All Questions
304 questions with no upvoted or accepted answers
25
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0
answers
1k
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Status of the Euler characteristic in characteristic p
In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...
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20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
17
votes
0
answers
1k
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Katz--Mazur for abelian varieties
Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...
- 18.7k
15
votes
0
answers
2k
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Why was it so difficult to define the relative de Rham-Witt complex?
In Illusie's original article, the de Rham-Witt complex is defined for a smooth scheme over a perfect characteristic $p$ base $S$, without reference to $S$. Some 25 years later, Langer and Zink ...
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15
votes
0
answers
779
views
Lifting varieties from char. $p$ to char. 0 after alterations
The question is related to this MO question:
Lifting varieties to characteristic zero.
Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an ...
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14
votes
0
answers
1k
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A slick proof (?) of Zariski-Nagata purity in characteristic $p$
I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...
- 2,663
13
votes
0
answers
509
views
Is there a simple proof that representations of GL(n,k) are determined by their restriction to diagonal matrices?
Let $k$ be a field of characteristic zero. The general linear group $\mathrm{GL}(n,k)$ has a subgroup $\mathrm{D}(n,k)$ consisting of invertible diagonal matrices. These are linear algebraic groups ...
- 22.3k
13
votes
0
answers
749
views
Rings whose Frobenius is flat
Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$.
I am interested in rings for which $F_R$ is flat (hence ...
anon1432
13
votes
0
answers
407
views
Bernstein-Zelevinsky classification: viewing a representation as a subrepresentation or a quotient
$\DeclareMathOperator{\GL}{GL}$
$\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible ...
- 6,180
13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
- 3,637
12
votes
0
answers
729
views
Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)
Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$.
I have seen another post on ...
- 3,539
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
11
votes
0
answers
316
views
Mysterious "raison d'être" of filtrations of congruence subgroups
I wonder for long why congruence subgroups seem to arise so naturally in certain filtrations. Everything below is on a local field $F_p$.
Filtration for $GL_n$. Casselman and later Jacquet, Piatetski-...
- 5,424
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
11
votes
0
answers
1k
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Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?
It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
- 1,764
10
votes
0
answers
371
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
- 1,335
10
votes
0
answers
373
views
Local Langlands Correspondence for unramified principal series representations
Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which ...
- 6,180
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
9
votes
0
answers
295
views
Why the hyperoctahedral group is a ``reductive'' group?
Sorry for the misleading title, I actually mean the following:
The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...
- 1,666
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
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9
votes
0
answers
1k
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Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
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8
votes
0
answers
228
views
Chevalley-Solomon formula and Weyl character formula
Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
- 3,446
8
votes
0
answers
331
views
$\mathbb G_{\mathrm{m}}$-gerbes are to (derived) Azumaya algebras as $G$-gerbes are to …?
Let $X$ be a quasicompact quasiseparated scheme over a field $k$. The connection between Azumaya algebras over $X$ and $\mathbb G_{\mathrm{m}}$-gerbes over $X$ is well-known: there exists an injection ...
- 455
8
votes
0
answers
267
views
A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$
I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
- 1,856
8
votes
0
answers
300
views
Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?
Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
- 2,356
8
votes
0
answers
688
views
An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich
Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$.
In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map
$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
- 2,368
8
votes
0
answers
265
views
Chevalley restriction theorem: group vs lie algebra version
Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as
(1) $k[G]^G \xrightarrow{\sim} k[T]^W$
and ...
- 689
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
- 6,180
8
votes
0
answers
471
views
Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k
8
votes
0
answers
381
views
Degeneration of wildly ramified local monodromy representations - near or far from Deligne?
Suppose we have a surface $S$ mapping to a curve $C$ and a finite cover $Y/S$ that is ramified at a divisor $D$. For each point $x \in C$ we get a ramified cover of the curve $S_x$, and we can study ...
- 149k
8
votes
0
answers
873
views
Resolution of singularities in positive characteristic
I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a ...
- 783
7
votes
0
answers
231
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 ...
- 537
7
votes
0
answers
291
views
What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of compact support)?
Let $G$ be a linear algebraic group over a number field $k$. Let $\mathbb A$ denote the adeles of $k$, $\mathbb A_f$ the finite adeles, and $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. Here are ...
- 6,180
7
votes
0
answers
374
views
Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
- 7,377
7
votes
0
answers
355
views
Are curves over imperfect fields defined over a smaller field?
Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
- 149k
7
votes
0
answers
294
views
Picard scheme of varieties over imperfect fields
Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
- 4,450
7
votes
0
answers
236
views
Invariant theory of $SL_2$ over a field of positive characteristic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...
- 605
7
votes
0
answers
286
views
Level p characteristic 2 modular forms and thetas
BACKGROUND
Let p be an odd prime. An element of Z/2[[x]] is "modular of level p" if it is the mod 2 reduction of a g in Z[[x]] with g the Fourier expansion of a modular form for gamma_0(p). In ...
- 5,422
7
votes
0
answers
207
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
- 1,199
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
6
votes
0
answers
95
views
Generic representations of $\mathrm{GL}_n(\mathbb{R})$
Let $F$ be a local field of characteristic $0$, $G=\mathrm{GL}_n(F)$.
When $F$ is $p$-adic, Bernstein and Zelevinsky classified the irreducible generic representations. The statement is:
Let $\delta_{...
- 709
6
votes
0
answers
173
views
Orlik-Solomon algebra and hyperplane complements in positive characteristic
Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$.
Given a ring $R$ ...
- 728
6
votes
0
answers
113
views
$S_n$-invariant polynomials on the dual of reflection representation
Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...
- 1,336
6
votes
0
answers
111
views
subalgebra of invariants for a reductive subgroup
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Spec{Spec}$Trying to understand some tannakian reconstruction, I've stumbled about the following problem in invariant theory. I guess it's something ...
- 887
6
votes
0
answers
469
views
How to compute (unipotent) radicals?
My question follows some previous one, essentially this one. I want to understand, given an algebraic group $G$ (say linear), how to compute its radical and unipotent radical. The (unipotent) radical ...
- 937
6
votes
0
answers
236
views
When is an irreducible unramified principal series representation $K$-spherical?
Let $G = \operatorname{GL}_n(\mathbb Q_p)$, $T$ the usual maximal torus of $G$, and $K = \operatorname{GL}_n(\mathbb Z_p)$.
Let $\chi$ be an unramified character of $T$, with $\chi(t_1, ... , t_n) =...
- 6,180
6
votes
0
answers
217
views
Dimension of space of K-fixed vectors
If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular,
(1) $H(G(...
- 91
6
votes
0
answers
308
views
Is there a list of the inner forms of the quasisplit groups over local and global fields of characteristic 0?
From what I've gathered from trying to learn the classification of reductive groups, the classification of semisimple groups over a local or global field $F$ of characteristic 0 proceeds in several ...
6
votes
0
answers
282
views
$G$ is quasisplit at almost all places
Let $G$ be a connected, reductive group over a global field $k$. I am trying to understand why $G_v = G \times_k k_v$ is quasisplit for almost all places $v$ of $k$. There are several equivalent ...
- 6,180
6
votes
0
answers
1k
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Definition of Admissible Representation
Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...
- 6,180