Questions tagged [p-adic-hodge-theory]

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Reference request: good reduction equivalent to crystalline étale cohomology

Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the étale cohomology of $X$ is crystalline, and $X$ has semistable ...
Richard's user avatar
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0 answers
91 views

Prime to $p$ monodromy of local system on rigid variety

Suppose $F$ is a finite extension of $\mathbb Q_p$, and $X$ is a rigid variety over $F$. I saw in proposition 3.7 of Oswal, Shankar, Zhu, and Patel - A $p$-adic analogue of Borel's theorem: "Let $...
Richard's user avatar
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2 votes
0 answers
140 views

What is the Galois representation structure of $B_{\text{cris}}^+/(t)$?

In $p$-adic Hodge theory, there is a nice exact sequence for quotients of $B_{\text{dr}}^+$. Denote by $t$ the typical uniformizer of $B_{\text{dr}}^+$ (the cyclotomic character), then there is a $G_{\...
kindasorta's user avatar
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1 vote
0 answers
76 views

The bound for zeros of the composition of polynomials and analytic functions

Suppose $K$ is a number field, and $A\in M_n(K)$. $v$ is a place of $K$, and $f_1,\cdots,f_n$ are analytic functions (one variable) on $m_v\mathcal O_{K,v}$, satisfying: $\frac{\mathrm d \bf {f}}{\...
Richard's user avatar
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5 votes
0 answers
521 views

Theorem 7.11 in Scholze's $p$-adic Hodge Theory

I was trying to understand the statement and proof of Theorem 7.11 in Scholze's paper "$p$-adic Hodge Theory for Rigid-Analytic Varieties". I'll reproduce part of the statement below: Let $...
Kush Singhal's user avatar
3 votes
0 answers
154 views

Algebraic properties of Witt vectors $W(K^{\flat\circ})$, $K$ a characteristic 0 perfectoid field

Let $K$ be as in the title with tilt $K^\flat$. $W = W(K^{\circ\flat})$ satisfies a universal property: it is the unique $p$-adically complete $p$-torsion free $\mathbb{Z}_p$-algebra $A$ with $A / pA \...
Vik78's user avatar
  • 477
1 vote
0 answers
209 views

Deformations over $A_{\inf}$

Setup: Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$. Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring. Let $\mathcal{X}$ be a flat, projective $\...
Kostas Kartas's user avatar
0 votes
0 answers
156 views

Kato's explicit reciprocity law paper

Does anyone have a copy of Kato's article Generalized explicit reciprocity laws in Advanced Studies in Contemp. Math which is used heavily in his paper constructing his eponymous Euler system? I used ...
xir's user avatar
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5 votes
0 answers
275 views

Geometry of syntomic cohomology

Deligne cohomology has a geometric interpretation. For example, $H^{2}_{\mathcal{D}}(X,\mathbb{Z}(1))$ is identified with the group $H^{1}(X,\mathcal{O}_{X}^{\ast})$ of isomorphism classes of line ...
Oli Gregory's user avatar
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2 votes
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170 views

Calculate $D_{\mathrm{cris}}(V)$ for a crystalline representation

$\newcommand{\cris}{\mathrm{cris}}$In my setting, $K/\mathbb Q_p$ is finite and unramified, and $V$ is a $2$-dimensional crystalline representation of $G_K$. Then we have $D_{\cris}(V)$, which is $2$-...
Richard's user avatar
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Calculation of de Rham cohomology of abelian varieties/ jacobian varieties

It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
Richard's user avatar
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1 vote
0 answers
121 views

Centralizer of Frobenius on filtered $\phi$ module

Suppose $K$ is an unramified extension of $\mathbb Q_p$ of degree $m$, and $\sigma$ is the $p$ power frobenius on $K$. Suppose $V$ is a $2$ dimensional admissible filtered $\phi$ module over $K$. I ...
Richard's user avatar
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2 votes
1 answer
213 views

About the filtration of crystalline cohomology

Suppose $K$ is an finite unramified extension of $\mathbb Q_p$ with residue field $k$, and let $Y$ be an proper smooth variety defined over $k$. We know if $Y$ admits a proper smooth lifting $X/W(k)$ ...
Richard's user avatar
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2 votes
1 answer
295 views

Irreducibility of Tate module (as a Galois representation) of elliptic curves with good reduction

This question is following the previous question. Definitions: Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is an elliptic curve defined over $F$ with good reduction. Denote ...
Richard's user avatar
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3 votes
1 answer
234 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
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3 votes
1 answer
308 views

Adic generic fiber of a small formal scheme in the sense of Faltings

$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic ...
user514790's user avatar
1 vote
0 answers
119 views

Is the Frobenius semisimple on the de-Rham cohomology?

Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
Richard's user avatar
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1 vote
0 answers
103 views

Is there a bound on the number of $p$-adic semisimple representations?

Faltings proved the following: Fix integers $w, d \geqslant 0$, and fix a number field $K$ and a finite set $S$ of primes of $\mathcal{O}_K$. There are, up to conjugation, only finitely many ...
Richard's user avatar
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2 votes
0 answers
72 views

Reference request: learning Fontaine-Messing theory

I am interested in learning about Fontaine-Messing theory. Besides the original papers, though, I don't know any good expository literature on this topic (crystalline representations, etc.). Can ...
cgb5436's user avatar
  • 221
2 votes
1 answer
232 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 ...
user471019's user avatar
5 votes
0 answers
183 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
1 vote
0 answers
202 views

$p$-adic étale cohomology group of open smooth varieties

Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $X$ be a smooth variety over $K$. Dr. Yamashita announced that he had proved the Galois representation of $p$-adic étale cohomology group $H^*_{\...
OOOOOO's user avatar
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5 votes
0 answers
319 views

Calculating étale fundamental groups from the usual fundamental group

$\newcommand{Spec}{\operatorname{Spec}}$Let $X$ be a connected affine smooth variety over $\mathbb{Q}$, with a point $x\in X(\Spec(\mathbb{Q})$. For any algebraically closed field $K$ of ...
Fernando Peña Vázquez's user avatar
3 votes
1 answer
433 views

p-adic period map in Lawrence and Venkatesh

In Lawrence and Venkatesh's paper on the Mordell conjecture, they prove that there are finitely many $K$-rational points on a hyperbolic curve $X$, where $K$ is a number field, by showing that there ...
kindasorta's user avatar
  • 1,373
3 votes
0 answers
223 views

Nygaard filtration on Fontaine's period ring

Let $K$ be a discretely valued extension of $\mathbb{Q}_p$ with perfect residue field $k$, and $\mathcal{C}$ a completed algebraic closure of $K$ with the ring of integers $\mathcal{O}_{\mathcal{C}}$. ...
user145752's user avatar
6 votes
0 answers
583 views

On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"

I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...
user141099's user avatar
11 votes
2 answers
1k views

What is the Perrin-Riou logarithm (or regulator)?

Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this workshop at the CRM in ...
Anton Hilado's user avatar
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4 votes
0 answers
157 views

base change property of Topological Hochschild homology

What is the "base change property" of topological Hochschild homology? In Proposition 11.10 of Bhatt-Morrow-Scholze's paper "Topological Hochschild homology and integral p-adic Hodge ...
K.M.'s user avatar
  • 91
3 votes
1 answer
292 views

Why is Fontaine's infinitesimal period ring $A_{\text{inf}}$ complete?

Fix a perfectoid field $K$ in mixed characteristic with ring of integers $\mathcal{O}$ and pseudo-uniformizer $\varpi$. Its tilt is the fraction field of $\mathcal{O}^{\flat}=\varprojlim_{x\mapsto x^{...
user141099's user avatar
8 votes
0 answers
395 views

Intuition for de Rham comparison theorem in $p$-adic Hodge theory

The de Rham comparison theorem from $p$-adic Hodge theory compares the etale cohomology of a variety with the de Rham cohomology of that variety. It says the following: Let $K/\mathbf{Q}_p$ be a ...
Adithya Chakravarthy's user avatar
8 votes
1 answer
653 views

$p$-adic comparison of cohomology with coefficients in $\mathbb{Z}_{p}$ and $\mathbb{B}_{\textrm{dR}}$ on general smooth algebraic varieties

This is something which I'm sure is well known to experts which I would appreciate some information about. In his paper [1], Scholze proves (e.g. Theorem 8.4, Theorem 8.8) that on a proper adic space $...
David Urbanik's user avatar
1 vote
1 answer
154 views

$p$-adic étale cohomology groups are not $\mathbb{C}_p$-admissible

It is stated in Caruso - An introduction to $p$-adic period rings (the remarks following equation (2)) that the $p$-adic étale cohomology groups of an algebraic variety $X$ over a finite extension $K$ ...
Tuvasbien's user avatar
  • 156
2 votes
1 answer
360 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
1 vote
1 answer
186 views

Trianguline representation

I have a problem in understanding the concept of trianguline representation. Maybe someone can enlighten me. Let $K$ be a finite extension of $\mathbb{Q}_p$ and $V$ be a $p$-adic representation of $...
Konstantin's user avatar
2 votes
1 answer
314 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_{...
Sunny's user avatar
  • 609
10 votes
0 answers
1k views

Roadmap for p-adic Hodge theory

I'd like to be able to start studying p-adic Hodge theory and hope that by posing this question, I can be better prepared to work towards it. I ask for a roadmap because I understand that I have a lot ...
Thigh High Crocs's user avatar
6 votes
0 answers
528 views

Are crystalline cohomology obsolete?

