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42
votes
12answers
4k views

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

The structure of the multiplicative groups of Z/pZ or of Zp is the same for odd primes, but not for 2. Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for 2. So ...
32
votes
2answers
4k views

current status of crystalline cohomology?

The great references given on Ilya's question make me wonder about the current status of the many conjectures and open questions in Illusie's survey from 1994 on crystalline cohomology. Obviously ...
29
votes
10answers
2k views

Which 'well-known' algebraic geometric results do not hold in characteristic 2?

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$. Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...
27
votes
6answers
4k views

Algebraically closed fields of positive characteristic

I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
25
votes
3answers
2k views

product of all F_p, p prime

Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements. Is it true that $R$ has a quotient by a maximal ideal which is a field of ...
22
votes
1answer
1k 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 ...
21
votes
2answers
1k 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 ...
20
votes
4answers
3k views

What would a “moral” proof of the Weil Conjectures require?

At the very end of this 2006 interview (rm), Kontsevich says "...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better ...
20
votes
3answers
2k 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 ...
20
votes
4answers
2k views

The Frobenius morphism

I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power". Generates a ...
20
votes
3answers
1k 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 $G_{Q_p}$ ...
19
votes
4answers
1k views

Simplest example of jumping of cohomology of structure sheaf in smooth families?

Using Hodge theory (and the ill-defined Lefschetz principle), one can show that in characteristic 0, given a proper smooth family $X \rightarrow B$, the cohomology groups of the structure sheaf of the ...
19
votes
4answers
2k views

Wild Ramification

The question is, loosely put, what is known about wild ramification? Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there ...
19
votes
4answers
2k views

Have people successfully worked with the full ring of diferential operators in characteristic p?

This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
19
votes
1answer
936 views

Have we ever proved any non-solvable case of reciprocity without the Langlands program ?

The reciprocity of the title is the following not completely well-posed problem: Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe" (in some sense) ...
18
votes
7answers
2k views

Etale covers of the affine line

In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
17
votes
3answers
1k 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 ...
17
votes
1answer
805 views

When does the relative differential $df=0$ imply that $f$ comes from the base?

Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of $B \otimes_A B \to B$, as in [Hartshorne II.8]. If ...
15
votes
5answers
2k views

Can we count isogeny classes of abelian varieties?

Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...
15
votes
5answers
2k 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 ...
15
votes
3answers
1k 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. ...
15
votes
3answers
737 views

What are supersingular varieties?

For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties. I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an ...
15
votes
1answer
609 views

Higher level analogs of Nicolas-Serre theory

NICOLAS-SERRE THEORY Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke ...
14
votes
2answers
579 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 ...
14
votes
1answer
762 views

Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?

I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
14
votes
2answers
943 views

Bertini theorems for base-point-free linear systems in positive characteristics

Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...
13
votes
1answer
532 views

Help wanted with Chebotarev condition in characteristic 2

Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field ...
13
votes
1answer
653 views

Coarse moduli spaces over Z and F_p

I would like to know to what extent it is possible to compare fibers over $\mathbb{F}_p$ of coarse moduli spaces over $\mathbb{Z}$, and coarse moduli spaces over $\mathbb{F}_p$. I ask a more precise ...
13
votes
2answers
958 views

Frobenius splitting and derived Cartier isomorphism

Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results. 1. If $X$ is ...
13
votes
0answers
395 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 ...
12
votes
4answers
1k views

Finite subgroups of $PGL_2(K)$ in characteristic $p$

Let $K$ be a field of characteristic $p$. What are the finite subgroups of $PGL_2(K)$ whose orders are divisible by $p$? And if $G$ and $H$ are two such subgroups that are isomorphic, can one say when ...
12
votes
2answers
738 views

Can a reductive group act non-linearly on a vector group?

Let $k$ be a field; I'm going to discuss linear algebraic groups over $k$. The question I'll pose is only interesting when the characteristic is $p>0$. 1. Some motivation A vector group is an ...
12
votes
2answers
667 views

Is the tangent space functor from PD formal groups to Lie algebras an equivalence?

The previous version of this question was rather badly broken, and I hope this version makes some sense. There have been a few questions on MathOverflow about how much representation-theoretic ...
11
votes
2answers
1k views

Why is one interested in the mod p reduction of modular curves and Shimura varieties?

Why is one interested in the mod p reduction of modular curves and Shimura varieties? From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the ...
11
votes
3answers
4k views

Finite extension of fields with no primitive element

What is an example of a finite field extension which is not generated by a single element? Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
11
votes
3answers
984 views

Representations in characteristic p

Let G be a finite group and let F be an algebraically closed field. If the characteristic of F is 0, then the number of irreducible F-representations of G is given by the number of conjugacy classes ...
11
votes
3answers
961 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 $W_k$ ...
11
votes
3answers
1k views

On Category O in positive characteristic

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of ...
11
votes
1answer
527 views

Obstructions to formally integrating vector fields in characteristic p?

Let $M$ be a smooth scheme over some field $k$ of characteristic $p$, and $\vec X$ a vector field on it. Equivalently, $\vec X$ gives a map $Spec\ k[\epsilon]/\langle \epsilon^2 \rangle \times M \to ...
11
votes
0answers
415 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 ...
11
votes
0answers
480 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 ...
10
votes
2answers
723 views

Weil Conjectures for nonprojective algebraic varieties

If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?
10
votes
2answers
677 views

Class groups of normal domains over finite fields

Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be ...
10
votes
1answer
837 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 des schemas" ...
10
votes
1answer
522 views

Frobenius splitting of Fano varieties

Dear MO, Question 1. Do you know of an example of a Fano variety which is not Frobenius split? Background (1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...
10
votes
2answers
357 views

Counting points on varieties of low codimension

The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...
10
votes
1answer
1k views

State of resolution in positive characteristic?

Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these programs: 1, 2. It would be great if someone who listenes ...
10
votes
2answers
745 views

Does a universal Frobenius map exist?

For any prime p, one has the Frobenius homomorphism Fp defined on rings of characteristic p. Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ...
10
votes
1answer
413 views

Do Richardson varieties have rational singularities in arbitrary characteristic?

The title basically asks the question. I'll review the relevant terminology and explain what I have and haven't found in the literature. Let $G$ be a reductive group. Let $v \leq w$ be elements of ...
10
votes
1answer
770 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 ...