All Questions
828 questions
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algebraic de Rham cohomology of toric varieties (reference request)
I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled:
...
- 1,142
5
votes
0
answers
217
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Exact differential forms in characteristic $p>0$
Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the ...
- 245
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0
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1k
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Some questions about cuspidal representations and automorphic representations
My reference is Daniel Bump's book, Automorphic Forms and Representations. $G$ is a connected reductive group over a number field $k$ (in Bump's book he takes $G = \operatorname{GL}_n$). Let $K = K_{...
- 6,180
5
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answers
197
views
Semisimplicity of the p-adic étale Tate module over $F_p(t)$
Let $k$ be a finitely generated field of positive characteristic p.
Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...
- 728
5
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0
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243
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Map associated to linear system onto curve is morphism
In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
- 61
5
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0
answers
568
views
Eisenbud-Goto conjecture in Positive Characteristic
Eisenbud-Goto conjecture predicted that the Castelnuovo-Mumford regularity
${\rm reg}(X)$ of a non-degenerate projective variety $X\subset \mathbb{P}^N$
is bounded by the $\deg(X)-{\rm codim}(X,\...
- 1,595
5
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0
answers
210
views
rational cohomology of classifying spaces of complex reductive Lie groups
I am looking for a reference or an ad-hoc proof of the following fact, which seems to be known to experts: Let $\mathbf{G}$ be a complex algebraic group with maximal (algebraic) torus $\mathbf{T}$ and ...
- 2,928
5
votes
0
answers
162
views
Definition of the homomorphism $\textrm{Gal}(k_s/k) \rightarrow \textrm{Aut}(\psi_0(G))$
Let $G$ be a connected, reductive group over a field $k$. Identify $G$ with its $\overline{k}$-points. Let $\Gamma = \textrm{Gal}(k_s/k) = \textrm{Aut}(\overline{k}/k)$. Let $B$ be a Borel subgroup ...
- 6,180
5
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answers
257
views
Symplectic leaves in positive characteristic
I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated ...
- 599
5
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0
answers
287
views
Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
- 163
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501
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How to think about non-connected reductive groups
Suppose someone knows well the theory of connected reductive groups, over an algebraically closed field or more generally over any field, say for instance most of the content of Borel-Tits.
Is ...
- 26.1k
5
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0
answers
223
views
Decomposition of k-split tori of p-adic reductive groups
Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$.
There is a group homomorphism :
$$...
- 991
5
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0
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387
views
Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?
In this question Joel Bellaiche constructed an algebra, M, of modular forms for
gamma_0 (N) in finite characteristic (which he called p, but I'll call ell) and asked to know its structure. Matt ...
- 5,422
5
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0
answers
253
views
Do there exist pseudo-reductive (but not reductive) groups of small dimension?
I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is ...
- 181
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0
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454
views
If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
- 4,487
5
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0
answers
234
views
Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
- 10.7k
5
votes
0
answers
530
views
Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?
Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$.
Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
- 3,897
4
votes
2
answers
923
views
What is the correct formulation of the CDE triangle?
The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
- 9,132
4
votes
1
answer
638
views
perfect fields in positive characteristic
Let $k$ be an infinite perfect field in positive characteristic $p$, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example:...
- 168
4
votes
1
answer
428
views
p-torsion in the Picard group of a regular projective curve
Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
- 2,051
4
votes
2
answers
552
views
Variety of conjugacy classes
Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some ...
- 1,190
4
votes
1
answer
272
views
How do non-trivial global differentials give non-trivial cohomology classes in positive characteristic
Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$.
If $k= \mathbb C$, there is a natural injective morphism of vector spaces
$$H^0(X,\...
- 49
4
votes
2
answers
604
views
Adem-Wu relations from Bullett-Macdonald identities
Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...
- 33.8k
4
votes
3
answers
267
views
References for $K$-orbits in $G/B$
Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
- 741
4
votes
2
answers
231
views
Weight of the defining function of a Bruhat cell in a simply connected semisimple group
Let $G$ be a connected semisimple simply connected group over an algebraically connected field. Let $w_0$ be the longest element in the Weyl group $W$ and $s_i$ be a simple reflection in $W$. Choose a ...
- 291
4
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1
answer
248
views
Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$
There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\...
- 3,779
4
votes
2
answers
332
views
Ring of invariants for $n$-tuples of Lie algebras
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\Tr{Tr}$Consider the diagonal action of $\GL(n,\mathbb{C})$ on the variety of $k$-tuples of matrices, $\M_{n\times n}(\mathbb{C}...
- 142
4
votes
2
answers
918
views
Katz's proof of Cartier's (descent) theorem
I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
- 121
4
votes
1
answer
976
views
Character determines the representation?
Consider a semisimple Lie group or a $p$ adic reductive group $G$.
To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?
