All Questions
2,494 questions
2
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3
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887
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Useful notion of unramified Galois representation
Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$.
Let $\...
- 45.7k
0
votes
0
answers
373
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Vanishing of motivic cohomology with finite coefficients in negative degrees
I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not.
STATEMENT:
Let $X$ be a smooth and projective scheme over a finite field $\...
- 945
2
votes
0
answers
473
views
$\sigma$-conjugate iff $p$-adically close
First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative $\mathbb{Z}...
CommunityBot
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1
vote
0
answers
192
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Special values of zeta functions and extensions of base fields.
Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements.
Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in |X|}\frac{1}{1-T^{deg_{k}...
- 945
8
votes
1
answer
475
views
How do you compute the primes of bad reduction?
Suppose that I am given a subscheme $Y$ of $\mathbf{P}^n_{\mathbf{Z}}$, flat over $\operatorname{Spec}\mathbf{Z}$ and with smooth generic fiber $Y_{\mathbf{Q}}$, defined by the vanishing of some ...
- 20.8k
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
3
votes
1
answer
390
views
Torsion of elliptic curves is finite
Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic ...
- 4,015
2
votes
2
answers
1k
views
Equation for simple Jacobian of a genus two curve
Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.
Can we write down an explicit equation for the abelian surface $J$?
I know $X$ can be ...
- 41
3
votes
1
answer
206
views
Detecting sections on an arithmetic variety
Let $S$ be Spec $O_K$ with $O_K$ the ring of integers of a number field $K$.
Let $X\to S $ be an arithmetic variety, i.e., an integral smooth quasi-projective $S$-scheme with generic fibre $X_\eta$ ...
- 4,195
5
votes
1
answer
865
views
Properties of subvarieties of a simple abelian variety
Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.)
Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension.
Suppose ...
- 6,253
3
votes
0
answers
671
views
An example of almost etale extension
In the paper of Faltings' "p-adic Hodge theory", Faltings showed an example of almost etale extension before he proved the almost purity theorem. The example is following:
Let $k$ be a perfect field ...
- 1,921
14
votes
2
answers
1k
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 ...
CommunityBot
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1
vote
1
answer
474
views
global section of vector bundle and reduction
Let $k$ be an algebraically closed field of char $p\neq 0$, $W_2(k)$ the witt vector of length 2. $C_1$ a smooth projective curve over $W_2(k)$, and $H_1$ a vector bundle over $C_1$. We denote $C_0$ ...
- 3,076
6
votes
1
answer
438
views
Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$
Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron ...
CommunityBot
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9
votes
1
answer
1k
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Top chern class in positive characteristic
Given a nonsingular, projective variety $X$ of dimension $n$ over an algebraically closed field $k$.
Over $k=\mathbb{C}$, the top chern class $c_n(T_X)$ of the tangent sheaf is the Euler ...
- 18.7k
17
votes
2
answers
2k
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Why does Tate's conjecture imply semisimplicity of crystalline Frobenius?
I'm trying to find a reference for the following fact:
If Tate's conjecture is true for all smooth projective varieties over $\mathbb{F}_p$, then the Frobenius endomorphism on the crystalline ...
- 306
3
votes
1
answer
600
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Grothendieck's section conjecture and base change: restricting sections
Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...
- 149k
9
votes
1
answer
549
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Varieties with infinitely many etale covers and rational points
Let $X$ be a (smooth projective geometrically connected) variety over a field $k$.
Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$.
...
- 30.5k
31
votes
1
answer
2k
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Do all curves have Néron models
Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.
Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?
By a Néron model, I mean ...
CommunityBot
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9
votes
2
answers
583
views
Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples?
The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over $...
- 26.1k
4
votes
0
answers
320
views
Dieudonné modules over rings of charateristic zero
Dear Colleagues,
would appreciate if you could recommend references, if such a theory exits, for the following question.
Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...
- 185
1
vote
1
answer
700
views
CM liftings of abelian varieties and liftings of Frobenius
It is well-known that if $A$ is an ordinary abelian variety over a finite perfect field $ k$ of characteristic $ p>0$ and $ W=W(k)$ is the ring of Witt vectors over $ k$, then the canonical ...
- 185
4
votes
1
answer
625
views
Does a curve over a number field have a finite etale cover of given degree
Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer.
Does there exist a curve $Y$ over $K$ with a finite etale $K$-...
- 4,008
1
vote
1
answer
445
views
What is the reduction of this hyperelliptic curve
Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.
Let $S$ be non-empty finite set of finite places of $K$ and suppose ...
- 30.5k
7
votes
2
answers
494
views
Jacobians defined over smaller fields
Let $L/K$ be an extension of number fields.
Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$.
In general, the Jacobian $J(X)$ probably doesn'...
- 3,147
9
votes
5
answers
2k
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The significance of modularity for all Galois representations
On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...
