All Questions
2,543 questions
3
votes
0
answers
47
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
6
votes
0
answers
130
views
Is there a (relevant) framework in which "$\mathrm{GL}_m \times \mathrm{GL}_n = \mathrm{GL}_{m+n}$"?
Generally one considers a vector space $k^n$, then $\mathrm{GL}_n(k)$ is realized as the set of automorphisms of $k^n$. We have $k^m\times k^n = k^{m+n}$, however under the natural embedding, $\mathrm{...
- 313
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
4
votes
1
answer
168
views
Semisimple elements and fixed points
The following statement seems to be well-known:
Let $X$ be a variety on which an affine algebraic group $H$ acts with
finitely many orbits and let $s \in H$ be semisimple. Then $H_s = \{h
\in H \mid ...
- 65
5
votes
1
answer
100
views
An algebraic group $G$ over $L^+$ such that $G_{\mathbb{R}}$ is compact for almost all embeddings
Does there exist a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is ...
- 127
5
votes
1
answer
248
views
Galois action on Borovoi's algebraic fundamental group
In Borovoi's paper Abelian Galois cohomology of reductive groups, the algebraic fundamental group of a connected reductive group $G$ over a field $K$ of characteristic zero is defined as
$$\pi_1(G, T):...
- 53
3
votes
0
answers
136
views
Galois cohomology and Levi subgroups
Let $F$ a field and $G$ a smooth connected reductive group with a Levi subgroup $M$. Under what assumptions is $H^1(F, M) \to H^1(F, G)$ injective? In the case $F$ is nonarchimedean local I believe ...
- 605
2
votes
0
answers
80
views
Question about lattice with dense projection
Let $H\subset \operatorname{GL}(n,\mathbb{C})$ be a connected, semisimple algebraic group defined over $\mathbb{Q}$. Fix a number field $K$ with $[K:\mathbb{Q}]=3$ that is not totally real. Denote its ...
- 305
2
votes
1
answer
185
views
Number of rational points of a quotient of connected linear algebraic groups
Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius ...
- 23
2
votes
0
answers
103
views
Finite groups of Lie type
Table $22.1$ Finite groups of Lie type from "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman:
For the type $A$: $G_{sc}^{F}=\operatorname{SL}_{n}(q)$ and $G_{ad}...
- 217
3
votes
1
answer
152
views
Locally nilpotent derivations and triangularizability
If $ k $ is a field of characteristic zero and $ \delta \in T_{\mathbb{A}^{n}_{k}/k} $, then $ \delta $ is triangular if $ \delta = \sum_{i=2}^{n} f_{i}(x_{1},\dots,x_{i-1}) \frac{\partial}{\partial ...
- 912
1
vote
1
answer
80
views
$p$-torsion related to algebraic groups
Definition $14.14$ from "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman:
A prime $p$ is a torsion prime for a linear algebraic group $G$ if the fundamental ...
- 217
5
votes
1
answer
264
views
Central isogeny, Shimura varieties and exceptional cases
For a simple complex Lie algebra $\mathfrak g$, its weight lattice is not equal to the root lattice (i.e. the center of its simply connected form is a non-trivial finite group) iff $\mathfrak g$ is of ...
- 6,622
4
votes
0
answers
269
views
Is a pro-algebraic group over $\mathbb{Q}_p$ with Galois action the inverse limit of Galois-equivariant quotients?
Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...
- 15.4k
2
votes
0
answers
165
views
Definition for "almost simple" linear algebraic groups
Proposition 2.18 from "Elementary abelian $p$-subgroups of algebraic groups" by R. Griess. used the term "simply connected almost simple linear algebraic group $G$" without ...
- 217
1
vote
0
answers
79
views
Invariant theory (first fundamental theorem) for a direct sum of two fundamental representations
Let $G$ be a simple reductive group over $\mathbb C$, e.g. $G=\mathrm{SO}(V)$ is a special orthogonal group.
Let $W_1$ and $W_2$ be two irreducible representations of $G$. Assume both $W_i$ are ...
- 6,622
0
votes
0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
3
votes
0
answers
164
views
Pro-algebraic fundamental groups
Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic zero and fix a point $x\in X(K)$.
We can associate to $X$ two Tannakian categories: the category of ...
