All Questions
25 questions
5
votes
0
answers
192
views
Image of $\gamma-1$ on etale $(\varphi,\Gamma)$-modules
Let $p\geq 3$ be a prime, $D$ be an etale $(\varphi,\Gamma)$-modules over the classical period ring $A_{\mathbb{Q}_p}=\mathbb{Z}_p[\![T]\!][1/T]^{\widehat{\phantom{xx}}}_p$ and $\gamma$ be a ...
- 549
8
votes
2
answers
631
views
Motivation of the construction of $p$-adic period rings
Let $B$ be either $B_{\text{dR}}$ or $B_{\text{crys}}$. For a $\mathbb{Q}_p$-representation $V$ of the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ of a $p$-adic field $K$ (a finite extension ...
- 1,428
3
votes
1
answer
475
views
To identify $p$-adic Tate module $T_p(G)$ of $p$-divisible group $G$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$
Let $k$ be a perfect field of characteristic $p>0$, $W=W(k)$ its ring of Witt vectors, $K_0=W(k)[\frac{1}{p}]$ and, $K/K_0$ be a totally ramified extension. Let $\pi \in K$ be an uniformizer.
...
- 930
3
votes
0
answers
232
views
$l$-adic Galois representations factor through a common finite quotient
Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$.
Does there exist a number field $E$ such that ...
user158636
1
vote
0
answers
125
views
galois deformation ring with type is union of irreducible components
Notation:
$K$ finite extension of $\mathbb{Q}_p$, $G_K$ absolute Galois group of $K$,
$E$ is finite extension of $\mathbb{Q}_p$ (coefficient field), $O_E$ is ring of integer in $E$.
In this paper of ...
3
votes
0
answers
230
views
Independence of $p$ of Hodge-Tate weights
Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\...
user145520
7
votes
1
answer
402
views
Irreducible global Galois representation with weights 0, 1, 3?
Fix a prime number $p$. Can there exist a continuous irreducible representation $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}_3(\mathbb{Q}_p)$ that is unramified at almost all primes, ...
user145520
4
votes
0
answers
419
views
Is the Fargues–Fontaine curve proper?
It is well known that Fontaine's curve $X=\bigoplus_{k\geq0}B_{\text{cris}}^{+,\varphi=p^k}$ is a Noetherian irreducible complete scheme of dimension $1$. And completeness means that the degree ...
user141691
2
votes
1
answer
281
views
An example that a $p$ adic Galois representation is crystalline but not $B_e$ admissible
$B_e=B_{\text{cris}}^{\phi=1}$, so if a $p$-adic Galois representation $V$ is $B_e$ admissible, then it is crystalline, so I want to know an example that $V$ is crystalline but not $B_e$ admissible.
...
user141691
6
votes
1
answer
297
views
$B_{\mathrm{dR}}=B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$?
$B_{\mathrm{cris}}\subseteq B_{\mathrm{dR}}$ and $B_{\mathrm{dR}}^+$ are well-known period rings in $p$-adic Hodge. I know $B_{\mathrm{dR}}=B_{\mathrm{dR}}^+[\frac{1}{t}]$ and $\frac{1}{t}\in B_{\...
user141691
6
votes
2
answers
1k
views
Topology on $p$-adic period rings in an article by Fontaine, part II
This is a follow-up to this question. See that question for background and relevant notation.
In the answer to that question, it is claimed, if I understand the answer correctly, that a basis of ...
- 217
4
votes
1
answer
888
views
Fontaine-Fargues curve and period rings and untilt
When I read the paper "THE FARGUES–FONTAINE CURVE AND DIAMONDS" of Matthew Morrow, I have a question on page 11.
Question: The arthur said that the de Rham and crystalline period rings implicitly ...
user141691
2
votes
1
answer
882
views
How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?
Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...
user141691
7
votes
2
answers
1k
views
Classify 2-dim p-adic galois representations
Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...
user141691
2
votes
0
answers
232
views
Berthelot’s comparison theorem and functoriality
Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.
Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
- 181
11
votes
1
answer
1k
views
Reference request: Newton above Hodge
Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given ...
- 619
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 ...
- 685
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(...
- 685
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 ...
- 806
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 ...
- 685
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{...
- 111
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 ...
- 527
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}, \...
- 1,503