Questions tagged [characteristic-p]

Fields of characteristic $p$, i.e., fields for which there is a prime $p$ such that $px=0$ for each $x$. Do not use this tag for questions on characteristic polynomials of a matrix.

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Is there a high-concept explanation for why characteristic 2 is special?

The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
Qiaochu Yuan's user avatar
34 votes
2 answers
3k views

The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
Kevin Buzzard's user avatar
19 votes
3 answers
2k views

Elkies' supersingularity theorem in higher dimension

The following is a theorem of Elkies: Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero. ...
David E Speyer's user avatar
8 votes
1 answer
717 views

Deligne's exterior power

In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism $$p : A^{\otimes n} \to A^{\otimes n}, ...
Martin Brandenburg's user avatar
7 votes
1 answer
5k views

Chevalley's Theorem on Constructible Sets

I'm having a hard time understanding the theorem in the title, more specifically the proof of the related fact that the image of a dominant morphism contains a dense open set of it's closure. (My ...
Confused's user avatar
6 votes
0 answers
337 views

Are all stabilizer groups of the co-adjoint action smooth?

Let $k$ be a (non-archimedean) local field of positive characteristic $p$ and $\mathfrak{n}$ be any finite-dimensional nilpotent Lie algebra over $k$ with nilpotence length $l<p$. It is well-known ...
m07kl's user avatar
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1 vote
1 answer
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locally closed orbits in metric Hausdorff topology

I learned the following fact from Bruhat and Tits's paper "Homomorphismes “abstraits” de groupes algebriques simples" Section 3.18 that Let $k$ be a local field. Suppose that a $k$-group $H$ acts $k$...
m07kl's user avatar
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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 ...
S. Carnahan's user avatar
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28 votes
3 answers
2k views

Intuitive pictures in characteristic p

This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...
Jesus Martinez Garcia's user avatar
25 votes
2 answers
2k views

Reference for de Rham cohomology in positive characteristic

It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...
R. van Dobben de Bruyn's user avatar
25 votes
0 answers
1k views

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 ...
Vivek Shende's user avatar
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21 votes
5 answers
4k views

Mirror symmetry mod p?! ... Physics mod p?!

In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that? (Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$...
Kevin H. Lin's user avatar
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19 votes
5 answers
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Equivalent statements of the Riemann hypothesis in the Weil conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
Brandon Levin's user avatar
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 $W_k$ ...
Xandi Tuni's user avatar
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15 votes
1 answer
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Number of curves over a finite field

Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$? In other words does there exists a formula for the number of rational points ...
Puzzled's user avatar
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15 votes
2 answers
791 views

Can the failure of the multiplicativity of Euler factors at bad primes be corrected?

Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all. If $X$ is a scheme of finite type over a finite field, then the ...
JBorger's user avatar
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13 votes
1 answer
1k views

Do varieties with ample canonical bundle have finite automorphism group in small characteristic?

Suppose $X$ is a smooth projective variety over a field $k$, with ample canonical bundle. If $\operatorname{char}(k)=0$ or $\operatorname{char}(k)>\dim(X)$ and $X$ lifts to $W_2(k)$ (thanks ...
Daniel Litt's user avatar
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12 votes
2 answers
875 views

Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$

This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
Huy Dang's user avatar
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11 votes
1 answer
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Are automorphism groups of hypersurfaces reduced ?

In the following article : "H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1964) 347-361", it is shown that in finite characteristic, automorphism groups of ...
Olivier Benoist's user avatar
11 votes
0 answers
1k views

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 ...
bhwang's user avatar
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11 votes
2 answers
885 views

On a proposition in Hartshorne's paper "Ample vector bundles on curves"

In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following: Let $A$ be an abelian variety [over an alg. closed field $...
Damian Rössler's user avatar
11 votes
1 answer
644 views

Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded Lie algebra" as explained first in Goldman-...
user47856's user avatar
  • 113
11 votes
0 answers
570 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 ...
paul Monsky's user avatar
  • 5,412
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 ...
Jim Humphreys's user avatar
9 votes
3 answers
2k views

Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?

Background/motivation It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
Andrea Ferretti's user avatar
9 votes
1 answer
769 views

Endomorphism ring of simple ordinary abelian variety

Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer ...
Damian Rössler's user avatar
9 votes
1 answer
448 views

Showing subgroups with equal Lie algebras are equal

Let $k$ be a field. It might as well be algebraically closed, but I do not want to assume that it has characteristic $0$. I will write "group" for "affine group scheme over $k$", ...
LSpice's user avatar
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8 votes
1 answer
729 views

A variant on characteristic $p$ de Rham cohomology

I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction. Let $k$ be a perfect field ...
David E Speyer's user avatar
8 votes
2 answers
1k views

Lefschetz on étale fundamental group for quasi-projective varieties

If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$ $$\pi^1(H)\to\pi^1(X)$$ is an isomorphism, ...
Giulia's user avatar
  • 483
7 votes
4 answers
708 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 ...
Lars's user avatar
  • 4,400
5 votes
0 answers
383 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 ...
paul Monsky's user avatar
  • 5,412
5 votes
0 answers
230 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, ...
numberjedi's user avatar
4 votes
1 answer
612 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 ...
darij grinberg's user avatar
4 votes
1 answer
862 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 \...
Justin Campbell's user avatar
3 votes
0 answers
88 views

Dimension of a kernel of a cocycle map

Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following: Compute the kernel (or at ...
Marcos's user avatar
  • 577
3 votes
1 answer
336 views

Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
paul Monsky's user avatar
  • 5,412
0 votes
2 answers
2k views

non discrete valuation ring [closed]

Hi, I am looking for examples of non-discrete valuation rings. Could you help me? Thanks
unknown's user avatar
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