All Questions
16
questions
2
votes
1
answer
101
views
Rationality of quasi-elementary group actions
I am learning the paper https://arxiv.org/pdf/1604.01005.pdf "Reductive group actions" by Knop and Krotz.
They defined that a linear k-algebraic group $H$ with unipotent radical $H_{u}$ is ...
- 121
4
votes
1
answer
302
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 ...
- 7,313
4
votes
1
answer
400
views
Universal covering groups of simple linear algebraic group schemes
Let $R$ be a Dedekind domain with fraction field $K$, and let $G$ be a smooth affine group scheme over $S = \text{Spec }R$ whose geometric fibers are connected and simple linear algebraic groups (i.e.,...
- 7,313
7
votes
1
answer
472
views
When must a set of sections which is Zariski dense in the generic fiber also be dense in some special fiber?
Let $f : X\rightarrow S$ be a flat finite type morphism of schemes with $S$ integral and Noetherian. Let $\eta\in S$ be the generic point.
Let $\{\sigma_i\}$ be a collection of sections of $f$ (...
- 7,313
2
votes
0
answers
116
views
Splitting of prime and order of reduction of point of infinite order in an abelian variety
I have already asked this question on stackexchange without much luck. I apologize if the question is too trivial to be asked here.
Let $A$ be an abelian variety defined over a number field $K$, $P \...
- 411
3
votes
0
answers
138
views
2-fold linear cover of reductive group of type A
Let $F$ be a nonarchimedean local field of characteristic zero. Let $G=\operatorname{Res}_{E/F}\operatorname{GL}_n$ or $\operatorname{Res}_{E/F}\operatorname{U}_n$, where $\operatorname{U}_n$ is any ...
- 833
2
votes
0
answers
114
views
relative rank two group: structure of parabolic subgroup-- high-level Jacobson--Morozov sl_2 triple
Given a parabolic subgroup $P=MN$ of a connected reductive group $G$ defined over a local field $F$, let $W_M$ be the relative Weyl group of $M$ in $G$, assume that the reduced roots relative to $M$ ...
- 301
2
votes
0
answers
253
views
Vector extension for p-divisible group
Background:
I am trying to understand a proof of Messing's book at Page 120. My goal it to understand the universal vector extention.
Reference:
Messing, The crystals associated to Barsotti-Tate ...
- 397
2
votes
0
answers
118
views
Explicit example for Display Theory for p-divisible group
Recently I am studying the display theory of formal p-div groups ([1] )by Zink.
I would like to study by working on an example. As far as I understood, the display theory is a generalization of ...
- 397
4
votes
0
answers
151
views
Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors
I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...
- 1,701
3
votes
0
answers
139
views
Cartan decomposition for $G[z]$
Let $G$ be a reductive group over complex numbers. Fix some maximal torus $T$. Let $\Lambda^{+}$ be the monoid of dominant coweights. It is known that one has a Cartan decomposition $$G((z))=\coprod\...
- 31
15
votes
1
answer
471
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
0
votes
2
answers
499
views
Is any F-stable maximal torus contained in some F-stable Borel subgroup? [closed]
Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
- 841
3
votes
3
answers
538
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 $...
- 7,682
1
vote
1
answer
649
views
Centralizer of elliptic elements in $GL(2)$
Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma \in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F_v$ of $F$ and $G_v = ...
- 11.1k