All Questions
1,203 questions
79
votes
12
answers
13k
views
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, ...
- 118k
48
votes
5
answers
15k
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 ...
- 12.6k
47
votes
2
answers
9k
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 (...
47
votes
1
answer
1k
views
Summing infinitely many infinitesimally small variables makes sense in algebra
There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:
Consider the ring of ...
- 3,772
37
votes
4
answers
12k
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 ...
- 24.1k
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 ...
- 45.7k
36
votes
2
answers
7k
views
Why polarization of abelian varieties?
Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange.
I could never understand the intuition behind polarization of abelian ...
- 2,227
36
votes
1
answer
9k
views
Fontaine-Mazur for GL_1
For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...
- 21.3k
35
votes
3
answers
5k
views
In which ways can the isogeny theorem fail for local fields?
Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}_l \to \operatorname{Hom}_G(T_l A, T_l B)$ is ...
- 30.5k
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 ...
- 41.4k
33
votes
5
answers
8k
views
Why no abelian varieties over Z?
Motivation
I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form
the set $\{...
- 15.1k
32
votes
10
answers
3k
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 $2$. ...
- 2,215
32
votes
4
answers
5k
views
Over which fields does the Mordell-Weil theorem hold?
According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...
- 20.8k
31
votes
1
answer
5k
views
Modern proof of Serre's open image theorem?
Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...
- 1,905
31
votes
4
answers
5k
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 ...
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 ...
- 2,215
27
votes
7
answers
6k
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 ...
- 52.7k
27
votes
2
answers
3k
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 ...
- 28.8k
27
votes
4
answers
3k
views
Have people successfully worked with the full ring of differential 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 ...
- 57.6k
26
votes
4
answers
6k
views
Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...
- 16k
25
votes
3
answers
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 ...
- 5,163
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 ...
- 8,723
24
votes
4
answers
2k
views
Is every abelian variety a subvariety of a Jacobian?
Let $k$ be an infinite field and $A$ be an abelian variety over $k$. Can $A$ be embedded into a Jacobian variety $J$ over $k$?
In these notes by William Stein this is stated without proof in remark 1....
- 373
24
votes
5
answers
6k
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 ...
24
votes
3
answers
3k
views
Products of primitive roots of the unity
Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th ...
- 8,838
24
votes
2
answers
2k
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) ...
- 26.1k
24
votes
1
answer
2k
views
When is "independence of l" known?
My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) Weil-...
- 5,363
23
votes
1
answer
1k
views
Is hyperelliptic cryptography "practical"?
Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...
- 16.9k
23
votes
1
answer
2k
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 ...
- 50.6k
22
votes
2
answers
4k
views
non principally polarized complex abelian varieties
I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an ...
- 3,968
22
votes
4
answers
4k
views
Which curves can be found on Abelian varieties?
We know that each genus 2 curve is embedded into its degree 1 Jacobian.
Under which conditions on $C$, $A$, $g$ and $n$ is it possible for a genus $g$ smooth curve $C$ to be embedded in an Abelian ...
- 23.4k
22
votes
3
answers
2k
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 $G_{Q_p}$ ...
- 1,680
21
votes
2
answers
5k
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 papers:
Kawanoue, Hiraku, Toward resolution of singularities over ...
- 10.8k
21
votes
5
answers
5k
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 $\mathbb{C}$...
- 21k
21
votes
4
answers
2k
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 ...
- 3,857
21
votes
1
answer
2k
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 $df=0$,...
- 27.9k
21
votes
1
answer
4k
views
Crystalline cohomology of abelian varieties
I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power ...
- 1,500
20
votes
5
answers
4k
views
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 ...
- 671
20
votes
4
answers
2k
views
Are there Néron models over higher dimensional base schemes?
Are there Néron models for Abelian varieties over higher dimensional ($> 1$) base schemes $S$, let's say $S$ smooth, separated and of finite type over a field?
If not, under what additional ...
user19475
20
votes
2
answers
2k
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 Frobenius ...
- 16.2k
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
19
votes
3
answers
2k
views
Bhargava's work on the BSD conjecture
How much would Bhargava's results on BSD improve if finiteness of the Tate-Shafarevich group, or at least its $\ell$-primary torsion for every $\ell$, was known? Would they improve to the point of ...
user92332
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.
...
- 156k
19
votes
1
answer
2k
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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 ...
- 4,450
19
votes
1
answer
977
views
Lang's Jacobian identity: slicker, elementary proof?
In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven:
Theorem 1. Let $p$ be a prime. Let $\...
- 33.8k
19
votes
1
answer
1k
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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 ...
- 3,625
19
votes
1
answer
2k
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Perfectoid approach to resolution of singularities in char $p$
Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...
- 199
19
votes
2
answers
3k
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 ...
- 20.5k
18
votes
2
answers
2k
views
Are Jacobians principally polarized over non-algebraically closed fields?
How does one define the Torelli map $M_g \to A_g$ of moduli stacks? On geometric points a curve maps to its principally polarized Jacobian.
So what I am asking is: if I have a curve $C$ over a non-...
- 10.5k
18
votes
1
answer
2k
views
Embedding abelian varieties into projective spaces of small dimension
Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into.
Is $d(A)$ uniform over all abelian varieties of a ...
- 4,164