All Questions
2,494 questions
2
votes
0
answers
321
views
CM abelian variety from an algebraic Hecke character?
Hi,
Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a
"rank 1 CM-motive" $M$ with
$\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the ...
- 2,842
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
CommunityBot
- 1
19
votes
3
answers
1k
views
Drawing planar graphs with integer edge lengths
It is well known that every planar graph has an embedding such that every edge is drawn as a straight line segment (Fáry's Theorem). Kemnitz and Harborth made the following stronger conjecture
...
CommunityBot
- 1
2
votes
0
answers
213
views
algebraic de Rham cohomology of hypersufaces
For a smooth hypersurface $X\subset\mathbb{P}^n_k$, where $k$ is an algebraic closed field of charactersitc $p>0$. How to compute its algebraic de Rham cohomology explicitly? or equivalently its ...
- 41
10
votes
3
answers
2k
views
Rational Isogenies of Prime Degree
Dear MO Community,
Let $N$ be a prime, and let $X_0(N)$ be the classical modular curve over $\mathbb{Q}$. We know ([1]) that, if there are noncuspidal points in $X_0(N)(\mathbb{Q})$, then $N \in$ {${...
7
votes
2
answers
516
views
Zograf's bound on the index of a modular curve for Shimura curves
I've been reading Voight's paper on Shimura curves and it prompted the following question; see http://www.cems.uvm.edu/~voight/articles/shimura-clay-proceedings-071707.pdf for which notes I'm talking ...
- 41.4k
0
votes
0
answers
82
views
Extending functions on curves to functions on abelian varieties
Say I have a function $f_g$ on the moduli space of curves $M_g$ for all $g\geq 1$. Is there some way of extending this to the moduli space of abelian varieties $A_g$ in a nice way?
What if I have ...
- 163
2
votes
1
answer
501
views
Etale group schemes over a local ring
Let $p$ be a prime number and $R$ be a Noetherian local ring of characteristic $p$ with residue field $k$. Let $G$ be a finite etale subgroup scheme over $R$ of order $p$. Suppose that the etale $k$-...
- 19.6k
16
votes
2
answers
2k
views
Period rings for Galois representations
I have some questions concerning period rings for Galois representations.
First, consider the case of $p$-adic representations of the absolute Galois group $G_K$, where $K$ denote a $p$-adic field. ...
- 3,867
14
votes
3
answers
2k
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 ...
- 6,357
2
votes
2
answers
530
views
branch points of modular parametrization of an elliptic curve
Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to \mathbf{...
- 1,209
5
votes
4
answers
518
views
What is the obstruction for a local set of models of a curve to come from a global model?
If $X_{\mathbb{Q}}$ is a curve over $\mathbb{Q}$, we get a curve $X_{\mathbb{Q}_p}$ over $\mathbb{Q}_p$ for every prime $p$.
My question is about the reverse process. Say we are given curves $X_{\...
CommunityBot
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7
votes
1
answer
756
views
$p$-adic uniformization not from the Drinfel'd spaces?
It seems that when we talk about the $p$-adic uniformization, we typically mean those uniformized by either the Drinfel'd upper spaces (for which we think of the examples of Mumford curves and some ...
- 45.7k
5
votes
2
answers
556
views
Existence of certain identities involving characteristic 2 "thetas"
Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated ...
- 5,422
4
votes
0
answers
124
views
Detecting linear dependence on multiplicative groups
Let G = $\mathbb{G}_m^2/\mathbb{Q}$ and let $\Gamma \subseteq G(\mathbb{Q})$ be a free abelian group of rank 2. Assume that the set of primes $p$ for which $\Gamma \mod p$ is cyclic has positive ...
- 527
13
votes
0
answers
1k
views
Effective proofs of Siegel's theorem using arithmetic geometry
This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is a ...
- 9,114
6
votes
0
answers
436
views
Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?
As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on '...