- 11.2k
4
votes
2
answers
339
views
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
- 16.9k
4
votes
3
answers
275
views
Does every nilpotent lie in the span of simple root vectors?
Let $G$ be a reductive group (for simplicity, we can work over $\mathbb{C}$). Let $N \in \operatorname{Lie} G$ be nilpotent. Does there exist a Borel pair $(B,T)$ of $G$ such that $N$ lies in the span ...
- 953
4
votes
1
answer
502
views
Motivic integration in positive characteristic: how much is known?
It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
- 16.9k
4
votes
1
answer
998
views
Dimension of irreducible representations in characteristic p
Hi, I know that over $\mathbb{C}$ the dimension of an irreducible representation of a finite group $G$ must divide the order of the group. I've read somewhere that if $p$ does not divide the order of ...
- 407
4
votes
1
answer
412
views
F_q-structures on schemes
Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the ...
- 5,243
4
votes
1
answer
198
views
Simple restricted but not restricted simple Lie algebras
Let $F$ be a field which has a positive characteristic $p \ge 2$ and $(\mathfrak{g},[p])$ be a restricted Lie algebras over a field $F$ where $[p]$ is a $p$-th power map on $\mathfrak{g}$. $(\mathfrak{...
- 145
4
votes
2
answers
340
views
On families of supersingular abelian surfaces over the projective line
Let $k=\mathbb{F}_q$. I recently learned that there are non-isotrivial families $f:X\to \mathbb{P}^1_k$ of supersingular abelian surfaces. In particular, the Kodaira-Spencer map of this family is non-...
- 43
4
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1
answer
358
views
Examples of perfect pseudo algebraically closed fields in positive characteristic
Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?
- 2,323
4
votes
1
answer
887
views
Reference request for $pro-p$ Iwahori subgroup of $GL_n(F)$
I am searching for a book/lecture notes/articles where I can find the definition and properties of the $pro-p$ Iwahori subgroup of $GL_n(F)$,(with examples if possible) the Iwahori decomposition of ...
- 685
4
votes
2
answers
394
views
Colon property of Gorenstein rings
Let $(R, \mathfrak{m})$ be a Gorenstein local ring of characteistic $p>0$. Let $x_1,...,x_d$ be a system of parameters of $R$. Let $I$ be an $\mathfrak{m}$-primary ideal containg $(x_1,...,x_d)$. ...
- 2,773
4
votes
1
answer
398
views
Properties of divisors when moving from char 0 to char p.
Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
- 141
4
votes
1
answer
171
views
Centralizer of conjugacy classes
Let $\mathrm{G}$ be a complex reductive group and let $\mathrm{O}_g$ be the adjoint orbit of $g\in \mathrm{G}$.
I wonder is the centralizer $\mathrm{C}_{\mathrm{G}}(\mathrm{O}_g)$ still a reductive ...
- 395
4
votes
1
answer
130
views
Do parabolic inductions share a composition factor if and only if the inducing data are associate?
Let $F$ be a local field of characteristic zero and $G$ a connected reductive group over $F$. Let us call an inducing datum a triple $(P,M,\sigma)$, where $P$ is a parabolic subgroup of $G$, $M$ is a ...
- 387
4
votes
1
answer
311
views
Maximal torus of linear algebraic group over a ring
Let $G$ be a linear algebraic group over a $k$-algebra $A$, where $k$ is an algebraically closed field.
Consider the structure morphism $G\rightarrow U={\rm Spec}(A)$.
Assume that for every $k$-point ...
- 1,595
4
votes
1
answer
321
views
Torsors in the analytic topology versus torsors in the etale topology
Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$.
Is every analytic $G$-torsor over $S$...
4
votes
1
answer
369
views
Gauss mapping in finite characteristic
Suppose that $X\subset\mathbb P^n$ is a $d$-dimentional smooth projective variety (not a linear subspace) over an algebraically closed field. If $\gamma\colon X\to\mathrm{Gr}(d,\mathbb P^n)$ is Gauss ...
- 1,939
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 3,605
4
votes
1
answer
683
views
Canonical rational form for $SL(n)$
The canonical rational form helps us to parametrize the conjugacy classes in $GL(n)$ over any commutative field.
How can we parametriize the conjugacy classes in $SL_n(k)$, where $k$ is an ...
- 11.2k
4
votes
1
answer
751
views
How to understand the representation theory of $SL(n)$ from $GL(n)$?
Let $F$ be a local field. Consider the group extension (split)
$$ PSL(n,F) \rightarrow PGL(n,F) \rightarrow F^\times / (F^\times)^n.$$
What knowledge about $PGL(n)$ is necessary in order to understand ...
- 11.2k
4
votes
1
answer
674
views
Inseparable Galois Cohomology
First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...
- 1,199
4
votes
2
answers
694
views
Ample line bundle and Frobenius morphism on smooth toric variety
Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...
- 85