I'm interested in $p$-adic cohomology theories now. I have learned that since de Rham cohomology behaves badly in char $p$, people invented crystalline cohomology in smooth cases and later rigid ...
Wilhelm's user avatar
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8 votes
1 answer
2k views

Some questions from the paper by Scholze-Weinstein

The following is from the paper by Scholze-Weinstein on moduli of $p$ divisible groups. My question is from a part of Lemma 4.1.7: If $R$ is a semiperfect ring, then the canonical map $W(R^{\flat}) \...
Ashutosh RC's user avatar
8 votes
1 answer
440 views

What is $TP(\mathbb{Z}_p)$?

Let $TP$ be periodic topological cyclic homology. What is $\pi_* TP(\mathbb{Z}_p)$? (i) I know that $\pi_* TP(\mathbb{F}_p) \cong \mathbb{Z}_p[v^{\pm 1}]$ with $v$ in degree $-2$ by IV.4.8 of Nikolaus-...
pink floyd's user avatar
3 votes
0 answers
269 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
4 votes
1 answer
348 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
  • 229
6 votes
0 answers
417 views

Quasi-syntomic descent and prismatic F-crystals

I am reading Bhatt and Scholze's paper on F-crystals, and they seem to be using the following result in the proof of Theorem 5.6: let $X \to Y$ be a quasisyntomic cover of formal schemes over $\...
Martin Ortiz's user avatar
2 votes
0 answers
183 views

$G_K$-fixed points of sections of affinoids on the Fargues-Fontaine curve

Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G_K=\mathrm{Gal}(\overline{K}/K)$ be its absolute Galois group. There are the Fargues-Fontaine analytic curves $Y_{FF}$ and $X_{FF}$ associated ...
xlord's user avatar
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10 votes
0 answers
673 views

Elementary aspects of The Fargues-Fontaine curve

To any pair $(E,F)$, where $E$ is a local field and $F$ is a perfectoid field, one can associate a curve $X^{\text{FF}}_{E,F}$, the so-called Fargues-Fontaine curve, which is unique up to Frobenius ...
QGravity's user avatar
  • 969
1 vote
1 answer
287 views

Exact sequence, de Rham representation

Let $k$ be a $p$-adic field and $G_k$ its absolute Galois group. Let $B_\text{dR}$ be the de Rham period ring with the usual filtration given by powers of $t$. For $i < j$ integers we have an exact ...
Konstantin's user avatar
1 vote
1 answer
125 views

Triangularizability of induced $(\phi, \Gamma)$-modules

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $L/K$ a finite unramified extension. Let $M$ be a $(\phi, \Gamma_L)$-module over the Robba ring of $L$ (with coefficients in some other $p$-adic ...
naf's user avatar
  • 10.5k
3 votes
0 answers
162 views

Smooth proper varieties over the integers that are not toric

Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric? By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
Asvin's user avatar
  • 7,608
12 votes
3 answers
3k views

Witt vectors, the cotangent complex, and a solid construction of $B_{dR}^+$

In a remarkable lecture delivered on October 29th: New Foundations for functional analysis, Dustin Clausen suggests at the 40 minute mark a remarkable new construction interpretation of Fontaine's ...
pupshaw's user avatar
  • 848
4 votes
0 answers
201 views

Grothendieck group of admissible $p$-adic representations

Let $K$ be a $p$-adic local field; $G = \mathop{\mathrm{Gal}}(\overline K | K)$; $\tau \in \{\text{HT}, \text{dR}, \text{crys}\}$, $B_\tau$ the corresponding period ring; $\mathop{\mathrm{Rep}}_{\...
Aoi Koshigaya's user avatar
2 votes
0 answers
129 views

Base change of Hodge-Witt cohomology

Let $k$ be a perfect field of characteristic $p$, and $L$ be a finite extension of $k$. For a smooth projective variety $X$ defined over $k$, we denote the base change $X \times_k L$ by $X_L$. In this ...
OOOOOO's user avatar
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