All Questions
2,495 questions
5
votes
2
answers
1k
views
Local factors of Hasse-Weil zeta function - what do they have in common?
Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
- 5,637
21
votes
0
answers
617
views
Bounding failures of the integral Hodge and Tate conjectures
It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...
- 21.4k
11
votes
2
answers
760
views
Families of Fano varieties over non-hyperbolic curves
Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve.
Let $f:X\to C$ be ...
- 9,497
6
votes
2
answers
430
views
paper by Nakata on 2-adic Galois representations
In Keisuke Arai's 2007 paper "On uniform lower bound of the Galois images associated to elliptic curves", which can be found on ArXiv, Arai makes the following citation:
K. Nakata. On the 2-adic ...
- 1,298
6
votes
3
answers
811
views
Torsion group of the following elliptic curve
Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.
Can one compute the torsion group $...
- 63
2
votes
1
answer
122
views
Does the isotropic group of local galois representation have finite index?
Let $\rho : G\to \mathrm{GL}_n(\mathbb Z_p)$ be a crystalline representation of $G=\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$. For any non zero element $a\in \mathbb{Z}_p^n$ , it spans a rank one ...
- 699
9
votes
1
answer
718
views
Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology
This question is inspired by the question: Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
Let $K$ be a number field (or finitely generated field of ...
- 5,504
5
votes
1
answer
760
views
log smooth curve vs pointed node curve
F.Kato has a statement said that a log smooth curve $f:(X, M,\alpha)\to (k,N,\beta)$ where $k$ is an algebraic closed field and $N$ is some fine log structure on $k$, is equvalent to pointed node ...
- 699
3
votes
0
answers
239
views
log structure on Witt ring and Frobenius map
I am a little confused about the log structure of Witt ring and its Frobenius map.
Let $k=\bar{\mathbb{F}}_p$, $W:=W(k)$ the Witt ring.
We know that to deal with the semistable reduction, i.e. scheme ...
- 699
4
votes
1
answer
164
views
Deforming curves to other curves over the field of rational numbers
Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.
If $k$ is an algebraically closed field of characteristic zero, there exists a connected ...
- 41
7
votes
2
answers
614
views
Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is
Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.
If $\mathcal X$ can be defined over $\overline{...
- 9,497
0
votes
0
answers
199
views
Question about the "middle" intermediate Jacobian
Suppose $Y$ is a smooth projective variety of dimension $2p-1$ over $\mathbb{C}$. I have a few questions about the $p^{~\text{ th}}$ intermediate Jacobian $J^p(Y)$ of $Y$.
Does it come from (i.e. is ...
- 299
10
votes
2
answers
761
views
Applications of alterations
In 1995, de Jong proved the existence of regular alterations in arbitrary characteristic. I would like to have a little survey of important applications of this theorem, e.g. things you could do if ...
- 1,072
2
votes
2
answers
636
views
L-functions of Calabi-Yau varieties
This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes ...
- 6,982
3
votes
1
answer
522
views
representation of algebraic fundamental group of projective line minus three point
everyone, I want to ask is there any result in the literature
similar to the following:
Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X_{\mathbb{Q}}$ ...
- 699
1
vote
0
answers
103
views
degree of isogenies between Jacobians and Abelian Varieties
Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...
- 11
15
votes
3
answers
2k
views
What is the intuition behind the definition of cuspidal representations?
Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(...
- 153
12
votes
2
answers
424
views
Existence of local sections
I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...
- 121
4
votes
0
answers
469
views
integral hard Lefschetz
I am looking for examples $(X,\eta)$ where the integral hard Lefschetz is an isomorphism:
$X/k$ is a smooth projective variety of dimension $d$ over an algebraic closure of a finite field and $\eta \...
user19475
1
vote
1
answer
244
views
Arithmetic property of a surface of general type
In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...
- 39
-1
votes
1
answer
419
views
What is the probability that a randomly chosen number from set of c.e.number is period(number)?
What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)?
What is the probability that a randomly chosen number from the set of computable numbers is ...
- 3,104
3
votes
1
answer
360
views
What is the relation between KC and height of rational number?
Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator ...
- 3,104
2
votes
1
answer
170
views
Determining existence of p-adic point on a plane curve
Is there an implemented algorithm that will take a polynomial $f(x,y)\in\mathbb Z[x,y]$ and a prime $p$, and determine whether the equation $f(x,y)=0$ has a solution over $\mathbb Q_p$?
- 1,021
1
vote
1
answer
284
views
Etale cohomology and restricted direct product
[migrated from math.stackexchange: http://math.stackexchange.com/questions/727896/etale-cohomology-and-restricted-direct-product]
$\newcommand{\h}{\operatorname{H}}$
Let $k$ be a global field, $A$ an ...
- 5,759
8
votes
0
answers
184
views
Can the failure of the multiplicativity of archimedean L-factors be corrected?
