All Questions
38
questions
7
votes
0
answers
134
views
Quasisplit forms of wonderful varieties
I will assume that $k$ is a characteristic $0$ non-archimedean field. A classical result of Tits [T] states that a quasisplit connected reductive group $G$ over $k$ is classified up to strict isogeny ...
- 121
1
vote
0
answers
143
views
Is the functor $\mathrm{Hom}(\mathrm{spec}\,k[x^{1/{p^\infty}}]/(x), -)$ from the category of finite commutative group schemes exact?
Question. Let $B \twoheadrightarrow C$ be a fully faithful homomorphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. ...
- 135
6
votes
1
answer
320
views
Exactness of the Weil restriction functor $\mathrm{Res}_{X/k}$
Question. Let $X$ be an Artinian scheme over a perfect field $k$. Consider the abelian category $\mathcal{C}$ of affine commutative group schemes of finite type. Is the Weil restriction $\mathrm{Res}_{...
- 135
3
votes
1
answer
247
views
Completely reducible subgroups over local field in terms of closed orbits
$\DeclareMathOperator\GL{GL}$Let $ \overline{\mathbb{Q}}_{p} $ be an algebraic closure of $ p $-adic numbers $ \mathbb{Q}_{p} $. A closed subgroup $ H $ of a general linear group $ \GL_{n}(\overline{\...
- 561
0
votes
0
answers
119
views
Does an isogeny between tori induce an isomorphism of the Lie algebras of their lft Néron models?
Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? Are there some interesting properties ...
- 617
4
votes
1
answer
370
views
Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero
I am reading a very nice paper of Newton and Thorne, Symmetric power functoriality for holomorphic modular forms, and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic ...
- 229
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
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
14
votes
0
answers
812
views
What goes wrong with this alternate proof of Dirichlet's Theorem?
I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
- 782
5
votes
3
answers
422
views
Variation of centraliser in $\operatorname{GL}(n,\mathbb{Z})$
$\DeclareMathOperator\GL{GL}$Let $n$ be a positive integer $\geq 2$. The setting is that $K \in \GL(n,\mathbb{Z})$, and people are interested in understanding the centralizer:
$$
C(K)=\{ B \in \GL(n,\...
- 145
14
votes
1
answer
1k
views
If it quacks like an abelian variety over a finite field
Consider smooth projective varieties over a finite field. If a curve "looks like" an elliptic curve (i.e. has genus $1$) then it can be made into an elliptic curve.
Is there something ...
- 117
1
vote
1
answer
167
views
Maps to additive group scheme
Let $\underline{\mathbb{Q}_p/\mathbb{Z}_p}$ be constant p-divisible group over $\mathbb{F}_p$. And let $\mathbb{G}_a$ be the additive group over $\mathbb{F}_p$. Let me prove
$$
Hom(\underline{\mathbb{...
- 397
6
votes
1
answer
511
views
How to compute Galois representations from etale cohomology groups of a generalized flag variety?
Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...
- 6,188
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
5
votes
1
answer
410
views
What is $\mathrm{O}_q/\mathrm{SO}_q$ if $q$ is a quadratic $\mathbb{Z}$-form which is degenerate?
Any binary quadratic $\mathbb{Z}$-form $q$ induces a symmetric bilinear form
$$ B_q(u,v) = q(u+v) - q(u) -q(v) \ \ \forall u,v \ \in\mathbb{Z}^2 $$
and it is considered non-degenerate (over $\mathbb{...
- 359