- 3,446
3
votes
0
answers
126
views
Parametrization of indecomposable modules via quiver varieties
Let $k$ be an algebraically closed field, $Q$ a quiver without oriented cycles and $m^\alpha (Q)$ the variety of quiver representations with dimension vector $\alpha$. There is a canonical algebraic ...
- 696
5
votes
0
answers
234
views
Avoiding Cartan subalgebra in a Lie algebra
Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation.
What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ ...
- 269
0
votes
1
answer
124
views
Horospherical type of a spherical variety
In the following, I will fix $k$ a characteristic zero algebraically closed field, and $G$ a connected reductive group over $k$, $B$ a Borel subgroup of $G$, $T\subseteq B$ a maximal torus, $X$ a $G$-...
- 121
1
vote
0
answers
73
views
Is the subgroup of $R$-trivial classes of an algebraic group an algebraic subgroup?
Let $G$ be an algebraic group over a field $F$. I'm willing to assume it is linear if that changes anything to what I'm going to say, and even reductive if that helps (but I don't think it should make ...
- 272
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
7
votes
1
answer
193
views
Constructing non-split $\mathbf{G}_m$-extensions of elliptic curves
I would like to find examples of abelian surfaces $A$ over a DVR $R$ with residue field $k$, so that the special fibre of $A$ is a non-split $\mathbf{G}_{m/k}$-extension of an elliptic curve over $k$.
...
- 265
5
votes
2
answers
344
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$,...
- 14.2k
5
votes
1
answer
290
views
Why does a line bundle on an abelian variety give a group extension only if it is algebraically trivial?
If $X$ is an abelian variety and $L$ is a line bundle, deleting the zero-section one obtains a diagram
$$
0 \to \mathbb G_m \to Y \stackrel{\phi}{\to} X \to 0
$$
where $\mathbb G_m = \phi^{-1}(0)$,
...
9
votes
2
answers
866
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,974
2
votes
0
answers
163
views
Equivariant Künneth formula for partial flag variety
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$. Let $P$ be a parabolic subgroup of $G$, $\mathscr{F}:=G/P$ the partial flag variety associated to $P$. For a $G$-variety $X$, ...
- 653
2
votes
0
answers
137
views
Irreducible non-reductive subgroup in GL(n) over a characteristic 0 field
Let $V$ be an $n$-dimensional vector space over a characteristic $0$ field $k$ (or better, let $V=\mathbb{A}^n_k$). I wonder whether the following is true:
Absolutely irreducible subgroups $H$ of $\...
- 455
4
votes
1
answer
418
views
Definition of Chow quotient
I am reading M. M. Kapranov's paper "Chow quotients of Grassmannians. I." (English) in Sergej Gelfand (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in ...
- 41
2
votes
0
answers
237
views
Obscure action of derivations on group schemes (SGA 3 Exp III)
In what follows, I will refer to prop. 0.8 in SGA 3 Exp. III which can be found for example at the link (https://webusers.imj-prg.fr/~patrick.polo/SGA3/). I will quickly introduce the notation without ...
- 21
2
votes
0
answers
82
views
Explicit $K$-basis of a Lie subalgebra
$\newcommand{\Kbar}{{\overline K}}
\newcommand{\Q}{{\mathbb Q}}
$I consider Example 5.7 of Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres of ...
- 14.2k
5
votes
1
answer
139
views
Representations with finitely many nilpotent orbits
Let $G$ be a reductive group over $\mathbb{C}$ and let $V$ be a finite dimensional representation of $G$. We can define the ``nilpotent cone'' of $V$ as
$$\mathcal{N}(V):=\{ v\in V\;: \; 0\in\overline{...
- 2,516
5
votes
0
answers
123
views
Algebraic groups and formal group laws in characteristic p
In characteristic zero, there is a well-known equivalence between Lie groups, formal group laws and Lie algebras.
Let $p$ be a prime. The equivalence between Lie groups and Lie algebras has an ...
- 413
4
votes
1
answer
165
views
Cohomology class of automorphism group of Galois twist
I've asked this question on Math.SE but didn't receive any answers.
Let $\Gamma$ be the Galois group of a finite Galois extension $K/k$ of fields of characteristic zero. Let $G$ be an algebraic group ...
