non-cocompact arithmetic group (e.g. $SL_n({\mathbb Z})$ with $n\geq 3$, or $Sp_{2g}({\mathbb Z})$ with $g\geq 2$). I have seen a result ...
5
votes
1answer
551 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 ...
5
votes
0answers
221 views

Generators for SL_2(R) for rings of integers R

Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Is $\text{SL}_2(\mathcal{O})$ generated by elementary matrices? If it isn't, is there any other natural generating set for it? ...
3
votes
1answer
217 views

Genus of arithmetic surface groups

It is known that for each genus, only finitely many points in the moduli space of hyperbolic genus g surfaces are arithmetic. I'm wondering if an existence result is known: for which g do we have ...
1
vote
1answer
274 views

Fixed points of the Borel-Serre compactification

Let $\Gamma$ be an arithmetic group and $X$ its symmetric space. Borel-Serre constructed a space $\bar{X} \supset X$ such that $\bar{X}/\Gamma$ is a compactification of $X/\Gamma$ [Corners and ...
15
votes
3answers
2k views

Generating the symplectic group

The too naive and vague version of my question is the following: given a collection of integer symplectic matrices all of the same size (say 2n by 2n), how can I tell if they generate the full ...
1
vote
2answers
388 views

Zariski density of conjugates of subgroups by arithmetic subgroups?

Let $G$ be a linear algebraic $\mathbb{Q}$-group, which is assumed to be connected, $\mathbb{Q}$-simple, and of adjoint type, such that the Lie group $G(\mathbb{R})$ has no compact factor defined over ...
2
votes
2answers
636 views

arithmetic groups VS. Zariski dense discrete subgroups?

Assume that $G$ is a semi-simple linear algebraic group defined over $\mathbb{Q}$, which is $\mathbb{Q}$-simple, and that $G(\mathbb(R)$ is non-compact, without $\mathbb{R}$-factors of rank 1. Then by ...
4
votes
0answers
198 views

links and interactions between different approaches to (super-)rigidity

By super-rigidity I mean some theorems concerning the arithmetic subgroups in semi-simple Lie groups. According to Margulis "Discrete subgroups of semi-simple Lie groups" (the book published by ...
4
votes
1answer
606 views

discrete subgroups in p-adic Lie groups?

It is known, from the works of G.Margulis, etc. that lattices in semi-simple real (algebraic) groups are "often" arithmetic subgroups, as long as the split rank is high enough. Here by a lattice in a ...
14
votes
1answer
763 views

Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?

I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
3
votes
1answer
626 views

Quotients of unipotent groups

Let $U (\mathbf{R})$ be the standard unipotent subgroup of $SL(3, \mathbf{R})$. So $U(\mathbf{R})$ is the group of 3 by 3 upper triangular matrices with 1s on the diagonal. I am interested in the ...
9
votes
0answers
351 views

A uniform bound for a “true” non-congruence subgroup

Before stating my question, let me recall the Congruence Subgroup Property/Problem: Given simply connected absolutely and almost simple algebraic group $G$ with fixed realization as a matrix group one ...
7
votes
4answers
968 views

cohomology of moduli spaces

Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...