My question is parallel to J. Borger' question:
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
As emphasized by Scholbach in his paper on special values of L-...
2
votes
0
answers
119
views
Bounds for the Tamagawa number of the Jacobian of a hyperelliptic curve
Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ and let $C$ be a hyperelliptic curve of genus $g$ defined over $K$ with Jacobian $J$.
Suppose that $C$ is given by ...
- 337
12
votes
1
answer
946
views
Inequality regarding sum of gaussian on lattices
When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...
- 804
12
votes
0
answers
215
views
Totally real points on curves
Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...
- 3,009
11
votes
1
answer
675
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-...
- 113
9
votes
1
answer
1k
views
Finite morphisms to projective space
Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...
- 91
11
votes
1
answer
775
views
Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring?
Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:
The Hodge-De Rham spectral sequence $E^{a,b}_1=...
- 699
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
11
votes
2
answers
1k
views
Representations of $\mathrm{SL}(2)$ in characteristic 2
$\DeclareMathOperator\SL{SL}$In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of $\SL(2)$-modules. In characteristic $p$, things are more complicated.
I am ...
- 605
4
votes
1
answer
390
views
Number field of degree 5
I am interested in field extensions of the rationals. About degree 3 extensions there are many refrences including the famous paper of Shanks "The Simplest Cubic Fields". In particular he gave ...
- 529
3
votes
1
answer
339
views
Potential good reduction of abelian varieties
In Corollary 3 on page 498 of the article "Good reduction of abelian varieties" it says that, under some specified conditions, the minimal subextension $L/K$ of $\overline{K}/K$ over which an abelian ...
5
votes
1
answer
859
views
Base change in crystalline cohomology?
Does one have a base change theorem in crystalline cohomology like in étale cohomology?
Suppose one has the following cartesian diagram
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}...
- 699
5
votes
1
answer
1k
views
What is "special" maximal compact subgroup of algebraig group over local field?
Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...
- 945
2
votes
1
answer
848
views
Compatibility of two definitions of elliptic elements in GLn
For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is equal ...
- 945
7
votes
1
answer
914
views
Explicit calculation of Weil Deligne representations
According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations.
However, given a galois representation, it is usually difficult ...
- 945
7
votes
2
answers
417
views
Is there a largest prime p such that J_0(p) completely splits into elliptic curves
The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...
- 2,224
8
votes
0
answers
318
views
The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor
As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...
- 16.9k
1
vote
0
answers
149
views
Covers of modular curves
I'm interested in covers of modular curves (especially cyclic covers) and I'm sure there's a lot of information out there available on this topic. However, I'm unable to locate any literature (on ...
- 1,841
2
votes
1
answer
218
views
A problem in intersection theorem
I'm reading the paper:
SGA 7 II, Intersections sur les surfaces regulieres.
In Papge 6 , I cannot understand why there is sign $-1$ in the formula (1.10.4):
Let $S$ be a trait, for any $\mathcal O_S$...
- 135
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
2
votes
1
answer
332
views
Pencil with desired Jet in Algebraic geometry(new!)
Let $k$ be an algebraic closed field.
Let $n$ be a positive integer.
Let $X$ be an irreducible, proper and smooth scheme over $k$ with an immersion $i:X\hookrightarrow E:=\mathbb P^N_k$ with $N$ ...
- 153
4
votes
1
answer
273
views
A problem on Jets in algebraic geometry
Let $k$ be a perfect field, let $n$ and $m$ be two positive integers.
Consider $X=\mathbb P_k^n\times \mathbb P_k^m$. Let $x_0=(1,0,\cdots,0;1,0,\cdots,0)\in X$ be fixed.
For any pair of integers $(...
- 153
7
votes
1
answer
440
views
When are Galois representations with open image attached to elliptic curves?
Let $K$ be a number field with absolute Galois group $G_K$.
Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in $GL_2(\hat{\mathbb{...
- 71
2
votes
1
answer
434
views
Explicit basis for the space of global sections of a twisted arithmetic ideal sheaf
Assume $x\in X=\mathbb{P}^1_{\mathbb{Z}}$ is a closed point with $f(x)=p\in Y$ where $f:X\rightarrow Y$, here $Y=Spec(\mathbb{Z})$. Assume $k(x)=\mathbb{F}_p$ and denote by $I_x$ the ideal sheaf of $x$...
- 251
2
votes
0
answers
120
views
Benchmark problems for computing rational points on varieties
Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties?
If so, could you please point ...
14
votes
1
answer
1k
views
The "Level N modular equation for delta" in characteristics 3, 5, 7 and 13
When $N > 1$, the modular forms $\Delta(z)$ and $\Delta(Nz)$ are algebraically independent over the complexes, and the same then is true of their expansions at infinity. But using the fact that
the ...
- 5,422