All Questions
Tagged with p-adic-hodge-theory arithmetic-geometry
75 questions
9
votes
0
answers
1k
views
Moduli interpretation of Fargues-Fontaine curve
The Fargues-Fontaine curve is, in his schematic version, a noetherian regular scheme $X$ of dimension 1 associated to a pair $(E,F)$, where $E$ is a local field (i.e. complete w.r.t. a discrete ...
- 1,386
8
votes
0
answers
581
views
On Topological Hochschild Homology
Nowdays, I hear talking about Topological Hochschild Homology more and more often, and I was wondering if someone could point out references to explain why it's important and interesting, and what ...
CommunityBot
- 1
2
votes
1
answer
315
views
Equivalence of vector bundles over $Spec(A_{\inf})$ and the punctured spectrum
I'm trying to understand the Lemma 4.6 of Bhatt-Morrow-Scholze's paper Integral $p$-adic Hodge Theory.
In the proof, for proving the restriction functor is fully faithful, it used a affine open cover $...
- 28.8k
18
votes
1
answer
1k
views
Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?
Let $X$ be a variety over a $p$-adic field $K$.
Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...
CommunityBot
- 1
2
votes
0
answers
357
views
Does the pro-étale local system defined over a p-adic period domains interpolate crystalline representations?
There is a Grothendieck-Messing period morphism of rigid-analytic spaces $\pi: \mathcal{M}_\eta^{rig}\to \mathcal{Fl}$ going from the generic fiber of an EL-type Rapoport-Zinks to a flag variety. The ...
- 435
32
votes
1
answer
8k
views
$p$-adic Hodge Theory for rigid spaces, after P. Scholze
I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties.
This question is around the "Poincaré Lemma" in the paper.
Throughout, let $X$ be a proper smooth rigid ...
- 21.3k
6
votes
0
answers
412
views
Two Definitions of Barsotti-Tate Representations
In different articles I have seen different definitions of Barsotti-Tate representations. I am wondering if and how these definitions are equivalent.
In Section 1.1 of Conrad-Diamond-Taylor they say ...
- 161
4
votes
1
answer
200
views
Smooth intertwining operators
Let $V$ be a crystalline irreducible representation of the absolute Galois group of $\mathbb{Q}_p$ with distinct Hodge Tate weights $(0,k-1), k \in \mathbb{Z}_{\geq 2}$.
Then $V$ is uniquely ...
CommunityBot
- 1
2
votes
1
answer
163
views
Locally analytic vectors of a quotient space
My question here is in connection with one of my previous question
"A definition of a (amalgamated) direct sum"
Following the notations there, my question is:
Why the locally analytic vectors of $B(...
- 6,924
3
votes
1
answer
412
views
Reference on a result on local Galois representation associated to classic modular form in p-adic Hodge theory
At the end of Fontaine’s rings and p-adic L-functions, P. Colmez states a Theorem 8.4.8 (click here) of Faltings-Tusji-Saito without references.
So I am wondering is there any references for this ...
- 17.6k
4
votes
1
answer
302
views
A definition of a (amalgamated) direct sum
I am wondering about a definition of a direct sum in page $31$ of this paper by R. Liu.
I am following the notations in page $31$ of the above paper. Let $V$ be a crystalline irreducible ...
- 685
2
votes
1
answer
367
views
p-adic representations of $GL_2(\mathbb{Q}_p)$
Let $L$ be a finite extension of $\mathbb{Q}_p$. Colmez defines here
the trainguline representations which are extensions of Robba rings of dimension $1$. Then, in this paper he contructs the ...
- 37k
2
votes
1
answer
282
views
Uniqueness of finite flat models over bases of low ramification via Breuil-Kisin modules
Let $R$ be a complete DVR of mixed characteristic $(0, p)$, let $K$ be its fraction field, and assume that the absolute ramification index $e$ of $R$ satisfies $e < p - 1$ and that the residue ...
- 2,713
8
votes
0
answers
259
views
Ramification for subgroups of Lubin-Tate formal group
Let $K/\mathbb{Q}_p$ be a finite field extension and $E/\mathcal{O}_K$ be the local N\'eron model of a CM elliptic curve with CM by $\mathcal{O}_F$ and let $G\subseteq E[p^n]$ be a subgroup over $\...
- 443
6
votes
0
answers
369
views
Conjecture on classification of $p$-divisible over the ring of integers of $\widehat{\bar{\mathbb{Q}}_p}$
I am reading the paper of Fargues Quelques résultats et conjectures concernant la courbe. In the end of this paper, there is a conjecture on the classification of $p$-divisible groups over $\...
- 193
3
votes
1
answer
404
views
Crystalline extension the $p$-adic cyclotomic character
Let $\epsilon_p$ be the $p$-adic cyclotomic character, $F$ be a real quadratic extension of $\mathbb{Q}$ in which $p$ splits, $\psi$ be an odd character of $G_\mathbb{Q}$ of finite image and with ...
- 4,008
5
votes
0
answers
677
views
Basic question on p-adic Hodge theory
I am starting to study the rudiments of p-adic Hodge theory and I have the following basic question. Let $\chi$ be the unramified quadratic character of $G_p = \mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{...
- 63.9k
9
votes
1
answer
546
views
Morphisms for good reduction are maps respecting filtration
Please see edits below!
So, let $A,A'/K$ be abelian varieties where $K$ is a $p$-adic local field with residue field $k$. Suppose further that they have good reduction with models $\mathscr{A},\...
CommunityBot
- 1
5
votes
2
answers
2k
views
Hodge-Tate weights of etale cohomology
Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction.
Question: What is the possible range of Hodge-Tate weights of the etale cohomology $H^i(X_{\overline K}, \...
- 1,066
16
votes
1
answer
2k
views
Applications of $p$-adic Hodge theory
I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...
- 1,404
2
votes
0
answers
389
views
Are there good properties of the divided power completion map?
Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed immersion)...
- 33
11
votes
0
answers
807
views
Torelli-like theorem for K3 surfaces on terms of its étale cohomology
Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?
For example: If $K\ne \mathbb{C} $ and $X\rightarrow \...
2
votes
1
answer
466
views
Minimal semistable model for K3-surfaces.
I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.
Also, if we have a semistable K3 surface with a log structure, ...
- 656
3
votes
0
answers
204
views
Hodge filtration over $\mathbb Z_p$
Let $p$ be a prime number.
Let $X\to\operatorname{Spec}\mathbb Z_p$ be smooth and proper. Is it true that
the map $H^i(X,\Omega^{\bullet\geq j}_{X/\mathbb Z_p})\to H^i(X,\Omega^\bullet_{X/\mathbb Z_p})...
- 2,842
8
votes
0
answers
315
views
simple proof of relation between H^1 crystalline and Dieudonne module?
Hi,
Let $k$ be a perfect field of characteristic $p > 0$. Let $A/k$ be an abelian variety. Then the first crystalline cohomology group of $A$ with respect to $W(k)$ (= Witt vectors) is canonically ...
- 2,842