- 383
2
votes
0
answers
154
views
A schematic representability of an algebraic space with group action
In the book "Néron Models" (BLR), there is a statement as follows (on page 164):
Let $S$ be a locally noetherian scheme and let $G$ be a smooth algebraic group space over $S$ with connected ...
- 291
7
votes
1
answer
287
views
What are the intermediate semisimple groups of type A?
Background: The first examples one sees of reductive groups over a field $k$ are $\text{GL}_n$, $\text{SL}_n$, and $\text{PGL}_n$. We all know the definitions of $\text{GL}_n$ and $\text{SL}_n$, and ...
- 835
0
votes
0
answers
84
views
is a linear algebraic group over an extension of $\mathbb{Q}_p$ a locally pro finite group?
Let $F$ be a non archimedean local field and let $G$ be linear algebraic group over $F$. I do not have a lot of experience with linear algebraic group, but it seems very obvious that $G$ inherits the ...
- 367
7
votes
1
answer
167
views
Counting matrices of bounded norm in SL_n(Z)
I'm looking for the asymptotic order of growth of the number of points in algebraic groups, such as $\mathrm{SL}_n(\mathbb{Z})$, of height/norm at most $X$, i.e. all entries are at most $X$ in ...
- 980
3
votes
1
answer
131
views
On Weil's theorem that a rational group action becomes regular action after some birational modification
People attribute the following theorem to Weil:
Any variety $X$ equipped with a birational action of a connected algebraic group $G$ is equivariantly birationally isomorphic to a variety $Y$ equipped ...
- 3,472
3
votes
1
answer
142
views
Relationships between the positive cone inside a root system and the dominant Weyl chamber
Let $G$ be a reductive group and fix a choice of positive roots inside the associated root system.
My question is about the relationship between the cone spanned by $\mathbb{Z}_{\geq 0}$-linear ...
- 33
6
votes
0
answers
139
views
For a proper scheme $X,$ the neutral component of automorphism group scheme $\mathrm{Aut}^0_X$ is an algebraic group
I am currently reading through M. Brion's "Notes on Automorphism Groups of Projective Varieties" found here. On page 5 he writes
We assume from now on that $X$ is proper, and we denote by $\...
- 201
4
votes
0
answers
101
views
Embedding of a nilpotent algebraic group in upper triangular matrices
Suppose we have a polynomial group law on $G=\mathbb{R}^n$ which gives it a structure of a nilpotent algebraic group.
Is it true that there exists an embedding of $G$ into the group of upper-...
- 728
1
vote
0
answers
138
views
Quotients of open subsets of the semi-stable locus
This is a rewrite of a deleted question. I've decided to focus on one particular example mentioned in that question. Below a point means a closed point.
Let $U$ be the set of irreducible non-cuspidal ...
- 23.5k
2
votes
0
answers
82
views
Product of a standard parabolic subgroup with the opposite one
Let $G$ be a semisimple algebraic group defined over an algebraic closed field. Fix a Borel subgroup $B$ and a maximal torus $T\subset B$. Let $\Delta$ be the set of simple roots. For a subset $\...
- 653
2
votes
0
answers
160
views
Centre of centralisers in connected reductive groups
Let $G$ be a connected reductive group over an algebraically closed field. Let $T$ be a maximal torus and $x\in T$. Let $G_x$ denote the centraliser of $x$ in $G$.
Question: What is an explicit ...
- 2,751
4
votes
2
answers
305
views
Is the property of being a torsor for a smooth affine group scheme detectable on the small étale site?
Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.
Let $h_X$, $h_G$ be the representable sheaves on ...
- 7,547
3
votes
0
answers
109
views
Does the Bruhat decomposition induces decomposition on integral points (on an open cell)?
Edit: both questions are resolved in comments. Let $F$ be a local field and $O$ its integral points. Let $G$ be a split reductive group over $O$. The Bruhat decomposition states that there is a ...
- 448
2
votes
1
answer
151
views
Closure of specialization of points of an affine group scheme with smooth generic fiber
Let $R$ be a henselian discrete valuation ring with residue field $k$, and let $G$ be an affine faithfully-flat finite type group scheme over $R$ with smooth generic fiber. Let $R'$ be the ring of ...
- 7,547
4
votes
1
answer
230
views
Extending Tannakian "dictionary" to gerbes
The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories".
Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...
- 363