All Questions
27 questions
5
votes
2
answers
343
views
Non-commuting elements of finite orders in a reductive group over a p-adic field
Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma:
Lemma. Assuming that $p$ is "good" for $G$,...
- 14.1k
2
votes
0
answers
98
views
Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
- 14.1k
7
votes
1
answer
633
views
Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
- 823
1
vote
1
answer
594
views
Some necessary condition for $\gcd(m,n) $ be a proper divisor of $\gcd(mk_2 +nk_1,mn) $ [closed]
Let $m,n,k_1,k_2 $ be natural numbers such that $(k_1,m)=(k_2,n)=1 $.
Statement 1: $\gcd(m,n) $ is a proper divisor of $\gcd(mk_2 +nk_1,mn) $, for every $k_1,k_2$ having the above property.
Statement ...
- 923
1
vote
0
answers
98
views
Existence of countable dense normal subgroups of global Galois group
Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
- 863
16
votes
2
answers
2k
views
One question on linear combinations of roots of unity
For $n \geq 1$, I want to find all solutions $x_i$ of the equation
\begin{equation}
\begin{array}l
x_i \in \mathbb{Z}, i=0,1,2\dotsc,n-1 \\
x_i^2 = 1, i=0,1,2\dotsc,n-1 \\
\...
- 696
8
votes
1
answer
711
views
Does this exact sequence split?
Let $K$ be a number field. $O_K$ be its ring of integers, so $O_K^*$ are the units.
We have sequence
$1 \rightarrow O_K^* \rightarrow K^* \rightarrow K^*/O_K^* \rightarrow 1$
Note that $K^*/O_K^*$ is ...
- 503
4
votes
1
answer
314
views
Criteria for Zariski density of subgroups of reductive groups
Let $G$ be a reductive group over a number field $K$. Let $\Gamma\subset G(K)$ be a subgroup.
My extremely naive question is - When can you deduce that $\Gamma$ is Zariski-dense? I'm looking for ...
- 7,527
5
votes
4
answers
1k
views
The integral closure $\overline{\mathbb{Z}}$ and the group $\overline{\mathbb{Z}}^{\times}$
Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\...
- 547
9
votes
1
answer
825
views
Forms of ${\rm SL}(2)$
I know all real forms of ${\rm SL}(2,{\Bbb C}$). They are ${\rm SL}(2,{\Bbb R})$ and ${\rm SU}(2)$.
Moreover, ${\rm SL}(2,{\Bbb R})$ is isomorphic to ${\rm SU}(1,1)$. Thus I can say that all real ...
- 14.1k
3
votes
0
answers
479
views
How are problems about number fields reduced to problems about their absolute Galois groups?
The article on Wikipedia about Neukirch–Uchida theorem claims right from the beginning the statement in my question. I have seen similar claims elsewhere before.
I am a little puzzled by this ...
- 99
5
votes
0
answers
360
views
Applications of Tits' alternative in algebraic number theory
I have recently studying Tits' alternative. The theorem statement goes like the following:
Tits' alternative: Let $G$ be any finitely generated linear group over a field. Then one of the following is ...
- 302
7
votes
0
answers
227
views
Automorphism group of poset of number fields
Consider the poset of number fields, partial order being defined by inclusion of fields. What is the group of order-preserving automorphisms of this poset? What if we take only Galois extensions of $\...
- 71
2
votes
0
answers
72
views
Centralizers of Cartan subgroups II
Let $K$ be an imaginary quadratic field and let $\mathcal O$ be its ring of integers. Suppose that $2$ is split in $\mathcal O$. Let $k$ be a positive integer. The multiplicative group $(\mathcal O/2^...
- 2,375
7
votes
1
answer
397
views
Cohomology of certain arithmetic groups
This is a question on literature about cohomology of arithmetic groups.
Let $M$ denote a quaternion algebra over $\mathbb Q$ and assume it is non-split over $\mathbb R$. Fix a maximal order $\Lambda$ ...
user1688
14
votes
1
answer
695
views
$\mathbb{Z}$-module structure of the subring generated by an algebraic number
Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
- 151
10
votes
1
answer
302
views
Finite subgroups of Lie group over algebraic ring of integers
I have frequently seen results like:
There are 4 isomorphism types of finite subgroups of $SL_2(\mathbb{Z})$, namely $\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_4,\mathbb{Z}_6$.
I wonder what is known of ...
- 1,846
7
votes
1
answer
850
views
Understanding the structure of unitary groups
I would like to understand precisely the structure of unitary groups.
Let $F$ be a global number field, $E$ a quadratic extension of $F$, and $U$ a unitary group on $E$ (i.e. the group of ...
- 5,424
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
15
votes
1
answer
474
views
Dirichlet's unit theorem for reductive schemes
Let $O_{K,S}$ be the ring of $S$-integers in a number field $K$. Dirichlet's unit theorem implies that the group of units in $O_{K,S}$ is a finitely generated group. In other words, the group $\mathbb ...
- 151
6
votes
0
answers
293
views
Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module
Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...
- 1,066
10
votes
2
answers
723
views
Does the Galois group of a Pisot polynomial contain the alternating group?
Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...
- 11.3k
6
votes
0
answers
221
views
Furtwangler's Principal ideal theorem in number fields
Does anyone know a simple proof, using cohomological method of the fact that the verlagerung from a finite group G. to its commutator subgroup G', i.e. $$G/G'->(G')^{ab}$$ vanishes?
The simplest ...
- 61
12
votes
1
answer
1k
views
Class field towers
It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of ...
user6976
7
votes
2
answers
1k
views
Galois groups and prescribed ramification
What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...
- 11.3k
5
votes
0
answers
267
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 ...
- 51
33
votes
5
answers
4k
views
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Apologies in advance if this is obvious.
Edit, 5/31/24: Since this question is getting some ...
- 118k