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Questions tagged [p-adic-hodge-theory]

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3
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0answers
203 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})\...
37
votes
0answers
1k views

What is Prismatic cohomology?

Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...
5
votes
0answers
145 views

Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$

$\DeclareMathOperator\Spa{Spa}$Fargues's Theorem for $\Spa(C,O_C)$ states that the category of (mixed characteristic) shtukas with one paw at $x_C$ is equivalent to the category of Breuil-Kisin-...
1
vote
0answers
133 views

Reference to a particular result of Scholl and Faltings

Let $f=\sum_{n\geq 1} a_n q^n$ be a normalized eigenform which is supersingular and crystalline at a prime $p$ and let $V_f$ be the associated crystalline representation, then it follows from the work ...
1
vote
0answers
123 views

A family of crystalline representations

Let $K$ be a number field and let $v$ be a finite place of $K$. Further, let $g \geq 1$ be a positive integer. Consider the family $F(K,v,g)$ consisting of abelian varieties $A$ of dimension $2g$, ...
2
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0answers
84 views

Generalizing characterizing crystalline representations of dimension 2 to certain special classes of crystalline representations of higher dimension

Let $A$ be an abelian variety defined over a number field $K$, and let $v$ be a prime of $K$ such that $A$ has good reduction modulo $v$. Let $\rho$ be the representation of $G_K = \text{Gal}(\...
6
votes
1answer
191 views

Integral Tate-Sen theory

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $C=\widehat{\overline{K}}$ be the completion of the algebraic closure of $K$. Let $\mathscr{O}_C$ be the ring of integers in $C$, and let $G_K$ ...
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0answers
91 views

Hodge-Tate weights of etale cohomology groups

Given a smooth algebraic variety $X$ over a number field $F$, its $p$-adic cohomology groups $H^i(X \times_F \bar F, \mathbb Q_p)$ carries an action of $\mathrm{Gal}(\bar F/F)$, which gives a ...
8
votes
0answers
292 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 ...
2
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0answers
163 views

Is there a Hodge structure for smooth proper varieties over $\mathbb{C_p}$? [duplicate]

For smooth proper varieties over $\mathbb{Q_p}$, we have several comparison theorems in p-adic Hodge theory, in particular a p-adic Hodge structure. Now for $\mathbb{C_p}$, is there any such results ...
4
votes
1answer
212 views

Reference Request: Specialization map in Huber's Context

The specialization map $sp:\mathfrak{X}_\eta\to \mathfrak{X}_{red}$ has an important role in rigid analytic geometry. I tried looking in Huber's papers ("Continuous Valuations", "A generalization of ...
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0answers
228 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 ...
3
votes
0answers
54 views

action of formal tori $I^\mathrm{ext}$

this is a question about the action of the formal tori defined in recent papers of Andreatta, Iovita and Pilloni. The notations are heavy, so I will follow the paper Triple product p-adic L-functions ...
3
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0answers
105 views

A question about Hasse Invariant and Modular curve

Let $N\geq4$ be a positive integer and p be a prime such that $(p,N)=1$, and $X=X_1(N)$ be the modular curve parameterizing (generalized) elliptic curves with $\Gamma_1(N)$-level structure. Base ...
4
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0answers
114 views

Complete characteristic p perfect Tate rings are uniform?

In Lemma 7.1.6 of his lecture notes on perfectoid spaces, Bhatt states that every complete characteristic p perfect Tate ring $A$ is uniform. In the proof he uses the Banach open mapping theorem on ...
11
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0answers
307 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 ...
8
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0answers
351 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 ...
4
votes
0answers
212 views

Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory

Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps $cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
2
votes
1answer
207 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 $...
8
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0answers
209 views

Failure of integral comparison between crystalline and de Rham cohomology over a highly ramified base

Let $K$ be a finite extension of $\mathbb{Q}_p$ with the ring of integers $\mathcal{O}_K$ and the residue field $k$. By a theorem of Berthelot and Ogus(https://link.springer.com/article/10.1007%...
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vote
1answer
223 views

A question about Kato's explicit reciprocity law

In the paper Iwasawa Theory and F-analytic Lubin-tate $(\phi,\Gamma)$-modules Prop 3.4.2 says that for any $x\in{S}$, there exists (not uniuqely) $f(T)\in{B_{rig,F}^+}$ such that $f(u_n)=\log_{LT}(...
2
votes
0answers
211 views

