All Questions
2,543 questions
7
votes
1
answer
1k
views
Chopping up Dynkin diagrams
Suppose I have a simple, simply connected (linear) algebraic group $\mathcal{G}$ over an algebraically-closed field $k$, which could have any characteristic. In fact, to keep things simple, let's ...
- 1,233
18
votes
3
answers
3k
views
Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 4,015
14
votes
2
answers
2k
views
Explicit cocycle for the central extension of the algebraic loop group G(C((t)))
Let $G$ be a simple Lie group and let $G(\mathbb{C}((t)))$ be its loop group.
The Lie algebra $\mathfrak{g}[[t]][t^{-1}]$ has a well known central extension
(see e.g.
Wikipedia) given by the cocycle
...
- 43.2k
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
2
votes
1
answer
939
views
Weyl group Invariants
What are the generators of $\mathbb C[V^m]^W$, where $W$ is the Weyl group
of type $E_6, E_7, E_8$, V^m denote 'm' (m > 1) copies of the Cartan subalgebra
and the action is the diagonal action?
Is ...
- 21
5
votes
3
answers
1k
views
Splitting of a division algebra with an involution of second kind
Let $k$ be a field, $K/k$ a separable quadratic extension,
and $D/K$ a central division algebra of dimension $r^2$ over $K$
with an involution $\sigma$ of second kind
(i.e. $\sigma$ acts non-trivially ...
- 14.2k
2
votes
0
answers
165
views
Connected components of the set of Hodge structures
Let $\mathbb S$ be the torus $\mathbb C^\times$ viewed as an algebraic group over $\mathbb R$. Let $G$ be any affine algebraic group over $\mathbb R$. The set $Hom(\mathbb S,G)$ of morphisms of real ...
- 21
8
votes
4
answers
3k
views
method of finding roots of polynominal equations with arithmetic operations and roots and other functions
Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on ...
- 1,626
0
votes
1
answer
470
views
Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology
The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let'...
- 1,783
33
votes
1
answer
1k
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 Conrad....
- 23.8k
15
votes
2
answers
3k
views
Are group schemes in Char 0 reduced? (YES)
A Theorem of Cartier (e.g. Mumford, Lecture 25) states that every separated, finite type group scheme $G/k$ over a field $k$ of characteristic $0$ is reduced. Does this result remain valid if we ...
- 3,284
7
votes
2
answers
2k
views
Jacobson-Morozov on the algebraic group level
Let $k$ be a field of characteristic 0 and $G$ a connected reductive algebraic group defined over $k$.
Let $g \in G(k)$ be a unipotent element. Is it true that there is a homomorphism $\varphi: SL_2(...
- 4,280
10
votes
1
answer
1k
views
Are there "reasonable" criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?
Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...
- 52.9k
8
votes
1
answer
696
views
Are groups in (Var/k, rational maps) necessarily algebraic groups?
Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant].
Let's consider a ...
- 23.4k
23
votes
1
answer
2k
views
Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf
Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...
- 50.7k
44
votes
2
answers
3k
views
What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?
Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped ...
- 54.6k
14
votes
2
answers
3k
views
How many ways are there to prove flag variety is a projective variety?
I am looking for references talking about different ways to prove flag variety $G/B$ is projective variety. Now I have some in mind:
There is a proof in Humphreys Linear algebraic groups, he first ...
- 5,515
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 839
12
votes
1
answer
1k
views
isomorphism of abelian varieties
Let $A, B, C$ and $D$ be abelian varieties (over $\mathbb{C}$) such that $A \times B \cong C \times D$, and $A \cong C$. From the irreducibility of abelian varieties, we can say that $B$ and $D$ are ...
- 363
3
votes
1
answer
252
views
Which algebraic groups are generated by (lifts of) reflections?
$\DeclareMathOperator\SL{SL}$The Cartan–Dieudonné theorem
states that each element $g \in \operatorname{O}(V)$, where $V$ is a quadratic space of dimension $n$ over a field of characteristic $\neq 2$, ...
- 4,280
2
votes
2
answers
1k
views
Automorphism group of bi-elliptic surface
$X$ = bi-elliptic surface (smooth and over $\mathbb{C}$),
Aut($X$) = the group of automorphisms of $X$,
Aut$^0(X)$ = connected component of the identity in Aut($X$).
Is Aut$^0(X)$ always an affine ...
- 363
2
votes
2
answers
584
views
Reference for Unitary Group attached to $E/k$
Unitary groups are very important objects in the setting of Langland's Conjecture because of the existence of Shimura Variety ( which I don't know) and also because people know how to attach a galois ...
8
votes
5
answers
939
views
Symplectic Steinberg group
I have several questions about Steinberg group and K2 for symplectic group:
Can I extend the definition of Steinberg symbols to symplectic case? Will they generate the center of Steinberg group?
Does ...
- 3,186
7
votes
2
answers
810
views
Is every homogeneous G-variety of the form G/H?
Let $G$ be an algebraic group over an algebraically closed field $k$. Then G/H is a quasi-projective homogeneous G-variety for any closed subgroup $H$. Now, several times I have seen something like "...
