All Questions
404 questions
5
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446
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More questions involving characteristic 2 theta series identities
In my answer to my earlier question, "Existence of certain identities involving characteristic 2 thetas", I established some curious identities when the thetas have prime "level" congruent to 1 mod 4 ...
- 5,422
5
votes
1
answer
461
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Given a branched cover with branch cycle description $(g_1,...,g_r)$, does $g_i$ generate some decomposition group?
Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...
- 5,929
5
votes
1
answer
1k
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Excellent schemes
I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...
- 53
5
votes
0
answers
139
views
Liouville property in the Bost theorem on foliations
Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
- 485
5
votes
0
answers
171
views
Finite simple groups of automorphisms of finite simple Lie algebras
I begin by briefly recalling some basic facts in order to pose my question in context.
According to the classification, the finite simple groups are cyclic of prime order, are alternating on $n \geq 5$...
- 177
5
votes
0
answers
148
views
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
views
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
5
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0
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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
votes
0
answers
243
views
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
votes
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
votes
0
answers
256
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
votes
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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votes
0
answers
387
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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
votes
0
answers
454
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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
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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
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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
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
1
answer
248
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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
918
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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
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
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
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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
votes
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
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
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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
673
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
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 $C, \eta$ ...
4
votes
1
answer
272
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Finiteness of cohomology with finite coefficients
Let $G$ be a finite abelian group and let $S$ be a variety over $\mathbb{F}_p$. It is natural (I think) to expect that the cohomology group $H^i(S,G)$ is finite. But with respect to which cohomology?
...
- 41
4
votes
1
answer
796
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Deformation space of non-ordinary abelian varieties
It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...
- 395
4
votes
1
answer
227
views
Compute de Rham-Witt sheaves
I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
- 309
4
votes
1
answer
269
views
Fourier transform on finite groups in characteristic $p>0$
Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
- 41
4
votes
1
answer
277
views
Jordan decomposition on the dual Lie algebra
$\newcommand\fg{\mathfrak g}\newcommand\gl{\mathfrak{gl}}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field, and let $G$ be a smooth, affine algebraic ...
- 12.9k
4
votes
2
answers
460
views
Are Chow groups invariant under universal homeomorphisms?
Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
- 16.9k
4
votes
1
answer
622
views
finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme
Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.
Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{...
user19475
4
votes
2
answers
1k
views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
- 3,362
4
votes
1
answer
221
views
Do permutation modules of solvable groups have self-dual socle in characteristic 2?
I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that ...
- 143
4
votes
1
answer
442
views
Solvable Lie algebra whose nilradical is not characteristic
Say that an ideal in a Lie algebra is characteristic if it is invariant under every derivation of the algebra.
It is well known that the nilradical of a finite-dimensional Lie algebra over a field ...
- 310
4
votes
1
answer
627
views
Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...
- 33.8k
4
votes
0
answers
108
views
Shafarevich conjecture for Abelian varieties over global function fields
Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
- 679
4
votes
0
answers
135
views
Nilpotent orbits in characteristic $0$ vs. positive characteristics
Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
- 2,751