- 156k
5
votes
1
answer
1k
views
Minimal Model Program for surfaces over algebraically closed fields of characteristic p
Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
- 656
10
votes
0
answers
404
views
Is there an algorithm which determines if a curve has good reduction outside a given set of primes
Fix a number field $K/\mathbf{Q}$, a finite set of places $S$ in $K$, an integer $g$ and a curve $X$ over $K$ of genus $g$.
Is there an algorithm which tells you if $X$ has good reduction outside $S$?...
- 4,593
3
votes
1
answer
483
views
Brauer-Manin obstruction and Hasse principle
I am looking for
varieties without $\mathbf{Q}$-rational points where the absence can be explained by the Brauer-Manin obstruction, but not by the absence of adelic points
varieties without $\mathbf{...
- 30.5k
8
votes
3
answers
570
views
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$
Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...
CommunityBot
- 1
6
votes
2
answers
945
views
Notation/name for "Artin-Schreier roots"?
If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.
Of course ...
CommunityBot
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2
votes
0
answers
464
views
understanding Milne's article "Duality in the flat cohomology of a surface"
I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf
see the "Alternatively" on p. 177, paragraph before ...
CommunityBot
- 1
13
votes
2
answers
2k
views
Why is the definition of l-adic sheaves so complicated?
I find the definition of constructible $\bar{\mathbb Q}_l$-sheaves (or their derived category) on a variety of positive characteristic quite involved and ad Hoc. Roughly it goes as follows:
First one ...
- 45.7k
3
votes
1
answer
805
views
Finite connected groups over a perfect field of characteristic p
In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, ...
- 163
1
vote
1
answer
363
views
Manin-Drinfeld and constructing a finite morphism with two given ramification points
Fix a smooth projective connected curve $X$ over $\overline{\mathbf{Q}}$ of genus $g\geq 1$ and distinct points $x,y \in X$ such that $x-y$ has infinite order in the Jacobian.
Can we always find a ...
- 45.7k
3
votes
0
answers
308
views
Invertible Hasse-Witt for non-ordinary curves
Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
- 8,467
12
votes
4
answers
2k
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 ...
CommunityBot
- 1
2
votes
0
answers
368
views
modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes"
Is there a modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" http://www.jstor.org/pss/1970644 i.e. using Grothendieck's language of schemes and in English?
CommunityBot
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1
vote
0
answers
115
views
singularities $\mathcal{A}_{g,d}$ in positive characteristic
Hi,
I would like to understand the singularities of the moduli spaces of abelian varieties with polarization of degree $d>1$ in characteristic $p>0$ when $p|d$. Do you know some good references?...
- 11
6
votes
0
answers
2k
views
project proposal: English translation of Deligne's "La conjecture de Weil : II" [closed]
First of all, I hope this "question" is appropriate here. If not, please delete it.
I would like to propose a translation project of Deligne's "La conjecture de Weil : II" 52_137_0">http://www.numdam....
CommunityBot
- 1
2
votes
1
answer
413
views
Does each finite morphism of curves have a model whose minimal resolution is semi-stable
Let $\pi:Y\to X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$.
Question. Does there exist a finite field extension $L/K$ and a regular model $\...
- 11.1k
0
votes
0
answers
282
views
well known facts on openness condition
Hi,
I would like to understand and prove the following two "well-known" facts:
1)
If $B$ is a scheme and $P$ a property for which I know:
i) if $B=Spec(V)$ where $V$ is a complete DVR, if $P$ ...
- 1
5
votes
0
answers
393
views
What is the shape of the zeta function of a singular hypersurface?
So let $X$ be a projective hypersurface inside $\mathbb{P}_{\mathbb{Z}}^n$ of degree $d$.
Assume that
(a) $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are irreducible,
(b) and that $X(\mathbb{C}...
- 7,609
24
votes
0
answers
816
views
Smooth proper schemes over Z with points everywhere locally
This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question.
