All Questions
2,494 questions
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
13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
- 3,637
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
2
votes
0
answers
137
views
Removing finitely many points from a Shimura curve
Let $X$ be a compact Shimura curve. If we remove finitely many points from this curve, do we neccessarily get a "non-compact Shimura curve"? I have some reasons to believe that the answer is negative, ...
- 637
4
votes
1
answer
221
views
Do permutation modules of solvable groups have self-dual socle in characteristic 2?
I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that ...
- 143
0
votes
0
answers
555
views
étale cohomology with values in the $\ell$-torsion of an Abelian scheme
Let $S/\mathbf{F}_q$ be a $d$-dimensional smooth projective variety and $A/S$ be an Abelian scheme. Is there an easy description of $H^0(S, A(\ell)(d-1))$?` ($A(\ell)$ = union of $A_{\ell^n}$)
- 227
2
votes
1
answer
202
views
In Riemann Existence, what is the interpretation of the space of complex-geometric points?
I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question:
Question
Riemann existence says that if we have a variety over $\mathbb{C}$, $X_{...
- 5,929
5
votes
0
answers
744
views
p-divisible groups of superspecial abelian varieties
Let $p$ be a prime and $F$ be an algebraic closure of the field with p elements. I will consider abelian varieties over F up to prime-to-$p$ isogeny. Principal polarizations will be $Q$-homogeneous ...
- 51
1
vote
0
answers
187
views
stack quotient question
Hi,
I have the following question:
let $k$ a field with $char(k)= p>0$, which we can assume to be perfect, $W(k)$ the ring of Witt vector, and $a,b$ positive integers.
Consider the ring $R=W(k)[x,...
- 11
3
votes
0
answers
302
views
Small primes as stepping stones
It is well-known that in his celebrated proof of Fermat's Last Theorem, Wiles made a crucial use of the result of Langlands and Tunnell to deduce modularity of the Galois representation on the 3-...
- 786
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
12
votes
0
answers
716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
0
votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
0
votes
1
answer
175
views
An inseparable lift of a regular variety.
Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...
- 16.9k
3
votes
0
answers
145
views
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:
...
- 1,921
0
votes
0
answers
263
views
Computing the function field of a curve given as a subvariety of the Jacobian of its cover or merely the degree of the covering
I read following paragraph from:
G. Tamme, Teilkörper höheren Geschlechts eines algebraischen Funktionenkörpers, Arch. Math. 23 (1972), 257--259
Here $C$ is a curve of genus $\ge 2$ and $J$ is the ...
- 601
3
votes
0
answers
483
views
Questions about Shimura curves
1: Suppose $A_3 $ is the moduli space of abelian varieties of dimension 3 .Is the union of all one dimension shimura varieties in $A_3 $ connected?
2: Given a Shimura curve (explicit construction), ...
- 709
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
3
votes
0
answers
281
views
What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?
I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...
- 4,265
1
vote
1
answer
213
views
flatness of a kernel
Hi,
let $A$ an abelian scheme over a curve $C$ and $n$ an integer greather than 3 coprime with the characteristic of the ground field. Do you know why the kernel of the multiplication by $n$ is flat ...
- 31
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
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
239
views
Torsion points on commutative $Z_p$-group schemes
Hi,
Let G be a smooth commutative $\mathbb{Z}_p$-group scheme of finite type and let $G_0$ be the $\mathbb{Q}_p$-fiber. We have an embedding $G(\mathbb{Z}_p)\subseteq G_0(\mathbb{Q}_p)$. My question ...
- 527
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
7
votes
0
answers
273
views
Do Scharaschkin's results on Brauer-Manin obstructions on curves generalize to non-projective curves?
Theorem: Let X be a smooth projective curve over a number field K, and let $\delta$ be the index of X (i.e., the minimal degree of a K-rational divisor on X). Then V. Scharaschkin proved in this ...
- 10.5k
1
vote
0
answers
409
views
surjectivity of reduction for schemes smooth over Henselian base?
If $S$ is a Henselian local scheme with closed point $v$ and $X$ a smooth $S$-scheme, then it is well known that the canonical map $X(S) \rightarrow X(v)$ is surjective.