- 5,243
4
votes
2
answers
1k
views
Are all connected solvable affine algebraic groups supersolvable?
The basic question is whether there is a notion of chief factor of a connected solvable algebraic group that matches my intuition. A few smaller assertions are sprinkled through the explanation, and ...
- 10.7k
6
votes
3
answers
3k
views
Whenever I read "centraliser of maximal split torus", I think of...
Inspired by this question
I'd like to ask something more specific:
In the theory of connected reductive groups over fields, one often reads about the centraliser of a maximal split torus. Here is ...
23
votes
3
answers
5k
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 ...
10
votes
4
answers
2k
views
Connected components of the orthogonal group O(2n) in characteristic 2.
I am looking for a reference for the following fact:
The orthogonal group $O_{2n}$ over an algebraically closed field of characteristic 2
has exactly two connected components.
To be more precise, let ...
- 14.2k
5
votes
1
answer
4k
views
Simply connectedness of algebraic group
$G$ is a semisimple algebraic group over $k$, if $G_{\bar k}$ is simply connected when we do base change to $\bar k$, can we descent the simply connectedness to $G$?
Here, simply connectedness means ...
- 1,091
5
votes
1
answer
407
views
Generating Classical Groups over Finite Local Rings
I am interested in classical groups (in particular $SL_n$, $Sp_{2n}$, $SO_n^{+}$) over finite
rings of the form $$R_k=\mathbb{F}_q[t]/(t^k)$$ for some prime power $q$ (where $q$ is odd in the
...
- 51
12
votes
0
answers
716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
7
votes
2
answers
2k
views
Reductive subgroup corresponding to a subdiagram of the Dynkin diagram of a simple group
I am looking for a reference for the following well-known fact: For any subdiagram $\Delta_0$ of of the Dynkin diagram $\Delta=D(G)$
of an adjoint simple group $G$ over an algebraically closed field $...
- 14.2k
8
votes
3
answers
2k
views
Cohomology rings of $ GL_n(C)$, $SL_n(C)$
Can anyone provide me with the reference for the following fact
(idea of the proof will be appreciated too):
Cohomology ring with $\mathbb Q$-coefficients of a group $G$ (I don't know precisely what ...
- 1,783
1
vote
1
answer
435
views
Definition of congruence subgroup for non-matrix groups
For an algebraic group that cannot be embedded into $GL_n$, is there a nice definition for congruence subgroup? Do we just define it as the compact open subgroup of $G(A_f)$, where $A_f$ is the finite ...
- 1,091
9
votes
0
answers
668
views
Role of nontrivial component groups in Springer Correspondence?
Set-up for classical Springer Correspondence:
$G$ = reductive group (usually assumed to be semisimple of adjoint type) over $\mathbb{C}$, with Borel subgroup and
maximal torus $B \supset T$, Weyl ...
- 52.9k
4
votes
2
answers
402
views
lower bound for torsion of abelian varieties
Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...
12
votes
3
answers
4k
views
Books on reductive groups using scheme theory
Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are ...
- 19.6k
11
votes
1
answer
2k
views
Realizations and pinnings (épinglages) of reductive groups
Let $G$ be a reductive group over an (say, algebraically closed) field $k$. Springer (in his book on algebraic groups) calls for a chosen maximal torus $T$ in $G$ a family $(u_\alpha) _{\alpha \in \...
- 5,243
3
votes
2
answers
732
views
If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...
- 5,243
2
votes
1
answer
414
views
generators of the ideal of an unipotent-generated algebraic group
Given any affine algebraic group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$ with a faithfull representation in $GL_n(\mathbb{F})$ . If one knows the generators of the ...
- 53
7
votes
2
answers
1k
views
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
- 3,758
5
votes
3
answers
739
views
Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 13.1k
3
votes
1
answer
424
views
Principal bundle for contractible group is weak homotopy equivalence for ind schemes
This is may be obvious, but I am not comfortable with ind-schemes.
I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular ...
- 156k
6
votes
3
answers
745
views
Quotient of a reductive group by a non-smooth central finite subgroup
I need a construction in linear algebraic groups which uses taking quotient by a central finite group subscheme.
My question is, whether it goes through in ``bad'' characteristics, when this group ...
- 14.2k
36
votes
3
answers
7k
views
What is the difference between PSL_2 and PGL_2?
Let $K$ be a field and $G:=SL_2(K)$, then $G$ is a $K-$split reductive group (to use some big words). These groups are classified by a based root datum $(X,D,X',D')$. Let $G'$ be group associated to $(...
- 4,280
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
- 4,450
22
votes
3
answers
2k
views
One dimensional (phi,Gamma)-modules in char p
I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ ...
- 1,680
9
votes
1
answer
763
views
Restriction theorems over finite fields
A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...
- 20.2k
19
votes
1
answer
2k
views
The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
- 4,450
6
votes
2
answers
507
views
Concerning the dimension of a complex variety modulo a prime
Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain ...
- 63