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Status of Fontaine-Mazur conjecture

In the language of Richard Taylor's 2004 (extended) ICM article (''Galois Representations'', Annales de la faculté des sciences de Toulouse (2004) Tome XIII, no. 1, 73-119), the conjecture is the ...
Stiofán Fordham's user avatar
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 ...
user10676's user avatar
  • 527
15 votes
1 answer
1k views

What is the classification of characters in $p$-adic Hodge theory?

Let $K$ be a $p$-adic field and $\chi : Gal_K \rightarrow \mathbb{Q}_p^\times$ be a character. I know that $\chi$ is Hodge-Tate of weight $0$ iff $\chi(I_K)$ is finite (by Sen's theory), and that it ...
user10676's user avatar
  • 527
15 votes
0 answers
592 views

Failure of local Fontaine Mazur

This question unfortunately has a very similar name to this one, but I what want to ask here is different. Let $K$ be a finite extension of $\mathbb{Q}_p$. It seems to be well known that the local ...
xlord's user avatar
  • 643
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 ...
S. Li's user avatar
  • 619
11 votes
1 answer
1k views

Hodge–Tate structures of modular forms

The title refers to the paper of Faltings: Hodge-Tate structures and modular forms. Math. Ann. 278 (1987), no. 1-4, 133–149. The main theorem in the paper says that the associated Galois rep to a ...
abvtmf's user avatar
  • 111
8 votes
2 answers
629 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 ...
User0829's user avatar
  • 1,428
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 ...
user avatar
7 votes
1 answer
401 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, ...
user avatar
7 votes
1 answer
368 views

How large is Dcris of certain twists of modular forms?

I want to determine $\mathrm D_{\mathrm{cris}}$ of certain twists of the Galois representations attached to modular forms. For one particular twist it is not clear to me how $\mathrm D_{\mathrm{cris}}$...
Michael Fütterer's user avatar
7 votes
0 answers
379 views

Local properties of Galois representations attached to torsion classes

$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}} \DeclareMathOperator{\Sym}{Sym}$ Let $F$ be a number field, and let $\Gamma$ be ...
Aurel's user avatar
  • 5,382
7 votes
0 answers
570 views

Which de Rham representations are trianguline?

Let $K/\mathbf{Q}_p$ be a finite extension, and let $V$ be an $n$-dimensional $\overline{\mathbf{Q}_p}$-vector space with a continuous action of $G_K$. Suppose $V$ is de Rham, so potentially ...
David Hansen's user avatar
  • 13.1k
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_{\...
user avatar
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 ...
DCM's user avatar
  • 217
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 ...
Misja's user avatar
  • 161
6 votes
0 answers
727 views

$p$-divisible groups and Breuil-Kisin modules with coefficients

Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and residue field $k$. Choose a uniformizer $\pi \in \mathcal{O}_K$ and $E(u)$ be the minimal (Eisenstein) ...
O-Ren Ishii's user avatar
6 votes
0 answers
197 views

classifying reducible 2-dimensional mod-p Galois representations

I want to classify reducible $2$-dimensional mod-$p$ Galois representations of a field $E$ of characteristic $p > 0$ (i.e. representations $G_E = \mathrm{Gal}(E^{sep}/E) \to GL_n(\mathbf{F}_p)$) $$ ...
user avatar
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}, \...
natura's user avatar
  • 1,503
5 votes
1 answer
486 views

Generalized Hodge-Tate weights of an arbitrary p-adic Galois representation

Let $V$ be a continuous representation of the absolute Galois group of $\mathbb{Q}_p$ with coefficients in $\mathbb{Q}_p$. The theory of Sen attaches to $V$ generalized Hodge-Tate weights which are ...
xlord's user avatar
  • 643
5 votes
1 answer
632 views

Psi operator on Phi-Gamma modules

This is a question about the base-rings appearing in the the theory of $(\varphi, \Gamma)$-modules in $p$-adic Hodge theory. Let $p$ be prime, $n \ge 1$, and let $$ \mathbf{A}_{\mathbf{Q}_p}^{\dagger,...
David Loeffler's user avatar
5 votes
1 answer
842 views

Reference for $p$-adic Hodge theory with coefficients

Let $K$ be a $p$-adic field and $L$ be a finite or infinite extension (maybe algebraic ?) of $\mathbb{Q}_p$. Is there a reference for $p$-Hodge theory for representations $\rho : Gal_K \rightarrow ...
user10676's user avatar
  • 527
5 votes
0 answers
194 views

When is the image of $\operatorname{Gal}(\bar K/K)$ open in $\operatorname{Aut}(V)$, where $V$ is the vector space coming from a $p$-adic Tate module?

Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $A$. Let $F$ be a $p$-divisible group and $T$ be the Tate module. Consider the vector space $V=T \otimes_{\mathbb{Q}_p} C$, where $...
Learner's user avatar
  • 195
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 ...
Yijun Yuan's user avatar
5 votes
0 answers
676 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{...
Michael's user avatar
  • 111
5 votes
0 answers
278 views

Tate's conjecture and symmetry of Hodge-Tate weights

I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf At page 10 he claims that an indirect ...
Bear's user avatar
  • 231
5 votes
0 answers
585 views

Bloch Kato Exponential as formal lie group exponential

Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism $$DR(V) \...
LMN's user avatar
  • 3,555
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 ...
MathStudent's user avatar
4 votes
1 answer
394 views

Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero

I am reading a very nice paper of Newton and Thorne, Symmetric power functoriality for holomorphic modular forms, and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic ...
babu_babu's user avatar
  • 241
4 votes
1 answer
152 views

Semistability of the $\ell$-adic representation of variety with semistable reduction

The question is in the title, but here's some quick background. It's easy to show (assuming smooth-proper base change) that the $\ell$-adic cohomology of a variety over the fraction field of a DVR ...
Nico's user avatar
  • 141
4 votes
1 answer
278 views

Irreducible local Galois representation with arbitrary Hodge-Tate weights

Let $p$ be a prime and $M$ be a finite multiset of non-negative integers. Does there exist a continuous irreducible de Rham representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\...
user avatar
4 votes
1 answer
887 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 ...
user avatar
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 ...
MathStudent's user avatar
4 votes
0 answers
293 views

Galois representation with infinite image but finite image everywhere locally

Fix a prime $l$. Let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$ be a semisimple continuous representation. Assume $\phi$ has finite image when restricted to $\mathrm{...
user avatar
4 votes
0 answers
418 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 ...
user avatar
3 votes
1 answer
329 views

Calculation of Frobenius on de Rham cohomology of elliptic curves with good reduction

I'm reading "An introduction to the theory of $p$-adic representations" by Laurent Berger. In the page 14 it says: Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is ...
Richard's user avatar
  • 775
3 votes
1 answer
674 views

Would it be a little but good exercise to construct or find out Breuil modules?

My question is about p-adic Hodge-Tate theory and p-adic Galois representation. One of the important semi-linear object in p-adic Galois representation is the $\text{Breuil Module}$. There are ...
MAS's user avatar
  • 930
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 ...
Bonbon's user avatar
  • 806
3 votes
1 answer
474 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. ...
MAS's user avatar
  • 930
3 votes
1 answer
184 views

Restriction of $(\varphi, N)$-modules

For any $p$-dic field $K$, we have an equivalence of categories $$D_{st}:Rep_{\mathbb{Q}_p}^{st}(G_K)\rightarrow MF_K^{ad}(\varphi,N),\quad V\mapsto (B_{st}\otimes_{\mathbb{Q}_p} V)^{G_K}$$ with quasi-...
curious math guy's user avatar
3 votes
0 answers
300 views

Galois invariant of Tate module

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $V$ be a de Rham representation of $G_K=\operatorname{Gal}(\overline{K}/K)$. By Corollary 3.8.4 of Bloch and Kato - L-functions and Tamagawa ...
Desunkid's user avatar
  • 247
3 votes
0 answers
148 views

$p$-adic Hodge theoretic properties of global Galois representations via $\ell$-Frobenii

Let $G_{\mathbb{Q},S} = \mathrm{Gal}(\mathbb{Q}_S/\mathbb Q)$ where $\mathbb Q_S$ is the largest algebraic extension of $\mathbb Q$ unramified outside a finite set of places $S$. Then the union over $\...
Ashwin Iyengar's user avatar
3 votes
0 answers
335 views

A question on the Robba ring

Notation is as in the question: https://math.stackexchange.com/questions/4090045/some-questions-about-the-robba-ring. We define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}...
lzhao's user avatar
  • 166
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 ...
user avatar
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{\...
user avatar
3 votes
0 answers
517 views

Condition on a Fontaine Laffaille module which prescribes the image of the associated Galois representation

The Setup: Let $m\geq 1$ be an integer, $\mathbb{F}$ be a finite field of characteristic $p$ and $W(\mathbb{F})$ the ring of Witt-vectors with residue field $\mathbb{F}$ and $\sigma:W(\mathbb{F})\...
user avatar
2 votes
1 answer
401 views

Crystalline when restricted to inertial subgroup

$\newcommand{\ur}{\mathrm{ur}}\newcommand{\cris}{\mathrm{cris}}$Let $K$ be a finite extension of $\mathbb{Q}_p$, $G_K=\operatorname{Gal}(\overline{K}/K)$ and $I_K \subset G_K$ its inertial subgroup. ...
Desunkid's user avatar
  • 247
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 ...
MathStudent's user avatar
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. ...
user avatar
2 votes
1 answer
243 views

$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline

Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight? Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing ...
user avatar
2 votes
1 answer
232 views

Local to global for semistable $G_{\mathbb{Q}_p}$-representations

Let $\rho_p:G_{\mathbb{Q}_p} \to \text{Gl}_n(\mathbb{Q}_p)$ be semistable representation. In local to global Galois representation, it was asked if one can find a geometric global Galois ...
curious math guy's user avatar