Suppose now that $Y$ is ...
- 646
2
votes
0
answers
387
views
Why is the Eisenstein quotient a quotient of the new part of the Jacobian?
Dear MO Community,
Let $X = X_0(N)_{/\mathbb{Q}}$, and $J$ its jacobian. Mazur defines the Eisenstein quotient of $J$, denoted $\widetilde{J}$, as
\[ 0 \rightarrow \gamma_IJ \rightarrow J \...
- 2,827
5
votes
0
answers
234
views
Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
- 10.7k
3
votes
0
answers
279
views
Tate-Shafarevich group of non-principally polarized abelian variety
Let $A/k$ be an abelian variety over a number field $k$ with a polarization of minimal degree $d>1$. (Assume all Tate-Shafarevich groups to be finite.)
What can one say about the order of $\mathrm{...
- 841
3
votes
0
answers
409
views
How looks the "land of Tamagawa numbers"?
Jonah Sinick's question here, other interesting ideas he mentioned, and Franz Lemmermeyer's remark make one think at Bloch and Kato's drawing + question. What's known or guessed about that "land" by ...
- 10.8k
2
votes
0
answers
131
views
why find a good field extension such that the curve has semi-stable model is important?
Hello everyone,
I'd like to ask some question about semi-stable reduction of curves.
The Deligne-Mumford theorem tell us "Let $A$ be an Dedekind domain, $K=K(A)$, for any smooth curve $X$ over $K$ ...
- 1,921
6
votes
0
answers
456
views
On periods of algebraic integers modulo rational primes
I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.
Let $K$ be a number field, which we may assume Galois if it ...
- 797
2
votes
0
answers
321
views
Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
3
votes
0
answers
193
views
rational points of component group of the special fiber of the Neron model
Let $A$ be an abelian variety over a number field $K$ and let $\mathcal{A}$ denote its Neron model over $\mathcal{O}_K$. Let $v \in M_K^0$ denote a finite prime of $K$, $k_v$ its residue field, $\...
- 841
4
votes
0
answers
390
views
Is there a reference that treats principal homogeneous spaces for (say) group varieties using schemes?
I was wondering if anyone could recommend a reference that discusses principal homogeneous spaces for general finite type group schemes over a field $k$ entirely in the language of schemes (or even ...
- 3,380
0
votes
0
answers
159
views
a question on the Poincar\'e bundle
Let $C$, a smooth curve. Let $J$ its Jacobian, consider the Poincar\'e bundle $\mathcal{P}$ on $J\times J$. Let $p: J\times J\rightarrow J$ the projection.
How can I compute the complex $R p_{*} \...
0
votes
0
answers
253
views
A misunderstanding about the Moret Bailly theorem
Sorry for the non-specific title, but this would be hard to fit in one line.
In "Groups as Galois Groups", theorem 10.27 says that there's some absolutely irreducible component of $\mathcal{H}_r^{in}(...
- 5,929
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
1
vote
1
answer
183
views
complete ring as union of finite type algebras
Hi,
why the completion of a local ring $R$ can be written as an increasing union of $R$-algebras of finite type?
- 141
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
3
votes
1
answer
185
views
How many linear terms are in the Hilbert set of H(z,t), a polynomial in 2 variables over a field k(s) of transcendence degree one over a finite field?
I am looking for a good reference for Hilbert's irreducibility theorem, and ofproperties of Hilbert sets besides Serres Lectures on The Mordell-Weil Theorem. In particular, I am interested it to the ...
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
1
vote
0
answers
155
views
indepence of Galois orbits on a product?
Let's take $X$ some fixed variety of a fixed base field $k$, which is assumed to be of char.zero for simplicity. Write $\Gamma_k$ for the Galois group of $k$. Given a point $x\in X(\bar{k})$, write $O(...
- 1,305
1
vote
0
answers
108
views
Why do subspaces of the space of Global holomorphic differentials of a function field correspond to its subfields
I'm asking this question as a follow up to the Felipe Voloch's answer to this question:
Subfields of a function field
which you can read it here:
Subfields of a function field
(I just didn't have ...
- 601