All Questions
2,494 questions
5
votes
3
answers
975
views
The historical development of automorphic geometry
Background:
Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...
- 1,704
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 ...
- 2,215
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 $S$?...
- 153
17
votes
2
answers
5k
views
why we need rigid geometry?
Hello, everyone.
I want to ask some questions about rigid geometry.
1.what is the motivation of rigid geometry?
2.what is the applications of rigid geometry for solving arithmetic problems, ...
- 1,921
10
votes
1
answer
1k
views
$\ell$-adic Weil cohomology theory
I have a reference or counterexample request. Suppose $k$ is a field and $\ell\neq char(k)$. There are several common references that show that $H^i_{et}(-, \mathbb{Q}_\ell )$ is a Weil cohomology ...
- 970
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{...
user19475
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 ...
user19475
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{...
- 23
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 ...
- 13.2k
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 ...
- 9,497
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 ...
- 395
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
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?
user19475
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....
user19475
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 $\...
- 9,497
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
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
3
votes
1
answer
459
views
Frobenius functor and length of local cohomology
Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the ...
- 3,101
9
votes
1
answer
423
views
finiteness of torsion points of an abelian variety over a totally real field？
Ribet has shown the following result: if $A$ is an abelian variety over a number field $E$, then the torsion subgroup of $A(E^{cyc})$ is finite. Here $E^{cyc}$ is the union $\bigcup_nE(\mu_n)$, $\mu_n$...
- 1,305
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 ...
- 4,487
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 $...
- 43
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 ...
- 75
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, ...
- 9,497
9
votes
1
answer
1k
views
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 ...
- 4,902
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 4,902
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 ...
- 395
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 $k$-...
- 6,348
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 ...
- 75
4
votes
1
answer
674
views
Inseparable Galois Cohomology
First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...
- 1,199
3
votes
2
answers
757
views
Almost Northcott properties for heights of abelian varieties
Let $h$ be a function on the moduli space of abelian varieties of dimension $g$ over $\overline{\mathbf{Q}}$.
Let $K$ be a number field and let $g\geq 2$ be an integer. Fix a real number $C$. Does ...
- 31
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 $X_d$ ...
- 9,497
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 ...
- 2,215
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!
- 2,842
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
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
9
votes
1
answer
683
views
Can we always find a curve which doesn't have semi-stable reduction
Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...
- 145
1
vote
0
answers
231
views
Lower bound for intersection number
The base scheme is an algebraically closed field.
Let $X\to \mathbf{P}^1$ be an arithmetic surface over $\mathbf{P}^1$ and let $P$ be a section of $X\to \mathbf{P}^1$. Let $D$ be an effective (edited)...
- 225
26
votes
2
answers
4k
views
A route towards understanding Shimura varieties?
I'm in the embarrassing situation that I want to ask a question that
was already asked, but (for complicated reasons) never answered. I'd
like to try with a blank slate.
Shimura varieties show ...
- 445
3
votes
0
answers
204
views
Computing Elliptic Curves of Conductor Divisible by a Large Prime Factor
A little while ago, I came across a paper (or slides from a talk or something) that seemed to suggest that the modular symbol method for computing elliptic curves over $\mathbf{Q}$ of prescribed ...
- 309
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 ...
- 601
2
votes
1
answer
268
views
Comparing heights of rational points on curves through covers
Let $a$ be a closed point in $\mathbf{P}^1_{\overline{\mathbf{Q}}}$.
Let $Y \cong \mathbf{P}^1_{\overline{\mathbf{Q}}} $ and let $\pi:Y\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ be a finite morphism ...
- 23
15
votes
1
answer
769
views
Crystalline realization of mixed Tate motives
Deligne and Goncharov, in their article of 2005, mention that the crystalline realization functor has yet to be worked out. What's the current state of the literature on this? And how big of an issue ...
- 313
4
votes
1
answer
628
views
Rationality of three-dimensional torus
Let $\zeta \in \overline{\mathbb{Q}}$ be a primitive 8th root of unity and let $K = \mathbb{Q}(\zeta)$. Let $N_{K/\mathbb{Q}}$ be the corresponding norm.
Consider the three-dimensional $\mathbb{Q}$-...
- 5,163
2
votes
1
answer
406
views
Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius
This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ (...
- 637
18
votes
2
answers
1k
views
Divisibility properties of Hurwitz numbers
Define numbers $H_k$ for integers $k\geq 4$ by $\sum_{x \in \mathbf{Z}[i]}x^{-k}=\frac{H_k}{k!} \omega^k$, where $\omega=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}$. These are nonzero when $4|k$, and Hurwitz ...
- 13.1k
15
votes
1
answer
1k
views
Lifting to Characteristic 0 not over W
I thought of this several months ago and forgot about it. Now I rethought of it again and I just can't find it anywhere in the literature, so I'll ask here.
Is it known whether or not there exists a (...
- 970
3
votes
1
answer
504
views
parabolic-Eisenstein decomposition of cohomology of modular curve
Hi,
Fix $N > 3$ and consider the modular curve $X(N)$ parametrizing elliptic curves with
full level N structures. Let $\pi : E(N)\to X(N)$ be the universal elliptic curve. Then
$V=R^1\pi_*\mathbf ...
- 2,842