2
votes
0answers
75 views

Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?

Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...
3
votes
2answers
202 views

Gelfand pair and double coset decomposition

Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus ...
2
votes
1answer
123 views

Indefinite orthogonal groups over p-adics

Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...
3
votes
2answers
252 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 ...
7
votes
0answers
219 views

Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...
0
votes
0answers
145 views

on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field. $F=k((t))$ and $\mathcal{O}=k[[t]]$. Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$. Let $E=k((t^{1/n}))$ with $n$ prime to the ...
6
votes
0answers
94 views

Rational points with small denominator in $U(n)$

Fix integers $n,d>0$. (I'm probably thinking about $n\leq 6$ and $d\leq 2000$.) Let $X$ be the set of matrices $A\in U(n)$ such that the entries of $dA$ lie in $\mathbb{Z}[i]$. Is there an ...
0
votes
1answer
135 views

Reference on elements of finite order in principal congruence subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$. The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...
9
votes
2answers
513 views

Rational orthogonal matrices

``everybody knows'' that an integral orthogonal matrix is a signed permutation matrix, so there are exactly $2^n n!$ such matrices in $O(n).$ Now, what if we ask for the enumeration of elements of ...
4
votes
1answer
336 views

Representations of reductive groups over local fields through parahoric induction

Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as ...
5
votes
1answer
517 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 ...
11
votes
3answers
1k views

Questions about the Bernstein center of a $p$-adic reductive group

Dear all, The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of ...
0
votes
0answers
114 views

A kind of orthogonal subgroup

Let $n$ a positive integer and $k \in \mathbb{Z}^n$ such that for all integer $a \geq 2$ and $h \in \mathbb{Z}^n$ we have $k \neq ah$. Here $\cdot$ is the scalar product. Is it true that $\{x \in ...
5
votes
1answer
741 views

An interesting double coset in the theory of automorphic forms

Dear all, Does anyone have some idea to describe the double coset $P(F)\\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , ...
1
vote
2answers
307 views

Reductive groups over non archimedean local fields.

I want to know if connected reductive groups over non archimedean local fields have a dense countable subset. I was thinking that this should be true because if $G(\mathbb{F})$ is such group where ...
10
votes
0answers
314 views

Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?

Let $R$ be a commutative ring, and, for $n\ge0$, ${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series $u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which $a_0\in R^\times$ and $u(x)\equiv ...
16
votes
4answers
2k views

Is strong approximation difficult?

Recently a colleague and I needed to use the fact that the natural map $SL_2(\mathbb{Z}) \rightarrow SL_2(\mathbb{Z}/N\mathbb{Z})$ is surjective for each $N$. I happily chugged my way through an ...
7
votes
1answer
506 views

Principal congruence subgroups in higher rank

I don't seem to have seen any explicit generators for the principal congruence subgroups of $SL(n, \mathbb{Z}),$ for $n>2,$ although it is known (Sury+Venkataramana) is that the number of ...
2
votes
1answer
335 views

Iwahori for PGL_2

What is the Iwahori subgroup for $PGL_2(F)$ where $F$ is a local field? I am also looking for the Levi subgroups but it seems that there is only 1 levi subgroup been the identity but this seems odd to ...
3
votes
0answers
358 views

L-function of an algebraic group defined over a function field

Let $G$ be an algebraic reductive group defined over an algebraic function field $K$ in one variable with a finite field of constants $\mathbf{F}_q$. For any prime $p$ of $K$, unramified in the ...
4
votes
0answers
162 views

Lattices in Hermitian spaces over local fields

Let $F$ be a $p$-adic field, $E / F$ a quadratic extension, and $n \ge 1$. Let $V = E^n$ with the obvious diagonal Hermitian form, $$ \langle (u_1, \dots, u_n), (v_1, \dots, v_n) \rangle = \sum_{i = ...
2
votes
5answers
649 views

Product of two algebraic subgroups of a (solvable) group = another algebraic subgroup?

Let $G$ be a linear algebraic group over a field $K$. (Say $K=\mathbb{F}_q$ or $K=\mathbb{C}$; do not assume $K$ is algebraically closed or of characteristic $0$.) Let $H_1$, $H_2$ be algebraic ...
5
votes
2answers
608 views

Unitary groups over number fields

When defining unitary groups over number fields, one usually takes $F$ to be a totally real number field, $E$ a CM quadratic extension of $F$, and $V$ a hermitian space attached to $E/F$. Then $U(V)$ ...
10
votes
1answer
523 views

Double coset spaces of reductive groups and integral representations of L-functions

Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space ...
11
votes
1answer
874 views

Fontaine's classification of p-divisible groups

Let k be a finite extension of $\mathbb{F}_p$, and W its ring of Witt vectors. Write W[F,V] for the Dieudonn\'e ring. Let G be a connected p-divisible group which is finite-dimensional over k, and ...
7
votes
1answer
278 views

Are affine groups over rings of integers finitely generated?

I'll begin by saying that I'm not sure what I want to ask specifically, but pretty sure what in general, so please don't hold my misunderstandings against me, but do comment on them. I know that the ...
6
votes
0answers
252 views

Real approximation for homogeneous spaces of linear algebraic groups

Let $X$ be a smooth geometrically integral variety over $\mathbf{Q}$, having a $\mathbf{Q}$-point. We say that $X$ has the real approximation property if $X(\mathbf{Q})$ is dense in $X(\mathbf{R})$. ...
2
votes
3answers
579 views

Upper bound for lowest common multiple of integers with (almost) fixed sum

For a positive integer $n$, let $f(n)$ be the maximum value of $\mathrm{LCM}(S)$ among multisets $S$ of positive integers satisfying $\sum_{i \in S} (i-1) = n$. What is known about upper bounds for ...
20
votes
2answers
2k views

Why are Tamagawa numbers equal to Pic/Sha?

For a connected algebraic group $G$ over a global field $K$ with adeles $A$, the Tamagawa number of $G$ is the volume of $G(A)/G(K)$. It is conjectured (and often known) to be rational, namely the ...
2
votes
1answer
370 views

Automorphism of algebraic group preserving a hyperspecial maximal compact

Suppose that $K/\mathbb{Q}_l$ is a finite extension, with ring of integers $\mathcal{O}_K$. Suppose $\mathcal{G}/K$ is a (linear) algebraic group (connected+reductive), and $\Gamma\subset ...
30
votes
1answer
840 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 ...
9
votes
3answers
2k views

Relation between Hecke Operator and Hecke Algebra

In the study of number theory (and in other branches of mathematics) presence of Hecke Algebra and Hecke Operator is very prominent. One of the many ways to define the Hecke Operator $T(p)$ is in ...
2
votes
1answer
227 views

Hyperspecial subgroup of a product of semisimple algebraic groups

Suppose that $F$ is a nonarchimedean local field, and that $G_1$, $G_2$ are connected, simply connected algebraic groups over $F$. Suppose moreover $G_1$ and $G_2$ are semisimple. Suppose $H$ is a ...
8
votes
2answers
661 views

number of irreducible representations over general fields

For a finite group, there are finitely many irreducible representations of complex numbers. What if the field is changed to some other fields? Like real numbers, p-adic field, finite field? In ...
8
votes
3answers
536 views

How to topologize X(R) when R is a topological ring?

Given a topological ring R, under what conditions and in what way, can one induce a topology on the R-points of a scheme X? For example, if X is P^n or A^n, one has natural topology on the R-points. ...