All Questions
36
questions
6
votes
1
answer
363
views
Interpreting group-theoretic sentences as statements about algebraic groups
Suppose we are given a sentence in the language of groups, e.g. $\phi=\forall x\forall y(x\cdot y=y\cdot x)$, and suppose that we are also given the data defining an algebraic group $G/k$. One can ...
- 333
4
votes
1
answer
405
views
Étale group schemes and specialization
If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
user501361
5
votes
1
answer
255
views
Torus gerbes over curves
Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected curve over $k$. Let $K(C)$ be the function field of $C$.
Tsen's Theorem implies that every $\mathbb{G}_m$-gerbe over $K(C)...
- 123
1
vote
0
answers
98
views
$K$-ranks of some algebraic groups in Lubotzky's "Discrete groups, expanding graphs and invariant measures"
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Let $G$ be a semisimple algebraic group and $K$ any field. Then
the $K$-rank of $G$ is the maximal rank $r$ of a $K$-splitting
torus $T \cong (K^...
- 5,780
2
votes
0
answers
98
views
Number of points of parabolic Springer fibres for general reductive groups
My question is the same as this post but for an arbitrary reductive $G$ instead of just $\mathrm{GL}_n$.
Let $G$ be a connected split reductive group over a finite field $k$.
Let $P$ be a parabolic ...
- 2,711
3
votes
1
answer
392
views
The Weil restriction of a simple algebraic group
Let $F$ be a number field, $G$ an $F$-simple affine algebraic group.
Then is the Weil restriction $\operatorname{Res}_{F/\mathbb{Q}} G$ $\mathbb{Q}$-simple?
I couldn’t find any references.
- 185
4
votes
0
answers
234
views
Number of homomorphisms from a group to $\mathrm{GL}_n(\mathbb{F}_q)$
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms ...
- 2,711
6
votes
1
answer
403
views
Number of points of parabolic Springer fibres
Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $u\in P$ a unipotent element. The parabolic Springer fibre associated to $(P,u)$ can be defined by
$$
\mathcal{P}_u:=\{gP\in G/P \mathrel\vert g^{...
- 2,711
5
votes
1
answer
170
views
Reductive groups over number rings
Let $F$ be a number field.
If $G$ is a reductive group over $\mathcal{O}_F$ then we can look where $G\otimes \mathbb{C}$ fits in the classification of complex reductive groups and get a "standard ...
- 111
3
votes
0
answers
138
views
2-fold linear cover of reductive group of type A
Let $F$ be a nonarchimedean local field of characteristic zero. Let $G=\operatorname{Res}_{E/F}\operatorname{GL}_n$ or $\operatorname{Res}_{E/F}\operatorname{U}_n$, where $\operatorname{U}_n$ is any ...
- 833
14
votes
1
answer
1k
views
If it quacks like an abelian variety over a finite field
Consider smooth projective varieties over a finite field. If a curve "looks like" an elliptic curve (i.e. has genus $1$) then it can be made into an elliptic curve.
Is there something ...
- 117
3
votes
0
answers
108
views
Deformation of p-divisible groups along nilpotent thickening
Let $S_0 \rightarrow S$ be a nilpotent thickening of schemes (no divided power provided) where $p$ is nilpotent, let $G$ be a $p$-divisble group over $S_0$, how to describe all liftings of $G$ to $S$ ...
- 6,188
7
votes
0
answers
226
views
Field extensions that preserve given cohomology classes
Let $G$ be a connected reductive group over $\mathbb{Q}$ and let $\operatorname{Ker}^1(\mathbb{Q},G) \subset H^1(\mathbb{Q},G)$ be the subset of classes that are trivial at all places. I am trying to ...
- 427
0
votes
0
answers
208
views
Endomorphism rings of flat group schemes
Let $R$ be a commutative ring and $X$ be a flat $R$-group scheme.
We call $\text{End}_R(X)$ the ring of endomorphisms of the $R$-group scheme $X$, defined over $R$.
Let $R\to S$ be a ring map ...
user147445
2
votes
0
answers
270
views
Étale group scheme exact sequence
Consider the exact sequence of finite flat group schemes over the $2$-adic integers ring $\mathbb{Z}_2$:
$$0\longrightarrow\mathbb{Z}/2\mathbb{Z}\longrightarrow A\longrightarrow\mathbb{Z}/2\mathbb{Z}\...
user141691