All Questions
73
questions
7
votes
1
answer
333
views
Shouldn't we expect analytic (in the Berkovich sense) étale cohomology of a number field to be the cohomology of the Artin–Verdier site?
Let $K$ be a number field. Consider $X=\mathcal{M}(\mathcal O_K)$ the global Berkovich analytic space associated to $\mathcal O_K$ endowed with the norm $\|\cdot\|=\max\limits_{\sigma:K \...
- 794
7
votes
1
answer
454
views
Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$
Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
user122877
3
votes
1
answer
234
views
$\mathbf{Z}$-points of quasi-projective schemes
Let $U\subset\mathbf{P}^n_{\mathbf{Z}}$ be an open subscheme such that the smooth morphism $U\to\text{Spec}(\mathbf{Z})$ is surjective. Suppose $U(\mathbf{Q})\neq\varnothing$ and $U(\mathbf{Z}_p)\neq\...
- 85
2
votes
1
answer
317
views
Rank of $\mathbb{Z}_{p}$-module $H_{et}^{i}(X,\mathbb{Z}_{p}(r))$
I want to ask the following question.
Let $X$ be a smooth projective variety of dimension $d$ over $p$-adic field $k$ ( i.e. finite extension of $\mathbb{Q}_{p}$). Is it true that etale cohomology $H_{...
- 609
6
votes
1
answer
351
views
Adèlic points and algebraic closure
Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number ...
user335418
15
votes
1
answer
1k
views
Is Mazur's analogy between arithmetic and topology formal, in any sense?
I preface my question by admitting I know no algebraic geometry nor algebraic number theory. I do know some algebraic topology. I'm a student.
Recently I learned about sheaf cohomology. Then a little ...
- 599
2
votes
1
answer
410
views
Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?
In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a ...
- 519
1
vote
0
answers
170
views
Does Lemma 5.4 in Deligne's Ramanujan paper generalize to Shimura varieties of PEL type?
It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety.
Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for ...
- 31
1
vote
0
answers
147
views
Image of pullback for Brauer groups
If a have a dominant morphism $\pi:X \rightarrow \mathbb{P}^{1}$ where $X$ is a projective, geometrically integral $k$-scheme. Then this gives rise to a pullback map
\begin{align*}
\pi^{*}:\text{Br}(k(...
- 481
4
votes
1
answer
301
views
Torsion points on $E/\mathbb{Q}$ with large coordinates
Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points.
What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?
- 41
4
votes
0
answers
268
views
Explicit computations of the fundamental groups of perfectoid spaces
If $X$ is a perfectoid space then it has the same étale site as its tilt $X^\flat$. This means that the fundamental groups (suitably defined) of $X$ and $X^\flat$ are isomorphic.
Can you give ...
- 41
3
votes
0
answers
489
views
A question on the Bombieri-Lang conjecture
Let $X$ be a variety of general type, defined over a number field $K$. Then the Bombieri-Lang conjecture asserts that the set of rational points $X(K)$ (or $X(L)$ for any finite extension $L/K$) is ...
- 25.9k
3
votes
2
answers
382
views
Galois stable elements of the Picard group of a curve and the rational divisors
Let $C$ be a (smooth,proper) curve over a field $k$. Let $\operatorname{Div}_C(k)$ be the free abelian group generated by the closed points of $C/k$ and $k(C)^\times$ be the group of rational ...
- 7,716
3
votes
0
answers
279
views
Generalizations of Artin–Verdier duality?
Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $Spec\ O_\...
- 2,380
3
votes
0
answers
138
views
2-fold linear cover of reductive group of type A
Let $F$ be a nonarchimedean local field of characteristic zero. Let $G=\operatorname{Res}_{E/F}\operatorname{GL}_n$ or $\operatorname{Res}_{E/F}\operatorname{U}_n$, where $\operatorname{U}_n$ is any ...
- 833