CommunityBot
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2
votes
1
answer
615
views
Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?
For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...
- 16.9k
3
votes
0
answers
406
views
Is the geometry of a variety determined by the counts of rational points?
In Diophantine Geometry: An introduction, Hindry and Silverman write "Geometry Determines Arithmetic" (pg. 2) and "Geometry Governs Arithmetic" (pg. 474).
On pg. 211 of the same book, the authors ...
- 7,072
1
vote
0
answers
192
views
"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
1
vote
0
answers
187
views
Fields over which cubic hypersurfaces are rational
All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...
- 3,779
8
votes
1
answer
759
views
Honda-Tate in families
Let $k$ be a finite field, say with $q=p^a$ elements. Honda-Tate theory states that there is a bijection between isogeny classes of simple abelian varieties over $k$ and $\mathrm{Gal}(\overline{\...
- 149k
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votes
1
answer
273
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local galois representation with higher coefficient
Suppose K is a local field , G is its galois group, V a fine dimensional Vector space over F, which is a sub field of K, and totally ramified over $Q_p$. Consdider the linear action of G on V (V is ...
- 20.5k
14
votes
1
answer
1k
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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 ...
CommunityBot
- 1
12
votes
4
answers
715
views
Behaviour of Zeta-function under Finite Morphism
Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...
- 47.4k
5
votes
1
answer
630
views
Special value of $L$-function
Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...
- 23k
4
votes
2
answers
349
views
Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group
A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
- 96.4k
4
votes
1
answer
604
views
A question about the Tannakian etale fundamental group of a curve
Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let $n\...
- 1,838
3
votes
1
answer
485
views
Group of connected components of the global Néron-Raynaud model of a torus
Let $K = \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$
defined over a finite field of constants $\mathbb{F}_q$. Let $T$ be a $K$-torus.
We choose one ...
CommunityBot
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3
votes
0
answers
315
views
Question about witt vector of some ring
Suppose $R=Z_p[t]$ , and $\hat{R}$ its p-adic completion, suppose we have Endormorphism $\Phi$ of $\hat{R}$, whose redution mop p is just the absolute Frobenius of $\hat{R}/p\hat{R}$. And $R_{\infty}=...
- 709
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0
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282
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Does semi-stable reduction behave well with Weil restriction of scalars
Let $A$ be an abelian variety over a number field $K$ with semi-stable reduction over $O_K$.
Does the Weil restriction $\textrm{Res}_{K/\mathbf{Q}}A$ of $A$ to $\mathbf{Q}$ have semi-stable reduction ...
- 1,213
11
votes
1
answer
761
views
Mordell-Weil group of the universal abelian scheme
Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
...
- 30.5k
0
votes
2
answers
336
views
Does the self-product of a $g$-dimensional abelian variety contain an abelian variety of dimension smaller than $g$ at some point
Let me be more precise than the title. (This will be my last attempt to do something with abelian varieties. Sorry for all the basic questions. The answers have been great!)
Let $A$ be a simple ...
- 35.2k
0
votes
1
answer
298
views
Is any simple abelian variety covered by a non-simple abelian variety
Let $A/k$ be a simple abelian variety.
Does there exist a non-simple abelian variety $B/k$ and a finite homomorphism $f:B\to A$ over $k$?
I don't need $f:B\to A$ to be etale.
- 1,213
3
votes
1
answer
492
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Are abelian varieties degree two covers of some projective space
Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$.
There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question.
Does there exist a finite morphism $A\to \mathbf{P}^...
- 4,111
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 ...
- 6,543
1
vote
1
answer
215
views
Is the number of twists of a curve with a section in a given field finite
Let $X$ be a smooth projective geometrically connected curve over a number field $k$ of genus $g\geq 2$.
Is the number of twists of $X$ always infinite? (The answer is no, because there aren't any ...
- 3,625
10
votes
6
answers
2k
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Proofs in the same vein as Ax-Grothendieck
I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
- 38.7k
6
votes
1
answer
302
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Is the class of $k$-gonal curves dominant
Before I start, let me make a note on terminology. Curves are always smooth projective connected curves over an algebraically closed field of characteristic zero.
Let $\mathcal C$ be a class of ...
- 27k
7
votes
2
answers
1k
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questions on Néron-Tate canonical height
I have three questions regarding height pairings:
In [Serre, Lectures on the Mordell-Weil theorem], p. 85 f., it is stated that the following function is a local height function:
"Let $V/R$ be a ...
CommunityBot
- 1
3
votes
0
answers
145
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Curves whose stable reductions do not contain rational curves
Let $X$ be a smooth projective curve over $K:=K(A)$. $A$ is a strict henselian ring, $A/m=k=\bar k$. Suppose $\cal X$ is a stable model of $X$, ${\cal X}_{s}$ is the special fiber.
My question is:
...
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