The Breuil-Mezard Conjecture and Generalizations (Survey)

What's the current state of the Breuil-Mezard conjecture? Has the original version (from the 2002 paper) been solved in its entirety? What are some of the new directions being explored?
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0answers
165 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 ...
6
votes
1answer
188 views

Injectivity of Frobenius on $A_{cris}$

I am reading Brinon, Conrad "Notes on $p$-adic Hodge theory" and I can't find any reference for the proof of Theorem 9.1.8, namely the injectivity of the Frobenius endomorphism of $A_{cris}$. Does ...
29
votes
1answer
3k 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 ...
4
votes
0answers
126 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 ...
10
votes
3answers
917 views

p-adic Poincaré Lemma

suppose $X$ is a proper and smooth rigid analytic variety over $\text{Spa}(k)$, with $k$ a non-archimedean field of characteristic zero. One has the de Rham complex of analytic differential forms on $...
8
votes
0answers
306 views

periodic cyclic homology and tilting in the sense of Scholze

Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost ...
4
votes
1answer
179 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 ...
2
votes
1answer
124 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(...
7
votes
1answer
238 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 $\mathrm D_{\mathrm{cris}}$...
3
votes
1answer
292 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 ...
4
votes
1answer
183 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 ...
6
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0answers
192 views

Universal property of $A_{\mathrm{cris}}/p^n$

It is well known that the ring $A_{\mathrm{cris}}$ of Fontaine is the universal $p$-adically complete divided power thickening of $\mathcal{O}_{\mathbb{C}_p}$ over $\mathbb{Z}_p$; in fact, this is one ...
2
votes
1answer
257 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 ...
2
votes
1answer
207 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 ...
8
votes
0answers
208 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 $\...
4
votes
1answer
226 views

Serre tensor construction on finite flat group schemes

Let $K/\mathbb{Q}_p$ be a finite field extension and $\mathcal{O}_K\subseteq K$ be its ring of integral elements. Let also $G/\mathcal{O}_K$ be a finite flat $p$-group scheme that is also an $\mathcal{...
5
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0answers
264 views

Kisin module for CM elliptic curve

Let $E$ be a CM elliptic curve with CM by the field $K$ and assume that $p$ is ramified in $K$ so that $\pi^2 = p \in \mathcal{O}_K$. In particular, then $E$ has supersingular reduction at $p$ and by ...
3
votes
1answer
297 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 ...
2
votes
1answer
203 views

extension of the universal cyclotomic character

Let $p$ be a prime number, $\psi:G_\mathbb{Q} \rightarrow \bar{\mathbb{Q}}_p$ be an odd character of conductor $N$ prime to $p$, with finite image and such that $\psi(p)=1$. Let $\mathcal{W}$ be the ...
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0answers
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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{...
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votes
0answers
294 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 $\...
2
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0answers
246 views

Comparison theorem between étale and de Rham cohomology for local system

This question is based on Milne "canonical models of Shimura varieties and automorphic vector bundles" Let $(G,X)$ be a Shimura datum, $(V,\xi)$ be a rational representation of $G$ (I guess it means ...
5
votes
0answers
405 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) ...
7
votes
0answers
520 views

Grothendieck's motivation of crystalline cohomology

Here Illusie mentions Grothendieck's observation that using Gauss-Manin connection one can give a non-canonical isomorphism between de Rham cohomology of smooth schemes over $W(k)$ with isomorphic ...
2
votes
1answer
165 views

Semistability of local Siegel Galois rep:

When are the $l$-local $p$-adic Galois representations of Siegel modular forms semistable? By this I mean $\rho_{f}: G_{\mathbb{Q}}\to \operatorname{GSpin}_{2n+1}(\overline{\mathbb{Q}}_p)$ restricted ...
7
votes
2answers
2k views

congruent number problem [closed]

I am studying the congruent number problem and I heard that there is a paper by Kazuma Morita which claims to solve this problem from my colleague. I saw the paper on his homepage but it is very ...
3
votes
1answer
355 views

Katz $p$-adic L function and ordinary condition

Let $H$ be a CM field and $F$ be the maximal totally real subfield of $H$. Can we construct a Katz $p$-adic L-functions of Hecke characters without the ordinary condition (i.e every prime of $F$ above ...
8
votes
1answer
462 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},\...