Question. Is there a smooth proper scheme $X\to\operatorname{...
CommunityBot
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5
votes
0
answers
454
views
If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
CommunityBot
- 1
7
votes
2
answers
863
views
Can we bound the minimal degree of a field extension required to obtain semi-stable reduction
Let $K$ be a number field and let $X$ be a smooth projective geometrically connected curve over $K$.
There exists a finite field extension $L/K$ such that $X_L=X\otimes_K L$ has semi-stable ...
- 9,497
4
votes
1
answer
399
views
references for theta characteristic
Hi,
I am looking for references on theta characteristics.
In particular I am interesting in understanding the isomorphism $\Omega_A^g\cong\mathcal{O}_A(\Theta)^2$ where $A$ is an abelian variety and $...
- 3,076
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 ...
- 1,298
3
votes
1
answer
736
views
Is every Weil divisor on an arithmetic surface Q-Cartier
This question is about a technical issue I ran into.
Let $S$ be a connected 1-dimensional Dedekind scheme, and let $X\to S$ be a flat projective integral normal 2-dimensional scheme. (For simplicity, ...
- 149k
5
votes
0
answers
834
views
Motivic Galois group and Shimura varieties
Hi,
Suppose that one has a Shimura variety $Sh(G,X)$ where $(G,X)$ is the corresponding Shimura datum and suppose that it can be interpreted as a moduli space of motives (e.g. PEL type Shimura ...
- 647
6
votes
0
answers
295
views
Does a lower bound for models of finite group schemes exist?
Let $R$ be a discrete valuation ring (as beautiful as you like) and set $K:=Frac(R)$. Let $G_K$ be a finite $K$-group scheme, $G_1$ and $G_2$ two affine and flat models of $G_K$ of finite type, i.e. ...
- 73
3
votes
2
answers
435
views
Does a curve have infinitely many $K$-rational points under these hypotheses?
The question was a bit long for the title, so let me explain what I mean here. Let $k$ be some field (ideally, a number field). Let $X$ be a curve that is defined over $k$, such that it has a $k$-...
CommunityBot
- 1
10
votes
1
answer
1k
views
Bad behaviour of perverse sheaves over 'general' bases?
Could one define $\mathbb{Q}_l$-perverse etale sheaves over more or less general (excellent, separated) base scheme by combining the results of Gabber and Ekedahl? Would their functoriality properties ...
- 1,576
1
vote
0
answers
204
views
Which rational subfields are corresponding to the two dimensional subspaces of holomorphic differentials
I implemented the algorithm that Felipe Voloch's suggested in his reply to the question:
Subfields of a function field
the algorithm is here:
Subfields of a function field
I considered the ...
CommunityBot
- 1
0
votes
2
answers
386
views
Zariski closures of one parameter additive maps in positive characteristic
Suppose we are in characteristic $p$, and that the field, $K$, that we are working over is imperfect. We have a map $\Theta: K \to K^{\delta}$ where each coordinate function $\Theta_i$ is an additive ...
- 30.5k
5
votes
1
answer
710
views
Log resolutions on surfaces and 3-folds in characteristic p
If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
- 10.5k
2
votes
1
answer
245
views
Is there an easier argument to prove that almost all of these curves have no semi-stable reduction
Fix a number field $K$ and a polynomial $F(x)\in K[x]$ of degree at least $4$. For a squarefree integer $d$, define the curve $X_d$ over $K$ by the equation $dy^2 = F(x)$. Note that the curves $X_d$ ...
- 11.1k
7
votes
1
answer
1k
views
References for bad reduction of Jacobians of modular curves?
Hi,
Where can I learn about the reduction of the Jacobians of modular curves
such as X_0(N) and X_1(N) at primes p dividing N?
Thanks!
- 57.6k
8
votes
0
answers
873
views
Resolution of singularities in positive characteristic
I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a ...
- 783